Preferences and Similarities (CISM International Centre for Mechanical Sciences)

CISM COURSES AND LECTURES

Series Editors: The Rectors Giulio Maier - Milan Jean Salençon - Palaiseau Wilhelm Schneider - Wien

The Secretary General Bernhard Schrefler - Padua

Executive Editor Paolo Serafini - Udine

The series presents lecture notes, monographs, edited works and proceedings in the field of Mechanics, Engineering, Computer Science and Applied Mathematics. Purpose of the series is to make known in the international scientific and technical community results obtained in some of the activities organized by CISM, the International Centre for Mechanical Sciences.

INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES COURSES AND LECTURES - No. 504

PREFERENCES AND SIMILARITIES

EDITED BY GIACOMO DELLA RICCIA UNIVERSITY OF UDINE DIDIER DUBOIS UNIVERSITY OF TOULOUSE RUDOLF KRUSE UNIVERSITY OF MAGDEBURG HANS-JOACHIM LENZ FREIE UNIVERSITӒT BERLIN

This volume contains 29 illustrations

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. © 2008 by CISM, Udine Printed in Italy SPIN 12510191

All contributions have been typeset by the authors.

ISBN 978-3-211-85431-0 SpringerWienNewYork




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Similarity of Fuzzy Sets and Dominance of Random Variables

3

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4

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11

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12

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Similarity of Fuzzy Sets and Dominance of Random Variables

13

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14

Bernard De Baets

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15

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16

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20

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24

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A Logic-Based View of Similarities and Preferences

25

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1L;O< ?N ;F  &H NBCM J;J?L Q? L?PC?Q NB? <;MC= C>?;M FCHECHA NB? G?NLC= =IH=?JN I@ MCGCF;LCNS ;H> NB? @OH>;G?HN;F =IH=?JNM ;H> JLI=?>OL?M I@ @OTTS FIAC= 4? NOLH NB?H IOL ;NN?HNCIH NI NB? JLI?HCHA G?NLC=M NB;N =;JNOL? NB? M?G;HNC=M I@ ; J;LNC=OF;L JLICHA NI >CM=OMM NB? L?F;NCIH MCGCF;LCNS 4? ?RN?H> IOL MCGCF;LCNS <;M?> @ILG;FCMG NI NB? JL?@?L?H=? L?F;NCIHM CHNLI>O=CHA NB? HINCIHM I@ ?RN?H>?> =IH>CNCIH;F ;H> G;LACH;F JL?@?L?H=? ;H> >?LCP? ; L?MOFN J?LGCNNCHA NI CH@?L G;LACH;F JL?@?L?H=? P;FO?M @LIG EHIQF?>A? I@ =IH>CNCIH;F JL?@?L?H=? P;FO?M ;H> L?F;N?> G;LACH;F JL?@?L?H=? P;FO?M #CH;FFS Q? JL?M?HN ;JJFC=;NCIHM I@ NB? =IH=?JNM I@ MCGCF;LCNS ;H> ONCFCNS NI =IHNLIF ;H> JF;HHCHA JLI? P?FIJG?HN I@ ; =IGJL?B?HMCP? ;JJLI;=B NI =IHNLIF <;M?> IH NB? >?HCNCIH ;H> MSHNB?MCM I@ @OH=NCIHM >?M=LC?MCL; ;M IJJIM?> NI CM=OMMCIH Q? QCFF =IHMC>?L ;FMI CMMO?M L?F;N?> NI NB? H??> NI B;H>F? ;H> ;AAL?A;N? GOFNCJF? =LCN?LC; NB? JL?@?L?H=?M I@ >C
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?L?HNC;N? JIMMC OH=?LN;CHNS HIN; NI JLIPC>? MIFC> =IH=?JNO;F @IOH>;NCIHM @IL @OTTS FIAC= NB?M? @ILG;FCMGM Q?L? NSJC=;FFS ;RCIG;NC= CH H;NOL? =IH=?HNL;NCHA IH NB? L;NCIH;FCNS ;H> =IHMCM

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26

E. Ruspini

N?H=S I@ NB? ?RN?HMCIHM I@ =F;MMC=;F M?N NB?ILS ;H> =F;MMC=;F FIAC= JLIJIM?> <S NB? NB?ILS L;NB?L NB;H IH CHN?LJL?N;NCIH I@ CNM <;MC= HINCIHM CH N?LGM I@ GIL? JLCGCNCP? =IH=?JNM FNBIOAB I@N?H ;M L?G;LE?> <S $CF?M  MJ?=C= @ILG;FCMGM B;> ?P?FIJ?> QCNB CGJFC=CN CHN?LJL?N;NCIHM CH GCH> NB? F;=E I@ =F?;L =B;L ;=N?LCT;NCIHM I@ @OH>;G?HN;F =IH=?JNM MO=B ;M NB? HINCIH I@ >?AL?? I@ G?G NI >IOF?S  I@ NB?CL ?P?HNO;F L?D?=NCIH CH @;PIL I@ JLI<;>CNCIH NI L?KOCL?G?HNM NI L?MIFP? NB?M? ?JCMN?GIFIAC=;F KO;H>;LC?M NB?L? Q;M ;FMI ;H CGJILN;HN JL;AG;NC= L?;MIH NI JLIPC>? ; ?L MN;H>CHA I@ @OTTS FIAC= =IH=?JNM IGJON?L <;M?> CGJF?G?HN;NCIHM I@ I@ JIMMCM JLI>O=? @OTTS M?NM ;H> JIMMCCMNLCCNS I@ NB?M? L?MOFNM ;H> NB;N I@ NB? JLIAL;GM NB;N JLI>O=? NB?G =IOF> HIN JIMMC OHF?MM =ILL?=NH?MM N?MNM Q?L? =F?;LFS MJ?=C?> 0O=B MJ?=C=;NCIH CM HIN JIMMCCNS ;H> NI ?MNCG;N? NB? =IHM?KO?H=?M I@ JIN?HNC;F ?LLILM 

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28

E. Ruspini

C@  CM HIN CH NB? ?PC>?HNC;F M?N NBCM G?;MOL? KO;HNC?M NB? ?RN?HN NI QBC=B =IGJ?NCHA ?RJF;H;NCIHM ;L? GIL? ;>?KO;N? >?M=LCJNCIH I@ NB? JIMMC
+CCHCFOINI  1BCM G?;MOL? I@ JIN?HNC;F NLONB I@ ; MCHAF? JIMMC  =;H CFS ?RN?H>?> NI ; G?;MOL? I@ H?=?MM;LS NLONB I@ ; HIH?GJNS MO<M?N I@ JIMMCM  ;M NB? ?RN?HN NI QBC=B  B;M NI NI ?H=IGJ;MM NB? ?PC>?HNC;F M?N  C? NB? MG;FF?MN G?NLC= H?CAB I@  NB;N CH=FO>?M 
?CHA ;O=NCIH I@ ; G?NLC= MNLO=NOL? NI G?;MOL? NB? MCGCF;LCNS CFS ?RN?H> NB? G;CH CH@?L?HNC;F JLI=?>OL? I@ 46+<: 765,5: CHNI CNM A?H?L;FCT?> =IOH N?LJ;LN JLIPC>CHA ;H CHN?LJL?N;NCIH @IL NB? *6476:0;065(3 9<3, 6- 05-,9,5*, IL .,5,9(30A,+ 46+<: 765,5: I@ 6;>?B  2H>?L NBCM CHN?LJL?N;NCIH NBCM LOF? M NB? >CMN;H=? @LIG NB? ?PC>?HNC;F M?N  NI NB? M?N 79 C? NB? =IH=FOMCIH IH NB? <;MCM I@ CH@ILG;NCIH I@ NB? MCGCF;LCNS I@  ;H> 79 NB? JL?GCM? ;H> NB;N I@ NB? >CMN;H=? 79 NB? =IH>CNCIH;F LOF? IL CGJFC=;NCIH &H INB?L QIL>M NB? MCGCF;LCNS FIAC=  L?A;L>M NB? GI>OM JIH?HM ;M ;H ?RJL?MMCIH I@ NB? CH=FOMCIH L?F;NCIH M I@ M?NM CH NB? MJ;=?  A?H?L;FCTCHA NB? =F;MMC=;F GI>OM JIH?HM QBC=B ?RJL?MM?M NB? NL;HMCNCPCNS I@ NB? =F;MMC=;F L?F;NCIH I@ M?N CH=FOMCIH &@ 79 CM ; MO<M?N I@ 79 ;H> C@ 79 CM ; MO<M?N I@ 79 NB?H 79 CM ; MO<M?N I@ 79 1BCM G?NLC= ;JJLI;=B NI =B;L;=N?LCT? NB? NLONB I@ MN;N?G?HNM CH N?LGM I@ MCGCF;LCNC?M M CM =IGJF?N?FS >C
?L?HN CH H;NOL? NI NB;N ?GJB;MCT?> CH JLI<;?H=?  CM >?H?> CH N?LGM I@ HOG?LC= G?;MOL?M I@ NB? ?RN?HN I@ IP?LF;J I@ NB? M?NM   79 ;H>  #%' %! $ 4? L?=;FF NB;N ; -?B ; IL A?H?L;FCT?> ?KOCP;F?H=? L?F;NCIH  CM ; @OH=NCIH   
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A Logic-Based View of Similarities and Preferences

29

NI G;RCGOG MCGCF;LCNS M;NCM@SCHA NB? =IHMNL;CHNM            

                   

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QB?L?   CM IH? I@ ; @;GCFS I@ IJ?L;NILM =;FF?> ;90(5.<3(9 5694: IL #5694: @IL MBILN QBC=B ;L? CGJILN;HN CH M?P?L;F ?F>M 0=BQ?CT?L ;H> 0EF;L  1LCFF;M ;H> 3;FP?L>?  &N CM CGJILN;HN NI L?G;LE NB;N CH GOFNCP;FO?> FIAC=M MO=B ;M @OTTS FIAC= 1 HILGM ;L? NB? L?MOFN I@ FIAC=;F =IHMC>?L;NCIHM ;M A?H?L;FCT;NCIHM @IL NB? NLONB N;?> NI JLIPC>? NB? L?F;NCIH  QCNB NB? M?G;HNC=M =IGGIHFS ;MMI=C;N?> QCNB =IH=?JNM MO=B ;M JLIRCGCNS IL L?M?G;F FIAC= %OAB?M ;H> L?MMQ?FF  PC?QJICHN NB? MCG CF;LCNS L?F;NCIH  J?LGCNM >?HCNCIH I@ ; A?H?L;FCT?> ;==?MMC;F MSMN?GM ;==?MMC?H?> M NB;N G;E? JIMMC 5,*,::0;@ &H  NB? MCGCF;LCNS L?F;NCIH  >?H?M ;H ?RN?H>?> AL;>O;F L?F;NCIH I@ ;==?MMCCHA NI HOG?LC= HINCIHM I@ JIMMC H?=?MMCNS 1B? @IFFIQCHA HINCIHM I@ AL;>?> GI>;FCNS ?RN?H> NB? OH;LS IJ?L;NILM  ;H> I@ GI>;F FIAC= NI NB? CHHCN? @;GCFS I@ OH;LS IJ?L;NILM  ;H> QB?L? CM ; L?;F HOG
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30

E. Ruspini

 %!  1B?     I@  <S  CM NB? P;FO?

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1B? >?AL?? I@ CGJFC=;NCIH    CM ; G?;MOL? I@ NB? MCT? I@ NB? GCHCG;F H?CAB I@ 79 NB;N CHN?LM?=N 79 CH NB? M?HM? NB;N C@
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MCGCF;LCNS  M 79 ;H> 79
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4? ;L? HIQ CH ; JIMCNCIH NI CHNLI>O=? NB? HINCIHM I@ A?H?L;FCT?> JIMMC H?=?MMCNS >CMNLC?AL?? I@ CGJFC=;NCIH ;H> >?AL?? I@ =IHMCMN?H=S @OH=NCIHM

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A Logic-Based View of Similarities and Preferences

31

&@  CM NBIOABN ;M NB? M?N I@ JIMMCM =IGJ;NC?H=? IL CH@ILG;FFS NB? M?N I@ JIMMC?L =IHMC>?L;NCIH NB?H NB? H?=?MMCNS >CMNLC I@ NB? MCGCF;LCNS ?HNC;F QILF> ;H> ; JIMMC QB?L?  CM NLO? ILL? MJIH>CHAFS JIMMCCMNLCM IH NB? MCGCF;LCNC?M I@ JIMMCM CH  ;H> CH 79 1I >?LCP? NB? HINCIHM I@ =IH>CNCIH;F H?=?MMCNS ;H> CH>?J?H>?H=? Q? QCFF H??> NI L?=;FF NB? >?HCNCIH I@ NB? =IH=?JN I@ NB? 9,:0+<(;065 IL 7:,<+605 =,9:, @OH=NCIH ;MMI=C;N?> QCNB ; NLC;HAOF;L HILG   3;FP?L>? 

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32

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?@IL? NI =B;L;=N?LCT? NB? Q?FF EHIQH HINCIH I@ J;LNCNCIH CH NB? =IHN?RN I@ JIMMCM GI>?FM

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CH>C=;N?M ; FIQ?L I@ CNM

A Logic-Based View of Similarities and Preferences

33

>?AL?? I@ =IHMCMN?H=? 1B? @ILGOF;  7
9 MN;N?M NB?L?@IL? NB;N ACP?H NB? ?PC>?H=?   CM NLO? ;N F?;MN NI NB? >?AL?? ;H> NB;N CN CM =IGJ;NC?AL??
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 7
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;CLM CH MIG? MCNO;NCIHM <S ?RN?HMCIH I@ JLIJIMCNCIHM EHIQH NI MIOL=? I@ MCG CF;LCNS G?;MOL?M &H NBCM L?A;L> CN CM CGJILN;HN NI HIN? LMN ; @OH>;G?H N;F >C
?L?H=? CMNLC NB? G?NLC= HINCIH I@ MCGCF;LCNS -LI<;?H=S I@ =?LN;CH MSMN?GM NI ?RBC @LIG ?RJ?LCG?HN;F >;N; ;M CN CM OMO;FFS >IH? QCNB MSMN?G P;LC; >?M=LCJNCP? J;L;G?N?LM 0CGCF;LCNS L?F;NCIHM IH NB? INB?L B;H> ;L? ;LO=?> NI G?;MOL? L?M?G?HCNCIHM ;FNBIOAB ;LM NI G;H;A? ;H;FSNC=;F =IGJF?RCNS CH NB? M;G? Q;S NB;N =F;MMC=;F J;LNCNCIHM QBC=B MCGCF;LCNC?M A?H?L;FCT? ;L? CHNLI>O=?> NI CG JLIP? MSMN?G OH>?LMN;H>CHA 0CGCF;LCNS G?;MOL?M >I HIN NB?L?@IL? L??=N

34

E. Ruspini

JLIJ?LNC?M I@ NB? L?;F QILF> GIMN CGJILN;HNFS ;MM?LNCIHM G;>? IH NB?CL <;M?M ?A 8NB? JL?MMOL? CM MG;FF  GOMN CHA NI ; =F?;L CHN?LJL?N;NCIH MO=B ;M NB;N ACP?H CNS I@ A?H?L;FCT?> JLIJIMCNCIHM 1B? MCAHC=;HN =IH=?JNO;F >C
?L?H=? MCGCF;LCNS MBIOF> ?LMNII> <S HINCHA NB;N NB? F;NN?L CM >? J?H>?HN IH ; QC>? P;LC?NS I@ =LCN?LC; NI DO>A? JLIRCGCNS QBCF? NB? @ILG?L CM ; @ILG;FCT;NCIH I@ NB? =IH=?JN I@ MN;NCMNC=;F @L?KO?H=S M HIN?> @IL ?R;GJF? <S 1P?LMES  C@ =IOHNLC?M ;L? =IGJ;L?> CH N?LGM I@ NB?CL MCGCF;LCNS NB?H ';G;C=; GCABN ?L?> NI O=?> @IL =IHP?HC?H=? NI J?LGCN ;H;FIAC=;F L?;MIHCHA ;H> NI ?RN?H> <S ; @ILG I@ FIAC=;F ?RNL;JIF;NCIH ?RCMNCHA EHIQF?>A? -LI<; ;L? ;H ?RJL?MMCIH I@ NB? FCE?FCBII> NB;N OH>?L =?LN;CH MJ?=C?> =CL=OGMN;H=?M ; MSMN?G QCFF CL?=N MSHNB?MCM I@ OM?@OF MCGCF;LCNS L?F;NCIHM CM HIN OMO;FFS MNL;CABN@ILQ;L> &H NB? @?Q MCNO;NCIHM QB?L? MCGCF;LCNS L?F;NCIHM G;S LOF?M G;JJCHA IH? G?;MOL? I@ MCGCF;LCNS CHNI ;HINB?L ;L? HIN ?;MCFS >?LCP?> 2NCFCNS ;H> JL?@?L?H=? G?;MOL?M BIQ?P?L ;JJ?;L H;NOL;FFS CH NB? =IHN?RN I@ MSMN?G ;H;FSMCM ;H> MSMN?G IJNCGCT;NCIH JLI @IL G;F L?F;NCIHM JL?M?HN?> MCGCF;LCNS L?F;NCIHM @;=CFCN;N? >?HCNCIH I@ NB? F;NN?L 

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A Logic-Based View of Similarities and Preferences

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36

E. Ruspini

 NL;HMCNCP?  JL?IL>?L CH  C? ; @OH=NCIH   
  7 9 MO=B NB;N       @IL ;FF  CH                    @IL ;FF     CH   1B? P;FO?     CM ; G?;MOL? I@ NB? >?AL?? NI QBC=B  CM JL?@?L;?AL?? I@ NLONB I@ NB? MN;N?G?HN 81B? MN;N? I@ ;
;CLM L?JL?M?HN?> <S  CM ;N F?;MN ;M JL?@?L;?MCL; <S   4? NOLH IOL ;NN?HNCIH H?RN NI NB? L?F;NCIHM MCG CF;LCNS L?F;NCIHM #%'   ##  1B? @IFFIQCHA L?MOFNM L?F;N? NB? CGJILN;HN HINCIHM I@ JL?@?L?H=? ;H> MCGCF;LCNS ;H> JLIPC>? NB? M?G;HNC= <;MCM NI L?A;L> MCGCF;LCNC?M ;M G?;MOL?M I@ JLIRCGCNS ;FIHA =LCN?LC; >??G?> NI B;P? ONCFCNS <S ; L;NCIH;F >?=CMCIH G;ECHA ;A?HN &@  CM ; JL?@?L?H=? L?F;NCIH NB?H NB? L?F;NCIH  >?H?> <S NB? ?KO;NCIH
     GCH          CM ;  MCGCF;LCNS  L?F;NCIH &H@ILG;FFS NBCM ?KO;NCIH G;S ;M MN;NCHA NB;N NQI JIMMCM ;L? MCGCF;L @LIG NB? FCGCN?> J?LMJ?=NCP? I@ ; MCHAF? JL?@?L?H=? L?F;NCIH  C@ ?CNB?L I@ NB?G CM HIN MCAHC=;HNFS  +IQ C@ CM ; @OTTS M?N CH  NB?H NB? @OH=NCIH  >?H?> <S NB? ?KO;NCIH           CM ;  NL;HMCNCP?  JL?@?L?H=? L?F;NCIH CH  3;FP?L>?  &@ NB? @OTTS M?N CM NBIOABN I@ ;M ; ONCFCNS @OH=NCIH C? ; L;HECHA I@ JIMMCM CH N?LGM I@ NB?CL >?MCL;?AL?? NI QBC=B  CM JL?@?L;     NB?H  CM JL?@?L;     G?;MOL?M NB? MNL?HANB I@ NB;N JL?@?L?H=? 1SJC=;FFS M?P?L;F JL?@?L?H=? G?;MOL?M IL HINCIHM I@ JLI NI ?L?> CH NB? =IHN?RN I@ ; JLI CH ; HOG?LCP? ;H ;AAL? A;N? JL?@?L?H=? L?F;NCIH  MCGJF? ;JJLI;=B NI JLI>O=? MO=B ;H ;AAL?A;N? SC?F>CHA ;  NL;HMCNCP?  JL?@?L?H=? L?F;NCIH CM ?RJL?MM?> <S NB? @ILGOF;      CH@          

A Logic-Based View of Similarities and Preferences

37

1BCM @ILGOF; L?KOCL?M NB;N NQI JIMMCM ?L?> MCGCF;L @LIG ; DICHN J?LMJ?=NCP? ,NB?L JIMMC? I
JL?@?L?H=? L?KOCL?G?HNM NI =IHMNLO=N MOCN;?  ;MMOL?M NB;N ;HS  NL;HMCNCP?  L?F;NCIH G;S
1B? JL?PCIOMFS G?HNCIH?> L?F;NCIH MCGCF;LCNS L? F;NCIHM F?;>M NI ; =IOHN?LJ;LN I@ NBCM NB?IL?G @IL MCGCF;LCNS L?F;NCIHM !#
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38

E. Ruspini

M Q;M ;FMI NB? =;M? QCNB MCGCF;LCNS L?F;NCIHM Q? L?=OL NI HINCIH I@ %;OM>IL
>CMN;H=? NI SC?F> NB? L?KOCL?> A?H?L;FCT;NCIH

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MOJ     

  

CM CM =;FF?> NB? ,?;,5+,+ 79,-,9,5*, 9,3(;065 I@  IL MCGJFS NB? ?RN?HMCIH I@  1B? P;FO?    G?;MOL?M NB? >?AL?? NI QBC=B  CM JL?@?L; CM ;N F?;MN JL?@?L; NI NB? >?AL??    ;M @IL ?P?LS  QILF>  NB? P;FO? MOJ     G?;MOL?M  

NB? >?AL?? NI QBC=B MIG?  QILF>  CM ;M JL?@?L; =IHM?KO?H=? I@ NB? >?HCNCIH I@ ?RN?H>?> JL?@?L?H=? L?F;NCIH !# 1L;HMCNCPCNS I@   )?N 




@IL ;FF  CH  

  

         @IL ;FF    CH  

M Q;M NB? =;M? QCNB MCGCF;LCNS L?F;NCIHM NB? CGJILN;H=? I@ NBCM L?MOFN FC?M IH CNM LIF? ;M NB? <;MCM @IL ; FIAC=;F =;F=OFOM I@ JL?@?L?H=?M B;PCHA >?MCL ; NB? CH@?L?HNC;F G?=B;HCMGM I@ =F;MMC=;F FIAC= 4? ME?N=B CH NB? L?G;CH>?L I@ NBCM 0?=NCIH MIG? MCAHC=;HN =IH=?JNM ;H> L?MOFNM 4? LMN CHNLI>O=? NB? HINCIH I@ G;LACH;F JL?@?L?H=? CHN?H>?> NI JLIPC>? NB? @ILG;F <;MCM NI >?H? ?RN?H>?> JL?@?L?H=? L?F;NCIHM IP?L ; =I;LM?HCHA I@ NB? OHCP?LM?  1B? =I;LM?HCHA CM GI>?F?> PC; ; J;LNCNCIH I@ NB? F;HAO;A? 

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        &H JL;=NC=? NB? J;LNCNCIH  CM OMO;FFS ; >CMDICHN J;LNCNCIH C? C@  ;H>  ;L? NQI ?F?G?HNM I@ NB? J;LNCNCIH NB;N ;L? HIN FIAC=;FFS ?KOCP;F?HN NB?H NB?CL ?RN?HMCIHM >I HIN CHN?LM?=N C? 79  79   1B? HINCIH I@ =IH>CNCIH;F JL?@?L?H=? CM ;G?H; IH NB? CHN?LJL?N;NCIH I@ ; =IH>CNCIH

A Logic-Based View of Similarities and Preferences

39

;M ?PC>?H=? NB;N NB? NLO? MN;N? I@ ;
;CLM CM ; JIMMC FSCHA CH ; MO< M?N  NB? ?PC>?HNC;F M?N I@ NB? OHCP?LM?   &H NBCM CHN?LJL?N;NCIH NB? ?PC>?H=?  B;M ?
?=NCP?FS ?FCGCH;N?> @LIG =IHMC>?L;NCIH ;FF JIMMCM IONMC>?  ?
?=NCP?FS G;ECHA NB? ?PC>?HNC;F M?N NB? H?Q @L;G? I@ L?@?L?H=? 4B?H ?MNCG;NCHA NB? ?RN?HN NI QBC=B  QILF>M ;L? JL?@?L;M CN G;E?M M?HM? CH NBCM PC?Q NI =IHMC>?L IHFS NBIM? QILF>M NB;N ;L? =IHMCMN?HN QCNB NB? ?PC>?H=? 1BCM PC?Q L?;>CFS F?;>M NI NB? @IFFIQCHA >?HCNCIH

?HCNCIH 0NLC=N -L?@?L?H=? )?N  F?N  ?H?> <S          
     HINB?L ;JJLI;=B NI NB? >?HCNCIH I@ =IH>CNCIH;F JL?@?L?H=?VMCGCF;L CH MJCLCN NI NB;N ?GJFIS?> CH NB? >?HCNCIH I@ MCGCF;LCNS L?F;NCIHMVMNCFF =IHMC> ?LM ;FF JIMMCM CH  MCH=? ?P?H QILF>M NB;N ;L? HIN MNLC=NFS =IGJ;NC?H=? G;S ?L NBCM CHN?LJL?N;NCIH =IH>CNCIH;F JL?@?L?H=?M ;L? ; G?;MOL? I@ NB? ?RN?HN NI QBC=B  QILF>M ;L? JL?@?LL?> NI  QILF>M ;M =IGJ;L?> NI NB? ?RN?HN NB;N  QILF>M ;L? JL?@?LL?> NI  QILF>M 1BCM PC?QJICHN MOAA?MNM NB? @IFFIQCHA >?HCNCIH

?HCNCIH /?F;NCP? -L?@?L?H=? )?N  F?N  ?H?> <S                 M Q;M NB? =;M? QCNB MCGCF;LCNS L?F;NCIHM NBCM >?HCNCIH F?;>M NI ;H CH@?L ?HNC;F JLI=?>OL? NI ?MNCG;N? P;FO?M I@    NB? =IH=FOMCIH @LIG P;FO?M I@    NB? JL?GCM? ;H> NB? L?F;NCP? =IH>CNCIH;F JL?@?L?H=?     NB? CH@?L?HNC;F LOF? B?IL?G $?H?L;FCT?> *I>OM -IH?HM @IL -L?@?L?H=? /?F;NCIHM )?N   F?N  ;H>     =IH>CNCIH;F JL?@?L?H=?M I@ NB?  JL?@?L?H=?  L?F;NCIH  1B? @IFFIQCHA L?F;NCIH NB?H BIF>M              

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40



E. Ruspini

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?L?HN >?AL??M I@ CGJILN;H=? @IL ?;=B I@ M?P?L;F CHN?L;=NCHA >?=CMCIH ;A?HNM 1B? @ILG;F HINCIHM >CM=OMM?> CH NBCM J;J?L B;P? ;FMI AOC>?> NB? >?P?FIJG?HN I@ G?NBI>M @IL NB? KO;FCN;NCP? >?M=LCJNCIH I@ =IGJF?R I /OMJCHC ; < 6QCL ?N ;F  ;H> @IL NB? ;JJLIRCG;N? G;N=BCHA I@ J;NN?LHM CH F;LA? >;N;<;M?M 4IFP?LNIH ?N ;F  

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1B? ;CNCIH;F MN;N?G?HNM 8&@ NB? MJ??> CM G?>COG IJ?H NB? NBLINNF? MFIQFS  >?M=LC?LCP?> <S ;H;FSMCM I@ =IHN?RN >?J?H>?HN I?=CMCIH ;H> =IHNLIF JLI <S NB? =IHNLIF I@ ;ONIHIGIOM LI ;H> NB;N I@ CNM IJ?L;NCIH;F ?HPCLIHG?HN &H NB?M? JLI?L ; HOG NI H??>M NI 9,(*; NI =B;HACHA =CL=OG MN;H=?M ?A 8H?;L ;H I<MN;=F? ;PIC> I<MN;=F?  CH NB? IJ?L;NCIH;F ?HPCLIH G?HN CH ;>>CNCIH NI NB? 7<976:0=, AI;FM OH>?LFSCHA NB? =IHNLIF JLI ?HN?L NB? I=?  &H NB?M? =;M?M CN CM H?=?MM;LS NI ?G JFIS ?
?=NCP? G?NBI>M @IL NB? >SH;GC= G;H;A?G?HN I@ GOFNCJF? I =IHNLIFF?LM LECH  <;M?> IH NL?;NG?HN I@ GOFNCJF? HINCIHM I@ JL?@?L?H=? B;P? ?P?FIJ?> ;H> CGJF? G?HN?> (IHIFCA? ?N ;F  NI =IHNLIF =IGG?L=C;F ;ONIHIGIOM LI MNLO=NOL?M >CM=OMM?> CH NBCM J;J?L 1B?M? =IHNLIF G?=B;HCMGM ?MM?HNC;FFS JLI>O=? ;AAL?A;N?> G?;MOL?M I@ ;>?KO;=S <S =IG
A Logic-Based View of Similarities and Preferences

41

NB? =IG /;C
;  QBCF? J?LGCNNCHA NI CHN?LJL?N NB? ;AAL?A;N?M ?GJFISCHA JIM MC?M=LC?MCL; NI =IHMNLO=N NB? =IHNLIFF?LM CM NB? HINCIH I@ >?MCL; CNM IJ?L;NCIH;F ?HPCLIHG?HN ;H> I@ NB? JI N?HNC;F =IHNLIF ;=NCIHM  NB;N G;S NI ?L?> ;N ;HS NCG? >OLCHA NB? =IHNLIF JLI=?MM ";=B MO=B @OH=NCIH CM OMO;FFS ;MMI=C;N?> QCNB ; ?> NI ;NN;CH MIG? MJ?=C= I?;M >CM=OMM?> CH NBCM J;J?L F?;>M NI ;AAL?A;NCIH @ILGOF;M MO=B ;M   NRN           

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42

E. Ruspini

 1B? J?LMJC=OIOM H;NOL? I@ NB? FIAC= <;M?> JL?@?L?H=? ;H> ;A AL?A;NCIH @OH=NCIHM J?LGCNM NB? ?RJFC=CN MJ?=C=;NCIH I@ G?=B;HCMGM NI CH ; QC>? P;LC?NS I@ MCNO;NCIHM 0;INNC ?N ;F  LMN ;JJFC?> NBCM ;JJLI;=B NI NB? =IHNLIF I@ N?;GM I@ ;ONIHIGIOM LIO=CHA M=B?G?M NB;N >C
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A Logic-Based View of Similarities and Preferences

43

NB? CHN?LJL?N;NCIH I@ @OTTS ;H> JIMMC NB? OH>?L MN;H>CHA I@ NB?CL LIF? ;M NIIFM @IL NB? MIFONCIH I@ =IGJF?R MSMN?G ;H;FSMCM JLI
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44

E. Ruspini

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A Logic-Based View of Similarities and Preferences

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"HLCKO? % /OMJCHC JJFC=;NCIHM I@ CHN?FFCA?HN GOFNCI?=CMCIH G;E CHA &H ,ES;S (;SH;E )IN  6;>?B OLB;H 1OLEM?H ;H> &GL? ' /O>;M ?>CNILM 647<;(;065(3 5;,330.,5*, "6-; 647<;05. (5+ 0;/ 7730*(;065: +1, 0& 0?LC?M 0?LC?M # IG JON?L ;H> 0SMN?G 0=C?H=?M 0JLCHA?L 3?LF;A  "HLCKO? % /OMJCHC ,H NB? M?G;HNC=M I@ @OTTS FIAC= 5;  6- 7796?04(;, !,(:6505. U  "HLCKO? % /OMJCHC ;H> #L;H=?M= "MN?P; &HN?LJL?N;NCIHM I@ @OTTS M?NM &H "HLCKO? % /OMJCHC -C?LI - IHCMMIH? ;H> 4CNIF> -?>LS=T ?>CNILM #/, (5+)662 6- LI 0;INNC (OLN (IHIFCA? ;H> "HLCKO? % /OMJCHC  GOFNCP;FO?> FIAC= ;JJLI;=B NI CHN?AL;NCHA JF;HHCHA ;H> =IHNLIF 9;0
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46

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48

J-F. Bonnefon, D. Dubois and H. Fargier

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An Overview of Bipolar Qualitative Decision Rules

49

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50

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Logical Approaches to Similarity-based Reasoning

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Logical Approaches to Similarity-based Reasoning

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121

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aSHIP WMQMPEVMX]FEWIH QSHEP PSKMG 3ERYWGVMTX ' WLSVXIRIH ZIVWMSR [EW TYFPMWLIH EW B' -aSHIP QSHEP PSKMG MR# 6VSG SJ 2SKMG )SQTYXEFMPMX] ERH 8ERHSQRIWW  )SVHSFE 'VKIRXMRE   A)EV C 8 )EVRET :VIEXEFMPMX] ERH QIERMRK $78;>B>?7H >5 '284=24  _  "  A)EVC 8 )EVRET (74 >6820; >D=30C8>=B >5 $A>1018;8CH ;RMZIVWMX] )LMGEKS 6VIWW )LMGEKS " A),"C 6 )LEXEPMG ERH ) ,VSMHIZEY\ 2EXXMGIFEWIH KVEHIH PSKMG# ' QYP XMQSHEP ETTVSEGL /R $A>24438=6B C7 >=54A4=24 >= )=24AC08=CH 8= AC8
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Proximities in Statistics: Similarity and Distance Hans –J. Lenz Institute of Statistics and Econometrics, Freie Universität Berlin, Germany

Abstract We review similarity and distance measures used in Statistics for clustering and classification. We are motivated by the lack of most measures to adequately utilize a non uniform distribution defined on the data or sample space. Such measures are mappings from O x O o R+ where O is either a finite set of objects or vector space like Rp and R+ is the set of non-negative real numbers. In most cases those mappings fulfil conditions like symmetry and reflexivity. Moreover, further characteristics like transitivity or the triangle equation in case of distance measures are of concern. We start with Hartigan‘s list of proximity measures which he compiled in 1967. It is good practice to pay special attention to the type of scales of the variables involved, i.e. to nominal (often binary), ordinal and metric (interval and ratio) types of scales. We are interested in the algebraic structure of proximities as suggested by Hartigan (1967) and Cormack (1971), information-theoretic measures as discussed by Jardine and Sibson (1971), and the probabilistic W-distance measure as proposed by Skarabis (1970). The last measure combines distances of objects or vectors with their corresponding probabilities to improve overall discrimination power. The idea is that rare events, i.e. set of values with a very low probability of observing, related to a pair of objects may be a strong hint to strong similarity of this pair.

1 Introduction 1.1 Frame of discernment First we present some types of objects which are considered later, i.e. tupels, vectors or records, sets and probability distribution functions. Of course, there exists a unified view in form of sets embedded into a Cartesian space, but it is helpful to look at them separately. Given a two-dimensional data space we can define several distances between two tupels.

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Fig.1: Distance d between two tupels

A slightly different situation is given when two sets are to be compared.

Fig.2: Distance dH between two sets

It seems worthwhile considering a further case where two probability distributions are given.

Fig.3: Distance d between two distribution functions

In the following we present some examples which will highlight some weakness of traditional proximity (“nearness”) measures when prior information is available. 1.2 Examples Our first example consists of a sample of five tupels xi R2, i=1,2,…,5 with x1T= (1, 21); x2T= (0, 20); x3T= (1, 1); x4T= (0, 0); x5T= (1, 30). The Euclidean distance dE is used as a distance measure, i.e. dE(x, y) = (x – y)T(x – y) with x, y R2. We can easily compute dE(x1, x2) |1,4 ; dE(x3, x4) |1,4 ; dE(x2, x5) |10.

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Fig. 4: Sample Space

163



Assume now that the data are normally distributed with X, Y ~ N(0, 1). We observe that observations x3, x4 are equal or „near” to the centre PT = (0, 0) of the distribution. Their distance is dE(x3, x4) |1,4. The same distance is computed for the first two tupels x1, x2, which may be considered as outliers due to their very large second component. However, if we compare another “outlier pair”, i.e. (x2, x5) we notice that the Euclidean distance dE increases to dE(x2, x5) |10. To the best of the author’s knowledge Skarabis (1970) was the first to point to this phenomenon. We present a further example to stress the necessity for including probabilities of observations besides of the dissimilarities when comparing pairs of objects. We consider only one (Boolean) attribute and n = 4 objects: x1= 1; x2= 1; x3= 2; x4= 2. The semantic of the random variable X (“Behaviour in winter time”) becomes important. X=1 is the code of “Take a bath in an icy lake in winter” while X=2 stands for “Take a warm coat”. Assume that each observation xi, i = 1,2,3,4, is linked to one and only one person. Pair wise comparisons of persons o1, o2 and o3, o4 by a Hamming distance dH(xi,xj) = _xi - xj_, say, delivers dH(x1,x2) = dH(x3,x4) = 0. We induce that both pairs are equally similar. However, the probabilities of both values are generally very different due to P(X=1) << P(X=2), i.e. only few people use to take a outdoor bath in the winter time but almost all wear warm clothes at winter time. It is exactly this low probability which makes the first pair more similar to each other than the second one. This effect motivated Skarabis (1970) to define a probability – based distance measure dw which should fulfil dW(x1, x2) < dW(x3, x4). As a further example, take text mining which is mostly corresponding to a classification problem. As Frakes and Baeza-Yates (1992) point out words that occur extremely rarely are likely to be of no statistical relevance and thus may be dropped. Now assume one detects in two documents about movies the term “superkalifragilistic exialidocious”. Such words are artificial, extremely low frequent but very informative because they give a strong hint to the movie “Mary Poppins” to which the documents may refer to. How can one incorporate such prior knowledge into a proximity measure?

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2 Similarity and Distance measures A common way to measure dissimilarity between objects or numeric vectors of measurements defined on a data space X is to use distance functions. A minimal characteristic is that a distance function d - as well as similarity function s – should have a non negative range d(x,y) t 0 and be self similar, i.e. d(x,x) = 0. Def.1: A measurable function d: X2oRt0 where X is an arbitrary set, is called a distance (metric) iff for all x, y, zX we have: 1. d(x, y) = 0 Ù x = y (reflexive) 2. d(x, y) = d(y, x) (symmetric) 3. d(x, z) d d(x, y) + d(y, z) (triangle inequality). In Def.1 we note that relaxing 3. by d(x, z) d max{d(x, y), d(y, z)} d d(,x y) + d(y, z) leads to ultrametricity which is commonly used in science, cf. Murtagh (2007). Hartigan (1967) published a list of twelve possible proximity measures which are reprinted in Cormack (1971) and Cox and Cox (2001). His list includes metric, real-valued, symmetric real-valued, Euclidean distances and various kinds of complete or partial orderings as well as partitions of X2. As these measures strongly depend on the underlying scales of the variables involved, we are going to present them accordingly, i.e. separated according to a nominal or even binary, ordinal and metric (ratio or interval) scale.

2.1 Nominal or Binary Scales Sneath and Sokal (1973), Hubálek (1982) and Cox and Cox (2001) give a comprehensive table of similarity coefficients for binary data. Roughly speaking, similarities are derivable from related distances. Using a parametric measure of similarity one can represent most of those coefficient by one generalized formula. For this purpose we consider the following frequency table: Object s Object r

1

0

1

a

b

0

c

d

Table 1: Frequency table of two binary attributes

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We use the parametric similarity coefficient as used by Steinhausen and Langer (1977). Let O, G  R, O t 0, 0 d G d 1 and S rG,,sO (a  Gd ) /(a  Gd  O (b  c) . For example, it follows the Jaccard-Tanimoto coefficient Sr,sJT= a/(a+b+c) for O=1, G=0, the Rogers and Tanimoto coefficient Sr,sRT= (a+d)/(a+2(b+c)+d) for O=2, G=1, the simple matching coefficient Sr,ssm = (a+d)/(a+b+c+d) for O=1, G=1 etc. 2.2 Ordinal Scales Variables are called ordinal scaled if the relational operators ‘=’, ‘z’, ‘<’, and ‘>’ are feasible. This implies that the operators ‘min’ and ‘max’ are applicable, too. There a three alternatives how to handle variables which have ordinal scales. First, reduce the ordinal scale to a nominal one. In this case one accepts a loss of (ordering) information. Second, one can use Gower’s (1971) general similarity coefficient (“average possible similarity”) Sr,sG = (6Zrsi srsi)/ 6Zrsi, where srsi is the similarity between the rth and sth object based on the ith variable alone and Zrsi is an indicator function with Zrsi = 1 if the rth and sth object can be compared on the ith variable Zrsi = 0 if not. Finally, one can use ranks instead of measurements for each variable l =1,2,…,p. The sample (xi)i=1,2,…,n is sorted in ascending order, i.e. x[1]d x[2] d…d x[n]. The rank ri is the position of object i in the order statistic (x[i])i=1,2,…,n, i.e. Rank(xi)= ri = k if xi = x[k]. Let yil = ril/nl appropriately scaled ranks with yil  [0,1] and nl is the number of ranks. The “Canberra-Metric” applied on ranks is proposed, cf. Steinhausen and Langer (1977), which is scale-invariant. p

d ( yi , y j )

¦

l 1

y il  y jl y il  y jl

(1)

2.3 Metric Scales If the measurements are metric scaled, i.e. ratio or interval scaled - dependent upon the existence of an absolute zero, one usually represents the measurement vectors (x1,x2,…,xn) of a sample of size n as tupels (“points”) in the pdimensional space Rp of real numbers. Like in the qualitative case a lot of dissimilarity measures are proposed. 2.3.1 Distance Measures in Statistics

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H-J. Lenz p

(¦ x l  y l

d k ( x, y)

k

)1 / k

(2)

l 1

The most important parametric family of distance measures is the Minkowski metric which is defined as follows: At least, three special cases are important: p

k=1: Manhattan distance (city block distance) d 1 ( x, y )

¦ xl  yl



l 1

§ p · ¨ ¦ x l  y l 2 ¸ ¨ ¸ ©l 1 ¹

k=2: Euclidean distance d 2 ( x, y )

1/ 2

kof: maximum distance df(x, y)=max l=1,...,p |xl - yl|. These distances correspond to different kinds of neighbourhood which is illustrated in the following diagrams.  







 















 















 



























     Fig.5a: k=1

Fig. 5b: k=2

Fig.5c: kof

The family of Minkowski metrics has the following main features: x translation invariance, i.e. dk is not affected by a translation x o x + c, where c  Rp not scale invariant nor dimensional-free x not sensitive to non-uniform probability distributions on Rp; invariant to orthogonal linear transformations. Skipping further distance measures like Canberra metric, divergence coefficient, cosine distance etc., cf. Cox and Cox (2001) , we turn to the Mahalanobis distance (1936). The distance measure is dimension-free, invariant with respect to orthogonal transformations and is further scale-invariant. Def. 2: Let x, y  Rp be random vectors and 6 the corresponding covariance d M ( x, y )

matrix. Then

( x  y )T  1 ( x  y )

(3)

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is called Mahalanobis distance. This measure is influenced not only by the difference (x – y), but by the variances and correlation of the corresponding variables, too. The covariance matrix is defined by μ

1 n

¦ XY

1 ¦ ( xl  P )( yl  P )T n 1 l

and the mean vector by

¦ xl . Evidently, the Mahalanobis distance depends upon the distribution l

PX on Rp only by the first two moments, i.e. the covariance matrix 6 and – indirectly - the mean P.. Contrarily, the Kullback-Leibler divergence dKLof two distributions P1 and P2 on p

p

R measures the mean information given observation x R to separate P1 and P2, c.f. Kullback (1959). Def. 3: Let P1 and P2 be two probability distributions with Radon-Nikodym dP densities i pi (x) for i=1,2. Then dOi d KL ( P1, P2 )

p1 ( x ) dO x 2 ( x)

³ ( p1 ( x )  p2 ( x )) log p

(4)

is called Kullback-Leibler divergence. The Mahalanobis distance and the Kullback-Leibler measure are equivalent under the specific assumptions of Gaussian distributions and equal covariance matrices, cf. Skarabis (1971). p Lemma 1: Let the data space R with x, y ~ N(Pi, 6i) { Pi for i=1,2. Furthermore assume that 61 = 62. Then d KL ( P1, P2 ) ( μ1  μ2 )T  1 ( μ1  μ2 ) . Instead of a distance measure it is often more convenient to use a similarity measure. It can be derived by any (non-linear) monotone non-increasing mapping. Analogue to distances one can define Def. 4: A measurable function s: X2oR[0,1] where X is an arbitrary set, is called a similarity measure iff for all x, y, zX x x x

s(x,x) =1 s(x,y) = s(y,x) s(xi,yj) = 1 Ÿ oi=oj

(self-similarity) (symmetry) (object identity)

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H-J. Lenz

x

s(x,z) d s(x,y) s(y,z) /(s(x,y)+s(y,z)) inequality)

(analogue to triangle

As is well known it is difficult in some applications to establish symmetry as well as transitivity of s. As intuitively a great similarity means a small distance between two objects there exist a large set of transformations for switching from distances to similarity and vice versa, cf. Steinhausen and Langer (1977) and Borgelt (2006). For example, d = 1 - s or d = (1-s)1/2 may be such mappings.

2.3.2 Point wise Distance Measures Another area of application of similarity or distance measures comes up when pairs of functions or probability distributions are to be compared. There exist two principles for deriving appropriate measures: One can make a point wise comparison (“supremum-norm”) or one can use the area (“integral norm”) between both functions. We start with the first group. Assume that we have to compare a given distribution function F0 with an empirical frequency distribution Fn which is defined by

By definition both functions are monotonically non-decreasing. Then the Kolmogorov-Smirnov distance measure dKS is given by d KS sup Fn ( x)  F ( x) max{D n , D n } where D n max( Fn ( x)  F ( x)) and D n

max( F n ( x )  F n ( x ) F ( x )) . Fig. 6 illustrates the KS-measure for

the case above.

Fig. 6: Kolmogorov-Smirnov distance

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Evidently, the distance is measured at one and only one point in the domain R, i.e. arg sup _Fn(x)-F(x) _|0.459. Another idea is to measure the distance between two non empty point sets of some metric spaces. This approach was proposed for instance to discriminate between membership functions used in fuzzy logic, cf. Kruse and Meyer (1987). Let (X , dX ) be a metric space. We first define the distance between a point x  X and the non empty set A Ž X by d h ( x, A) inf d x ( x, A) . Then applying aA dH to both non empty sets A, B Ž X we get Def.5: Let A, B Ž X . The distance measure d H ( A,B ) max{d * ( A, B ), d * ( B , A )} where d * ( A, B ) sup d H (a , B ) is called Hausdorff metric.

(5)

a A

There exists a draw back of dH when being applied to two parameterised curves. As mentioned above dH measures the distance between sets of some metric space. So it discards the parameterisation and is applied to the range or image set of both curves. This can lead to the nasty side effect of a low distance between two curves while they appear to be quite different from visual inspection as shown in fig. 6, cf. the distance value G.

Fig. 7: Hausdorff metric dH applied to two curves

In 1906 Frechét introduced a pseudo-metric which is more appropriate for functions with a finite number of parameters. The Fréchet distance can be illustrated as follows. Suppose a man is walking his dog and that he is constrained to walk on a curve and his dog on another curve on a plane. Let both the man and the dog control their speed independently. They are not allowed to go backwards. Then, the Fréchet distance of the curves is the minimal length of a leash that is necessary. More formally, we have, cf. Godan (1991):

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Def.5: Let f: AoX and g: BoX be continuous mappings , (X, dX) a metric space and let A, B Ž Rd be homeomorphic to [0,1]d for fixed dimension dN. Let D: AoB be an orientation preserving homeomorphism. Then d F ( f , g ) inf sup d F ( f ( x), g (D ( x))) (6) D xA

is called Fréchet distance of f, g.  As an example we take d = 1 and consider two (polygonal) curves. We use a linear function D. The Fréchet distance dF is marked.

dF

Fig. 8: Fréchet distance dF

The following point wise distance measure receives a lot of attention in the asymptotic theory in Statistics. It is originated by Levy (1937) and is specially tailored to measure distances between not strictly but monotonically increasing functions, i.e. probability distribution functions. Def. 6: Let F, G be continuous distribution functions. Then d L ( F, G ) inf{H F ( x H )  H d G ( x) d F ( x  H )  H for all x  R}

(7)

(7) is called Levy distance between F and G. Rewriting (7) we get d L (F, G ) sup inf ^H ! 0 : G(x - H - H d F(x) d G(x  H )  H ` . It can be easily shown that —2 dL(F,G) is the maximum distance between F and G taken under an angle of 45o, cf. Fig. 9.

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171

—2 dL(F,G)

Fig. 9: Levy distance between distribution functions F, G

2.2.3 Distances based on Integral Norms While the distance measures discussed so far measure the distance between two monotone increasing functions F and G “point wise” there exists a broad spectrum of competitors which make use of “ integral norms”. Def.7: Let F be a proper distribution function with related density function f, Fn a sampled corresponding empirical distribution function and \ a finite-parameter weighting function. Then an integral based distance is given by 2

f

d Int ( F , Fn )

³ >Fn ( x)  F ( x)@ \ ( x) f ( x)dx

(8)

f

It is worthwhile considering at least two special cases: Selecting the weighting function \ ( x) [ F ( x)(1  F ( x))] 1 one gets the Anderson - Darling distance A 2

( Fn ( x)  F ( x)) 2 f ( x)dx , and using \(x)=1 F ( x)(1  F ( x)) R

n³

the Cramér-von-Mises distance Q

n ³ ( Fn ( x)  F ( x)) 2 f ( x)dx . R

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3 W-Distance Now, we come back to the example presented in the introductory section 1.2. The main message of the example was that two people who go swimming in an icy lake are more similar than a pair who is wearing warm winter clothes during winter time. Roughly speaking, two objects or people are more similar than another pair if their features coincide with values which have a low probability. Or the other way round: Observing rare values of objects is a stronger indication of similarity between a pair than to observe values which always happen. Evidently, the utilisation of such prior information should improve to determine an appropriate similarity or distance coefficient. As proposed by Skarabis (1970) the first step is to construct a complete quasi order (X2, %| ) based on a distance d and a probability distribution P, where X § § x k · § xi · · §x · x ¨ ¨ ¸, ¨ ¸ ¸  L d,P œ §¨ k ·¸ %| ¨ i ¸ ¨x ¸ ¨xj ¸ ¨ ¨© x l ¸¹ ¨ x j ¸ ¸ © l¹ © ¹¹ © ¹ ©

R and |X| < f:

Having constructed Ld,P appropriately, one can compute the predecessor set Mij of each pair (xi, xj) M ij

­°§ xk · § xi ·½° 2 § xk · ®¨¨ ¸¸  X : ¨¨ ¸¸ %| ¨¨ ¸¸¾ °¯© xl ¹ © xl ¹ © x j ¹°¿

The distance of a pair wise observation (xi, xj )X2 is measured by the probability of the set of all its ’ %| ‘- predecessors. We can now define the Wdistance, a shorthand notation for a probability (“Wahrscheinlichkeit”) based distance. Def. 8: DEF.: Let X  R and |X| < f. Then dW: (X2, %| ) o R[0, 1] with dW(xi, xj) = PX(Mij) is called W-distance on X. This measure has the following properties, cf. Skarabis (1970): x

symmetric: dW(xi, xj) = dW(xj, xi)

x

normalized: dW(xi, xj) d 1

x

the triangle inequality not (generally) fulfilled – depending upon PX

x

dimension free.

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3.1 Unordered Set X We apply now the dW to example no. 1 which is characterised by the triple (X, PX, dX). We let X = {1, 2} where 1{„take a bath“, and 2{„take a coat“ (exclusive values). The distance dX is simple indicator function, i.e. dX(x, y)=1 if x=y and dX(x, y)=0 else. The probability distribution PX on X is given by xX

x1 = 1

x2 =2

PX(x)

0,01

0,99

Table 2: probability distribution PX on X L1

­°ª§ xk · § xi ·º 2 2 ®«¨¨ ¸¸, ¨¨ x ¸¸»  X u X : xi °¯«¬© xl ¹ © j ¹»¼

L2

­°ª§ xk · § xi ·º ½° 2 2 ®«¨¨ ¸¸, ¨¨ x ¸¸»  X u X : xi z x j and xk , xl any value ¾ °¯«¬© xl ¹ © j ¹»¼ °¿

x j and xk

½° xl and P(xi ) t P(xk ) ¾ °¿

Then the quasi order (X2, %| ) can be determined as follows where Ld,P = L1 ‰ L2 . The W-distance value for a pair who takes an icy bath is small and is given by dW (x1 , x1 )

§ ­°§ x ·½° · PX ¨ ®¨¨ 1 ¸¸¾ ¸ ¨ °© x1 ¹° ¸ ¿¹ ©¯

P( M 11 )

1 100 2

0,0001

Contrarily, the W-distance value for a pair who wears a warm coat in winter dW (x2 , x2 ) P ( M 22 )

§ ­°§ x · § x ·½° · 1 99 2 PX ¨ ®¨¨ 1 ¸¸, ¨¨ 2 ¸¸¾ ¸  # 0,9802 ¨ °© x1 ¹ © x2 ¹° ¸ 100 2 100 2 ¿¹ ©¯

time is high By definition, all other pairs (xi, xj) with unequal observations, xi z xj, have dw(xi, xj) = 1. Moreover, due the small cardinality of X the sets Ld,P = L1 ‰ L2 can be completely enumerated. L1

­°ª§ x1 · § x2 ·º ½° ®«¨¨ ¸¸, ¨¨ ¸¸» ¾ °¯¬© x1 ¹ © x2 ¹¼ °¿

L2

­°ª§ x1 · § x1 ·º ½° ®«¨¨ ¸¸, ¨¨ ¸¸» ¾ °¯¬© x2 ¹ © x2 ¹¼ °¿

Finally, the complete quasi order (X2, % | ) is given by ­§ x1 · § x2 · § x1 ·½ ®¨¨ ¸¸, ¨¨ ¸¸, ¨¨ ¸¸¾ ¯© x1 ¹ © x2 ¹ © x2 ¹¿

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H-J. Lenz

One can summarize the computation of the W-distance for unordered sets X as follows: ­ ¦ PX2 ( xk ) for all x x ° i j dW ( xi , x j ) ® k pred ( xi ) (9) for all x z x i j °¯ 1 where the set of predecessor pred ( xi ) ^k  N : PX ( xk ) d PX ( xi )` for all i=1,2,….,n .

pairs

of

xi

is

given

by

3.2 One-dimensional Metric Space X We use as basic distance measure d a special form of a city block distance, i.e. d(x, y) = _x - y_ for all x, y X=R. Therefore such problems are described by the triple (R, PX, d). As done before a (complete) preorder (R 2, %| ) is constructed for induced similarity of pairs (x, y) of observations, however, this time the information from the basic distance measure d as well as from the distribution PX is utilized as follows: The dW-distance is given by

d W ( xi , x j ) L1

L2

­°§ x k · § x k · § x i · °½ PX ®¨¨ ¸¸ : ¨¨ ¸¸ %| ¨¨ ¸¸ ¾ °¯© x l ¹ © x l ¹ © x j ¹ °¿

­° ª§ xk · § xi · º ½° 2 2 ® «¨¨ ¸¸, ¨¨ ¸¸ »  X u X : d kl  d ij ¾ °¯ ¬«© x l ¹ © x j ¹ ¼» °¿

­° ª§ xk · § xi · º 2 2 ® «¨¨ ¸¸, ¨¨ ¸¸ »  X u X : d kl °¯ «¬© xl ¹ © x j ¹ »¼

½° d ij and P( >xk , xl @ ) d P( xi , x j ) ¾ °¿

>

We see that the W-distance is just the probability of the predecessor set of the current pair (xi, xj) including this pair itself. Because of the preorder (X2, %| ) we have again with Ld,P=L1‰L2. §§ xk · § xi ·· §x · x ¨ ¨ ¸, ¨ ¸ ¸  Ld,P œ §¨ k ·¸ %| ¨ i ¸ . ¨x ¸ ¨xj ¸ ¨ ¨© x l ¸¹ ¨ x j ¸ ¸ © l¹ © ¹¹ © ¹ © Therefore the pair (xk, xl) is more similar to each other than this is the case for (xi, § x · § xi · xj) if ¨¨ k ¸¸, ¨¨ ¸¸  L1 or L2. This means that either dkl < dij or if the basic (city © xl ¹ © x j ¹ block) distances are equal, i.e. dkl = dij, then the inequality of probabilities, PX((xk, xl)) d PX((xi, xj)) becomes true.

@

Proximities in Statistics: Similarity and Distance

175

We close with a final example. We consider the space X={0,2,3} R. Assume for simplicity that we can apply on X 2 the (basic) city block metric d(xi,xj) = dij = _xi - xj_. Then we determine the following distance matrix D = (dij) for all possible pairs of observations:

Table 3: Basic Distance Matrix D

Assume that the following non uniform joint distribution PX 2 on X2 is given:

Table 4: PX 2 on X2

We can compute in a straighforward manner the complete preorder (X2, %| ) on X2

. From D and PX 2 we can easily derive distances like dW((3,3)) = 1/10 < dW((2,2)) = 2/10 < dW((2,3))=1. The last inequality is true because tupel (2,2) z (2,3) and §§ 2· § 2·· therefore ¨¨ ¨¨ ¸¸, ¨¨ ¸¸ ¸¸  L1 . ©© 2¹ © 3¹¹ 3.3 W-distance in Xp We turn finally to the case where all observations xi, i=1,2,…,n, are p-tupels, not necessarily real vectors. A response or measurement vector gets relative to a prototype a small distance value if all its components are relatively small. Therefore we can define the W-distance analogue to the univariate case for all dW ( x i , x j )

­°§ x · § x · § x i ·½° PX ®¨¨ k ¸¸ : ¨¨ k ¸¸ %| ¨¨ ¸¸¾ °¯© x l ¹ © x l ¹ © x j ¹°¿

176

H-J. Lenz

pairs of p-tupels, (xi, xj)T Xp x Xp: As Skarabis (1970) proposed we can construct a complete preorder (Xp x Xp, %| ) in order to determine the set of predecessors: § § xk · § xi · · §x · x ¨ ¨ ¸ , ¨ ¸ ¸  Ld, P œ §¨ k ·¸ %| ¨ i ¸ ¨x ¸ ¨xj ¸ ¨ ¨© x l ¸¹ ¨ x j ¸ ¸ © l¹ © ¹¹ © ¹ ©

If we define for each vector xi its l-th element by xi l , l =1,2,…,p, we can consider component-wise W- distances d ijl ( x il , x jl ) for all l=1,2,…,p. Of course, the basic distance used to construct dW strongly depends upon the scale of measurements, i.e. nominal, ordinal or metric. Aggregation of the W-distances d ijl ( x il , x jl ) is performed by multiplication as Skarabis (1971) proposed. p ­°ª§ x k · § x i ·º 2p 2p l ®«¨¨ ¸¸, ¨¨ ¸¸»  X u X : – d kl d l 1 °¯«¬© x l ¹ © x j ¹»¼ Evidently, other combination rules can be thought of.

Ld ,P

p

½°

l 1

°¿

– d ijl ¾

References Borgelt, Ch., Prototype-based Classification and Clustering, Habilitationsschrift, Ottovon-Guericke-Universität Magdeburg, Magdeburg, 2005 Cormack, R.M. , A review of classification (with Discussion), J.R.Stat. Soc., A, 31, 321367 Cox, T.F. and Cox, M.A.A., Multidimensional Scaling, 2nd. Ed., Chapman & Hall, Boca Raton etc., 2001 Frakes, W.B. and Baeza-Yates, R., Information Retrieval: Data Structures and Algorithms, Prentice Hal, Upper Saddle River, 1992 Godan, M., Über die Komplexität der Bestimmung der Ähnlichkeit von geometrischen Objekten in höheren Dimensionen, Dissertation, Freie Universität Berlin, 1991 Gower, J., A general coefficient of similarity and some of its properties, Biometrics, 27, 857-874 Hartigan, J.A., Representation of similarity matrices by trees, J.Am.Stat.Assoc., 62, 11401158, 1967 Hubálek, Z., Coefficients of association and similarity based on binary (presence-absence) data; an evaluation, Biol. Rev., 57, 669-689 Kruse, R. and Meyer, K.D., Statistics with Vague Data. D. Reidel Publishing Company, Dordrecht, 1987 Kullback, S., Information Theory and Statistics, Wiley, New York etc., 1959 Jardine, N. and Sibson, R., Mathematical Taxonomy, Wiley, London, 1971 Mahalanobis, P.C., On the Generalized Distance in Statistics. In: Proceedings Natl. Inst. Sci. India, 2, 49-55, 1936

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Murtagh, F., Identifying and Exploiting ultrametricity. In: Advances in Data Analysis, Decker, R. and Lenz, H.-J. (eds.), Springer, Heidelberg, 2007 Skarabis, H., Mathematische Grundlagen und praktische Aspekte der Diskrimination und Klassifikation, Physika-Verlag, Würzburg, 1970 Sneath, P.H.A. and Sokal, R.R., Numerical Taxonomy, Freeman and Co., San Francisco, 1973


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Similarity Relations and Independence Concepts

195

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196

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! 1 ,-      1  '- ,    2  #1- -  1#-A  -, -  D % 1    1 ,-    ,  -, 1 ,,  +-   2 !, 1- -   7!, 1- -  = KCC@8 + - = P0A ,--    ,  - -   -, - + (   -  + ,   - #   -     1-D #   #  = -,  - =  1 ,-   -   DDD 2#  -     1 ,- DT &  1-, -  1#- -,  1-  --, =  1 ,-   '   2-    ,  -,- '  -A2- -  = --,  -, -  = -

    = -  #   11  1      -  2-  -  ;,  ' #   D -     -A , -  #  - #- - ' -1    1 ,-    2$   # ,   2,1 1- D % -1 1- 2 2-   1 - --,        2$  2 - #2$ ' $# 1 1-  2 - 2'D &  1#- -, -  =  1 ,-  '-,# -  2-  #  -  1 ,-  1-#D   1 ,-  1-#  - 1-   F     = +    # ' -   - -,A'-,# '-, +  '-,#  -   1 ,-    2$   D #,   -(  2  '-, UJ K?  +  - '-,# J - K  - 1  1-, - 1-0 1-,  1 ,- 

198

T. Sudkamp

2+ 2$D ! 1 ,-  1-# #   ,  -, 1-  -   - 1 ,    -    1 , #2$ '  1 ,-  $# 1D ! 1 ,-  1-# 1- 2 ; -0 1- -,,  1- 2 2- 

 # ', 2 -  --,         2$  

1- D # ' - - - #, 2-  -# 1-    - 12

   1-  2$  - 1  -D %  1 - -= +  -     -0  1  ,- ;- =    2$  2  2 2 -   - 2# -  1 ,-   1  2  1-     #12  11  -   -#D 3 2$ - -  - +     - 1  -=  1 ,-   1-  1- 2 2-  ,  1  - # L ,       -=  1   1 ,-      2$D % -0 1-  - - 2  2  #,-  1-1- -,    - #  ', -#       1 ,- D % -0 1 #, A ,#     # - >0 ' = 11= - '- # '   -A   ' D  #  - - ;  -0 1      - --2,  1 ,-  1-#D  ,, +    - -= %'( 7%'(= KC<<8 ',A    - 1 ,  1  1  -  1 ,-  - 2 1 ,,

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

Imprecision and Structure in Modelling Subjective Similarity

199

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

  % ,- -, - -   1 ,-  -1 - 2-    11 A -,   - 2#  2$    0 1  +  2$ - 12



 - 1  -D &   -=  1-  2$ -  2 2 -   - 2#  -#
     

= + - - 2#
 - - -  A -   '-,#       
D 3    - = - 2$  - 2  2 - A 1 -, '   N U        
? +  
 D    #    1 ,-  -1= - 2$    

 2  1,, --  2  - 2# -   D -   1 ,-    0 1   - 1,  2 -  -  - -, #1A 2 +  - - 2# '-,# 7 = KCVC8D 3    -  - = - 2$  ; 2  - 2#   2 - '  U7 8 7 8     7
8?  
= + 7 8   #1  '-,# -  - +    - 2#  D % # -1-, -#1  2   0 1  2-  1 ,-  -1  -  1 ,-  2-' - - '  -I   1 ,-   2$

- -  - 2+  -  -   -D &  0 1 A2-  1 ,-  -1=  -  #, 1-A # 2 -  ( +( 1 

  7  8 N


 

  

7 8  7 8



200

T. Sudkamp

+  KD %      1 ,-   7  8 N K   7  8  +    1-0 1#1 - 2+ + 2$   1- D % --- -= 7  8 N

K 7 8  7 8 

7K8

- 2    - - 1-#  -0  1  12,- 7 -  - "-A   = KCS@8 +  2$ -  2 -,A'-,# ' D 0 1-, #, -' -,,  #   '-,   -,,   1  -0 1   1 ,,  #2$ '  1 ,-  $# 1D /#1-, 7/#1-,= KC<@8   #   +   1 ,-  $# 1   #A    1  1-,   = 7  8  7 8D % - #    A - 2+ 1  1-,  - #1-  1 ,-  $# 1=   A   - (1  1 , -   -   2#   

1-  2$  - - 2-  1 ,-  1 ,D 0 1 -' -,  + - #2$ '  1 ,-  $# 1 -   - , 11  7%'(= KC<
 1 ,-  # W #2  -#F  1 ,-  $# 1  #    11  -# + ,  1 1-  -# -' - - A >#   1 ,-  $# 1 7 = KC<SI %'(= KC<<8D #   ' 2+ #2$ '  1 ,-  $# 1 -

 0 1  2- 1-#= + + ,, # -# 1-  -  2-      1 ,-  1-#       -D &   1 ,-  1-#    2 *--   -0  1  ,- ;- = 2$ - ; 2 -A  2# - -#1   +   2, '-,#F   -2 7*-- = KCJ@8D  # K ,,#- + -# '  +  A- 2# -A # '-,#= +  -  -    - -2  ',D % -  #   2, # 1     1-    - 2#F 7 8= 7 8= 7 8= - 7 8D    1,   + ,   =   =   =   

   #12  - 2# - -   2   -  7-,,

Imprecision and Structure in Modelling Subjective Similarity









201







                     



                     

  -# - 

   ' 1-8=    - -2   = -2   -    = - -2   -  7-,, - ' 1-8D )      - =  %((  + - 2 +  7  8 N   7   9    9   8

7O8

% *--  0    ,    ' 1-=   -  - ' 1-  # 7  8 N

7   9   8   

7Q8

+  - 2   -  , ' ,(  5 7! (-, -

!-= KCVQ8D % ( , '   1 ,-  7R8  - --1  -, -    *--  0 + - ! -  - - -  #-,  JD % #    -,, -(  2  - -, = 2# 1- 2 -  #  F O  UJ 8 -  -

 ' ' $  D
(  - 1 ,= %'(6  ( , ' # +    -  1 -     -# ,-  D &   - 1 ,=   1 ,-    -    ' 2 7  8 N " 7  8  7  8  ! 7 8

7S8

+ "  ! -  - -   #    -1 -   - 1 ,D % +  >  ,- ' -,    -# 2   1-    -,   -  1-  D % ,   +     - - - 1 , - 2 #   #  A11   1 ,-  #  - 1 ,   -,   #2$ ' $# 1D

202

T. Sudkamp

 -, -    #   1 ,-  1-#  1       ' 1-= - ' 1-= - 1 1- +-   7 -  -,D= OJJKI  - -  = OJJQ8= +  -, ,- ;  1-# 2-    2 # - 2-' #= >0 ' = - 1     D % - 1 ,=  - 1 ,= -   1 ,-  1-# # 

 7 -  -,D= OJJKI  - -  = OJJQ8 -  +      1-  ,-  D  -,- ' - -   -  +    - 2# 1,'D & -0  1  ,- ;-   - 2 # -    ;-  - -#  #, 2 -W 2  #   #    #,-  7-12--= KCVSI !-= KCVS8D & !  S + + ,, -  +   - 2#  1 ,     2+ -#D

!

    

%-0  1  ,- ;-  # - 2# '-,# -     -2  - -#D & 1  -,  2,1=  1-    -A  2# -      + '-,#D % 1- -   A-2

  1 #    1 ,-  1 ,= %'(     '  #A 1  '-,#  # -2, + A'-,# - D & -

   1#, , '-,# - #1  - 2#= #2$ '  1 ,-  $# 1  #    2 ,  - - - 2# 1- 2 - -,, - ; 2 - 2$D %   1 ,, 2 1   - 2# '-,#  2 # 112  '-,#    # D % A-2 1-  -- 1 - 2 0   - 2# +  '-,# 1-  - -, - -   -  -  #  ' - -1 ,  , #   1D 0   -A # 1-   1 ,-  -1  # - 2# -' 2   

2 !-  - *-  7!-  - *- = KCCC8 - % , -  -,D 7% , -  -,D= OJJK8D % - - # #  -    1#  - -,         =   =   =   D % 1 1, 

2 !-  - *-  '  ,-+   -   + ,  ,- + ( #  - -  $#  -  1,1-  -   2-   - -, 112  '-,#      # D     1#- -, - '--  -# 1-  1 ,  - - - 2#   1-  ' #-,,D % #,   1 ,-  '-,#  2-  2 #11   +  #12  -   1- -  A 1-D %     1-        -# -,, + #      0   1-   - -, 112  - #    , - 2#D %+ -# '-,# -  1,, 1- + ,, 2 - 

'-,# K= - J + ,,  - -    1-  2+  '-,#D - -, 112  '-,#  2    +  - #  -

Imprecision and Structure in Modelling Subjective Similarity



203

   

    

   - -   - -, 112 

 - ; D   0-1,= - #  - # - P-  -  T 1- 2 # - - - 2#  2   ,D % '-,# K     - 2#  -  1, - -     -= J  - - = - '-,# 2+ J - K  - -, - - D % #      A-2 -   - - -,   - -    '   # OD %  - - -,   - A - =  P,,T    -#  '   #2,,D    #2,,   - -  -#    - ,-  = +    #12  #2,,D &  # O  ; '    #2,,  '  2 - - 2#   -  -  2$ - ;  -#   D
    1 ,-   - -, # '-,# # 2     ' - -A  ' 1-D &  112  '-,# - 2  =   -#  #,

 1,, 1-D %#    1-   -# +  112A   '-,# - !  K    !D % A-2  1-     - -, 112   #  -1 0     - 1 , -    2 !-  - *-  7!-  - *- = KCCC8D  #2$ ' $# 1 1- -,  # - #  - - - 2# '-,#D    -#   2 2 , #   1 # - (''=

204

T. Sudkamp    

 

  - -, - -  1-    

 

 

                  

  #  1- 

("(= -  D  #2$ ' '-,#-   -  6   1-  # - #  DRG('' 9 DVG("( 9 JG D %  #   1      1-  -  , 112  '-,# - 2 -   - 1   # D % - 2# ,,  #2 '       -   , #   1 -   # RD -   #2  + ,,   2# - 1-0 1#1  K     1-    -#= +   #12  , #   1D   - - 2# ; 2 - , #   1         D &  '-,#   - 2#   2$  -  -   9


9   - !  9


9 !  =  ',=     1-    - 2#    K 





 ! 



K NK  



 ! 

1-   #    # R +  - #  DKG('' 9 DVG("( 9 DQG   # - 1-  D@D

"

   


%   -,   -#A2-  1 ,-  -1   ,   -  -#D % - 1,    -(= %'(   -  1 ,

Imprecision and Structure in Modelling Subjective Similarity

205

1# 0- -  1 , P- , 1  ,   ,'- -#   2-   +  +  1  #  -(DT   1 ,-  $# 1=  +'= 1-   # ,  -,,   ,'- -#D   0-1,= -# # -  = -  1-#-#= 1-(= 1 ,=  = -1   = 1 ,-= # ,= ,-A - ; = D= 1- 2 ,'-     1-    -# 1 2 ,D  -1    1= P2-    = -= - 1 ,-=  -# 1 2 , - 01,  1 ,-=T 0,  ,     #   - #,-  1-  D &     +     12    1 ,-  $# 1  1 WA  2'D % $# 1 1- 2 2-   1 #1- -1  -  1  D &   -= +   -#1 -  -A 1 # ,   #,,   -#D %   2 ,     ,  1  1-,  ;   #     -  12    1 ,-  -1F   1 ,-  1-#  -   2, 1- - - -# 1-D %  1, 1-   X  7Q8 + ,, 2 # -  # ,   1 ,-  1-#D 3 ,  7  8    1 ,-  # 2 2'-   -

  #   2'-      D %   2 ,   -   7  8 N  - -  -  1- -  7KJJ
85   -# - -  1-     -# - ( +    1-D %#  1,  1-   ,#  - -   -, 1-    2 ,    1 ,-  -1D %   +  W ,',  -        '    # S  - -# 1 1- +  7  8 N -  7  8 N !D % ,- 1 ,-2,, - !   --  ( +    -, -# 1- D %   + '  ,,#- -  12 -    1 ,-  -  #  1  1-, -# 1 1- 71-0 1-,   2 , 8D %  # +  --  1 1-    - #2  -    '  '-D % -1 # 1 1- -# 2 2     1 ,-    1-07K   K  !8     2,    1 ,-   1 7  !8D % 1-0 1-, -# 1 1- # +  1 1-  1  -

 -'  1  1-,   -#   11 D %   ,,#-   , + -   '    # SD % #,    2, 1-   1-07 9 !K J8D      12 -    1 ,-  -1 2 =    U1-07 9 !  K J8 1 7  !8? +  7  8 N -  7  8 N !D %- =  2    +    1-     ' #-, $# 1=   12 -  1#  , - '-,# 2+ 
#(-:E+  - 1  1#1 A  1    1   1 ,-  D )       A 1=    -#1 - 2 A -,    #     -      12 -   1#, ,

206

T. Sudkamp 


  



   





 

   

    



   





 

 

  12 -    1 ,- 

 1 ,-  $# 1D % 2 #   # 2   12 -    1A ,-  -1  7  8 N =  7  8 N != -  7  8 N # - U1-07 9 ! 9 #  O J8 1 7  ! #8?D & + ,   -    A '-, U  ?=   -  - 2  #  - - '-, #  - - ; 8  F K 7  8 N 7  8= J 7  8 N J 8  11#- ' F  7  8  7  8 N  7  8  7  8 8 -  - ' F 7 7  8  7  88  7  8 N  7  8 7 7  8

 7  88 & 1   12 -     1 ,-   1- = -  - '  -

 11#- '  1, -  #,  2 #  -         +    ' #-,  1 ,-   -  12  D 3     -    1 ,-  -1  -  - $ =   12 -   #    , '-,#   7  8 N

9 !  K

   1 1-    1# - , 2 $ D #,=  , + 2 #      2 ,   - ' D

Imprecision and Structure in Modelling Subjective Similarity

#

207

  


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
7#=,-8=  %7#8= &%!7,-=#8= -
7#,#=, 8=  %7#,#8= &%!7, =#,#8D &     2'- = ## 1-    7-(-1 - = KCCQI " , -( - %-- = KCC<8 -   - - # 7" - = KCCSI = KCCS8 + ', 

 1-  ,- -, -   -      ##-, --,,, 2+  1-    1   -  1 ,- D  -  - 7 -  -= KCC@8  #      1 ,-  2-    , ,- 2 ,- -, - 2$  1 A ,- D (' ('  1 ,-  # +     ;- ,- -, 'A ,- 2+  1 2   1- - , , 2$  1 ,- D % -,,  --,  -, -    1    -  - 1-  2+ 2$   2- 1-  -  - 1- D
('  1 A ,-  #  2  ,- -, - 2$  1 ,- D 1-  - ,,     '  - 0-1,  ,1(

((  1 ,- = +     , 2$    -   ,- D % 1 -  ,-    1 ,-  -1     

 A 2$ ,-  D -    ,-  2+ - 2# 1- 2   ;-  1    ,- '  1 ,-    2$D &     + # +  - - -       - 2#    -     -# 1-    , -,  1 ,- D % 1 ,  #  !  Q    -,, - 2#  2  #-,   ;-D " +'=  $#   ,- '  1 ,-   1 - 2# 1- ,- - 1  1 -  , - D &  2$ '    1- ' , 2-    # -2 ,    ,- -1 , =    ' , 7!).=

208

T. Sudkamp

-  +- =  -= 8 -   - 1    ;- -    -  ,-     # ,D %  1-   -   -   ,- '  1 ,-   ' ,  1  2   -1        -#D % , +    1 -W  ,- '  1 ,-  , +       -#   W - 2+  + D 3 + ,,   -      1 ,-  -1 2 -   +   - 2#    ,-D %         -  1  '-    ,   +  + ,, 2 ,,#- 2      ,- '  1 ,-     2$ =  = - D &  - 1  1- 2+   -  -1       - 2#=  7  8  7 8 - ,   ,-   2+ , +    - 2#D &   -  -'  -1   1-         - 2#=   ,-   2+  ,- '  1 ,-   7  8 - 7 8 - 1  2    +   , +    - 2# -D %    -  +  - -         2  2 -       - 2#     ,- 
         +   - 2#   -'      -     =  N J       KD 3 + ,,  # - +        ,-   # -  - 1         -   D % +    
-  - J - K=  ',D %  
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Imprecision and Structure in Modelling Subjective Similarity

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Imprecision and Structure in Modelling Subjective Similarity

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D
D /#1-,D    -, -2 ,    1  1 ,   1 A ,-  --F % ,-   2+  1 ,-  - - -,  D  ' (' "&= @S7S8FRRSBRVQ= KC<@D D D
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216

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Defensive Forecasting

217

GD 5C?JGRW DMJJMUQ RFC QRP?RCEW  " "

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?LB 6ICNRGA DMJJMUQ RFC QRP?RCEW   GD    "  GD    RFCL 6ICNRGA K?ICQ ? NPMR MD ?R JC?QR  ML CTCPW PMSLB   UFCL   ?LB  UFCL    6SPNPGQGLEJW '?UGBQ AMSLRCPCV?KNJC RM RFC NMQQG@GJGRW MD EMMB NPM@? @GJGRW DMPCA?QRGLE GQ LMR ?Q U?RCPRGEFR ?Q GR PQR ?NNC?PQ 7FC UC?I NMGLR MD 6ICNRGAQ ?LB 5C?JGRWQ QRP?RCEW GQ RF?R GR PCOSGPCQ RFCK RM @C ?@JC RM RCJJ UFCRFCP  GQ CV?ARJW COS?J RM  ?LB RM QUGRAF RFCGP KMTCQ UGRF GL LGRC ?@PSNRLCQQ BGQAMLRGLSMSQJW ?Q  QFGDRQ DPMK  RM QMKCRFGLE CTCP QM QJGEFRJW JCQQ %CA?SQC RFC GBC? MD ?L GLLGRCJW ?@PSNR QFGDR JGTCQ GL ?L GBC?JGXCB K?RFCK?RGA?J UMPJB UC QFMSJB @C U?PW ?@MSR BP?UGLE NP?ARGA?J AMLAJSQGMLQ DPMK GR 7M QFMU RF?R '?UGBQ AMSLRCPCV?KNJC BCNCLBQ ML RFC ?PRGAG?JGRW MD RFC K?RFCK?RGA?J GBC?JGX?RGML GR QSACQ RM QFMU RF?R GR BGQ?NNC?PQ UFCL UC FM@@JC 6ICNRGA ?LB 5C?JGRW GL QK?JJ U?WQ RF?R ?PC PC?QML?@JC MP GLAMLQC OSCLRG?J GL NP?ARGAC 7FGQ F?Q @CCL BMLC GL RUM BGCPCLR U?WQ  :C A?L FM@@JC 6ICNRGA ? @GR @W PCOSGPGLE RF?R FC DMJJMU ? QRP?RCEW RF?R K?ICQ FGQ @CR ? AMLRGLSMSQ DSLARGML MD RFC DMPCA?QR   7FGQ GQ LMR ? NP?ARGA?J GKNCBGKCLR @CA?SQC ? AMLRGLSMSQ DSLARGML A?L AF?LEC UGRF ?LW BCEPCC MD ?@PSNRLCQQ QFMPR MD GLLGRC ,R A?L C?QGJW @C QFMULZ, PCNPMBSAC GL  @CJMU RFC NPMMD EGTCL GL 9MTI CR ?J @ZRF?R )MPCA?QRCP A?L @C?R ?LW AMLRGLSMSQ QRP?RCEW $LB ?Q , CVNJ?GL GL  RFCPC ?PC AMLRGLSMSQ QRP?RCEGCQ DMP 6ICNRGA RF?R K?IC ? JMR KMLCW SLJCQQ )MPCA?QRCPQ NPM@?@GJGRGCQ NCPDMPK UCJJ ,L N?PRGASJ?P 6ICNRGA F?Q AMLRGLSMSQ QRP?RCEGCQ RF?R K?IC ? JMR MD KMLCW SLJCQQ )MPCA?QRCP GQ UCJJ A?JG@P?RCB GL RFC QCLQC RF?R ?@MSR  MD RFC CTCLRQ DMP UFGAF  GQ LC?P  UGJJ F?NNCL ?@MSR  MD RFC CTCLRQ DMP UFGAF  GQ LC?P  UGJJ F?NNCL CRA %W NJ?WGLE ?E?GLQR ?LB @C?RGLE ? AMLRGLSMSQ QRP?RCEW DMP 6ICNRGA RF?R K?ICQ KMLCW GD )MPCA?QRCP BMCQ LMR NCPDMPK UCJJ )MPCA?QRCP GQ QSPC RM NCPDMPK UCJJ LM K?RRCP UF?R 5C?JGRW BMCQ 7FGQ K?W QCCK QSPNPGQGLE @SR )MPCA?QRCPQ NJ?W RSPLQ MSR RM @C OSGRC L?RSP?J 7M ES?P?LRCC A?JG@P?RGML DMP CV?KNJC FC ?JU?WQ AFMMQCQ ? T?JSC MD  UFCPC FC F?Q @CCL D?GPJW UCJJ A?JG@P?RCB QM D?P QM RF?R

218

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Defensive Forecasting

219

 ,D RUM MP KMPC @CRRGLE QRP?RCEGCQ PGQI RFC Q?KC ?KMSLR MD KMLCW RFCL RFCGP ?TCP?EC UFGAF ?JQM PGQIQ RF?R Q?KC ?KMSLR MD KMLCW UGJJ KSJRGNJW GR @W ? J?PEC D?ARMP GD ?LW MD RFC QRP?RCEGCQ @CGLE ?TCP?ECB BM QM 6M UC A?L AMK@GLC E?KCRFCMPCRGA RCQRQ @W ?TCP?EGLE! QCC  DMP BCR?GJQ  ,L K?LW A?QCQ UC A?L ?TCP?EC CLMSEF @CRRGLE QRP?RCEGCQ RM M@R?GL MLC RF?R GQ OS?QGSLGTCPQ?J GL RFC QCLQC RF?R GR KSJRGNJGCQ RFC A?NGR?J PGQICB @W ? J?PEC D?ARMP UFCLCTCP MSP NPM@?@GJGQRGA NPCBGARGMLQ D?GJ GL ? U?W RF?R GQ GKNMPR?LR RM SQ! QCC  DMP BCR?GJQ 'CDCLQGTC DMPCA?QRGLE NJ?WQ ?E?GLQR ? OS?QGSLGTCPQ?J QRP?RCEW 

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

220

G. Shafer

NJ?WGLE ?E?GLQR SQ QRP?RCEGA?JJW 6CC 9MTI ?LB 6F?DCP   6F?DCP  :S ?LB 6F?DCP  DMP CV?KNJCQ GL UFGAF ?L GLTCQRMP NJ?WQ RFC PMJC MD 6ICNRGA UFGJC ? L?LAG?J K?PICR UFGAF BCRCPKGLCQ RFC PCRSPLQ ML FGQ GLTCQRKCLRQ NJ?WQ RFC PMJCQ MD )MPCA?QRCP ?LB 5C?JGRW 7FC L?LAG?J K?PICR GQ LMR ? QGLEJC NCPQML @SR RFC NPGACQ GR QCRQ ?PC GLSCLACB @W K?LW NCMNJC UFM ?PC TCPW GLRCLR ML R?IGLE ?BT?LR?EC MD RFC GLTCQRMP



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6GLAC (KGJC %MPCJQ NPMMD MD RFC QRPMLE J?U MD J?PEC LSK@CPQ GL  K?RF CK?RGAG?LQ F?TC SLBCPQRMMB RF?R KC?QSPC RFCMPW A?L NPMTGBC ? PGEMPMSQ DMSLB?RGML DMP RFC AJ?QQGA?J RFCMPCKQ MD NPM@?@GJGRW 'CDCLQGTC DMPCA?QRGLE GQ K?BC NMQQG@JC @W RFC PC?JGX?RGML RF?R E?KC RFCMPW NPMTGBCQ ?L COS?JJW PGEMPMSQ ?LB CTCL KMPC ECLCP?J DMSLB?RGML :GRFGL RFC E?KCRFCMPCRGA DP?KCUMPI ? AJ?QQGA?J NPM@?@GJGRW RFCMPCK @CAMKCQ ? RFCMPCK ?QQCPRGLE RFC CVGQRCLAC MD ? @CRRGLE QRP?RCEW RF?R KSJRGNJGCQ RFC A?NGR?J GR PGQIQ @W ? J?PEC D?ARMP SLJCQQ MSRAMKCQ Q?RGQDW ? EGTCL AMLBGRGML ,D AMKNJCRC NPM@?@GJ GRW BGQRPG@SRGMLQ ?PC EGTCL ?LB UC ?PC ?JJMUCB RM @SW ?LW P?LBMK T?PG?@JC DMP GRQ CVNCARCB T?JSC RFC E?KCRFCMPCRGA RFCMPCKQ A?L @C RFMSEFR MD ?Q ? U?W MD PCQR?RGLE RFC AMPPCQNMLBGLE KC?QSPCRFCMPCRGA RFCMPCKQ %SR RFC E?KCRFCMPCRGA RFCMPCKQ ?JQM ?NNJW UFCL MLJW QMKC T?PG?@JCQ ?PC NPGACB ?LB MCPCB DMP Q?JC 7FC E?KCRFCMPCRGA DP?KCUMPI F?Q GRQ PMMRQ GL UMPI @W -C?L 9GJJC GL RFC Q 9GJJC  9GJJC UMPICB SLBCP RFC ?QQSKNRGML RF?R AMKNJCRC NPM@?@GJGRW BGQRPG@SRGMLQ ?PC EGTCL ?LB DMP RFC KMQR N?PR FC AMLQGBCPCB MLJW RFC A?QC UFCPC UC QCCI @CRRGLE QRP?RCEGCQ RF?R KSJRGNJW RFC A?NGR?J PGQICB @W ?L GLLGRC D?ARMP ?AFGCTGLE RFGQ AMPPCQNMLBQ RM RFC F?NNCLGLE MD ?L CTCLR MD XCPM NPM@?@GJGRW P?RFCP RF?L KCPCJW RFC F?NNCLGLE MD ?L CTCLR MD QK?JJ NPM@?@GJGRW 0MPCMTCP FC QNCJJCB MSR FGQ PC?QMLGLE DSJJW MLJW DMP RFC A?QC UFCPC MSRAMKCQ ?PC @GL?PW %SR CVNJ?GLCB DSJJW RFC GBC? MD NPMTGLE RFCMPCKQ @W AMLQRPSARGLE @CRRGLE QRP?RCEGCQ ,L  , PCTGCU 9GJJCQ RFCMPCK UFGAF CQR?@JGQFCQ RFC COSGT?JCLAC @C RUCCL RFC F?NNCLGLE MD ?L CTCLR MD QK?JJ NPM@?@GJGRW ?LB RFC QSAACQQ MD ? @CRRGLE QRP?RCEW ,L  , PCTGCU FMU 9GJJC RFMSEFR ?@MSR K?PRGLE?JCQ GL RFC @GL?PW NPM@?@GJGRW NPMRMAMJ ,L  , BGQASQQ ? K?PRGLE?JC RF?R 9GJJC SQCB RM NPMTC RFC QRPMLE J?U MD J?PEC LSK@CPQ GL RFC QNCAG?J A?QC MD FGQ @GL?PW NPMRMAMJ GL UFGAF QSAACQQGTC CTCLRQ ?PC GLBCNCLBCLR ?LB ?JJ F?TC RFC Q?KC NPM@?@GJGRW  , RFCL RSPL RM RFC ECLCP?JGX?RGML RM RFC A?QC UFCPC LM HMGLR NPM@?@GJGRW BGQRPG@SRGML DMP UF?R UC K?W M@QCPTC GQ EGTCL ,L  , PCRSPL RM RFC NPM@?@GJGRW DMPCA?QRGLE NPMRMAMJ UC AMLQGBCPCB GL  , BGQASQQ RFC UC?I

Defensive Forecasting

221

?LB QRPMLE J?U MD J?PEC LSK@CPQ DMP RFGQ NPMRMAMJ ?LB ?Q ?L GJJSQRP?RGML , QICRAF ? TCPW QGKNJC NPMMD MD RFC UC?I J?U ,L  , BGQASQQ ? NPMRMAMJ GL UFGAF NPCBGARGMLQ  MD OS?LRGRGCQ " ?PC GLRCPNPCRCB ?Q NPGACQ ?R UFGAF 6ICNRGA A?L @SW MP QCJJ RFC "  ?LB , CVNJ?GL FMU UC A?L M@R?GL J?UQ MD J?PEC LSK@CPQ CTCL GL RFGQ A?QC 

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9GJJC AMLQGBCPCB MLJW RFC GLLGR?PW A?QC UFCPC RFC E?KC AMLRGLSCQ DMP ?L GLLGRC LSK@CP MD PMSLBQ ?LB 6ICNRGA RPGCQ RM @CAMKC GLLGRCJW PGAF @CA?SQC FC AMLACGTCB FGQ UMPI ?Q ? APGRGOSC MD 5GAF?PB TML 0GQCQQ LMRGML  MD ? AMJJCARGTC TML 0GQCQ   ! RFC RGRJC MD FGQ @MMI U?Q 9:)* (7.9.6:* )* 1& 349.43 )* (411*(9.+ )MP TML 0GQCQ ?L CTCLR F?Q ? NPM@?@GJGRW  MLJW GL RFC AMLRCVR MD ?L GLLGRC QCOSCLAC MD CTCLRQ UFMQC JGKGRGLE DPCOSCLAW MD MAASPPCLAC GQ  ?LB UFGAF M@CWQ RFC DSPRFCP AMLBGRGML RF?R ?LW PC?QML?@JC PSJC DMP QCJCARGLE ? QS@QCOSCLAC UGJJ NPMBSAC MLC UGRF RFC Q?KC JGKGRGLE DPCOSCLAW 7FGQ J?RRCP AMLBGRGML TML 0GQCQ RFMSEFR UMSJB ICCN ? E?K@JCP DPMK ECRRGLE PGAF @W @CRRGLE ML QMKC MD RFC CTCLRQ ?LB LMR MRFCPQ 9GJJC QFMUCB @W CV?KNJC RF?R TML 0GQCQQ AMLBGRGML GQ GL?BCOS?RC GL?QKSAF ?Q GR BMCQ LMR PSJC MSR RFC E?K@JCPQ ECRRGLE PGAF @W T?PWGLE FGQ @CR 9GJJC RFCL QFMUCB RF?R PSJGLE MSR ?L ?P@GRP?PW QRP?RCEWQ K?IGLE ? E?K@JCP GLLGRCJW PGAF CTCL GD RF?R QRP?RCEW BMCQ T?PW FMU KSAF ?LB ML UF?R QGBC RM @CR GQ HSQR RFC Q?KC ?Q PSJGLE MSR RFC F?NNCLGLE MD ?L ?P@GRP?PW CTCLR MD NPM@?@GJGRW XCPM 8LDMPRSL?RCJW 9GJJCQ UMPI U?Q J?PECJW MTCPJMMICB GL ECLCP?J @CA?SQC MD RFC CAJGNQC MD RFC )PCLAF QAFMMJ MD NPM@?@GJGRW ?DRCP :MPJB :?P ,, ?LB GL N?PRGASJ?P @CA?SQC MD RFC QSAACQQ MD -MQCNF / 'MM@Q PCDMPKSJ?RGML MD 9GJJCQ LMRGML MD ? K?PRGLE?JC GL KC?QSPCRFCMPCRGA RCPKQ 'MM@  'GQASQQGMLQ MD TML 0GQCQQ DPCOSCLRGQK QRGJJ SQS?JJW MTCPJMMI 9GJJCQ CKCLB?RGML MD GR ?LB 9GJJCQ RFCMPCK GQ QRGJJ LMR UCJJ ILMUL ,L MPBCP RM QR?RC 9GJJCQ RFCMPCK AMLQGBCP ? QCOSCLAC        MD @G L?PW P?LBMK T?PG?@JCQ UGRF ? HMGLR NPM@?@GJGRW BGQRPG@SRGML 3 6SNNMQC DMP QGKNJGAGRW RF?R 3 ?QQGELQ CTCPW LGRC QCOSCLAC "      " MD Q ?LB Q NMQGRGTC NPM@?@GJGRW QM RF?R GRQ AMLBGRGML?J NPM@?@GJGRGCQ DMP  EGTCL T?J SCQ MD RFC NPCACBGLE T?PG?@JCQ ?PC ?JU?WQ SL?K@GESMSQJW BCLCB )MJJMUGLE 9GJJC AMLQGBCP ? E?K@JCP UFM @CEGLQ UGRF  ?LB GQ ?JJMUCB RM @CR ?Q FC NJC?QCQ ML C?AF PMSLB NPMTGBCB RF?R FC LCTCP PGQIQ KMPC RF?L FC F?Q 7FC AMLBGRGML RF?R FC LCTCP PGQIQ KMPC RF?L FC F?Q GQ LCCBCB RM ES?P?LRCC RF?R RFC GLGRG?J A?NGR?J MD  PCNPCQCLRQ RFC RMR?J A?NGR?J FC PGQIQ ,D FC UCPC ?J JMUCB RM @MPPMU GL MPBCP RM K?IC J?PECP @CRQ FC UMSJB @C PGQIGLE KMPC :C A?L DMPK?JGXC RFGQ UGRF RFC DMJJMUGLE NPMRMAMJ UFCPC @CRRGLE ML  GQ

222

G. Shafer

PCNPCQCLRCB ?Q @SWGLE QMKC LSK@CP  NMQQG@JW XCPM MP LCE?RGTC MD RGAICRQ RF?R AMQR 3 "  " "       " "  ?LB N?W   '%)%%"   ""
(  $'- &'% " )- &'%)%%"  "  )25  "      6ICNRGA ?LLMSLACQ    5C?JGRW ?LLMSLACQ "     "    " 3 "  " "       " "  !-.,%.%*) *) &!+.% 6ICNRGA KSQR AFMMQC RFC  QM RF?R FGQ A?NGR?J GQ ?JU?WQ LMLLCE?RGTC     DMP ?JJ  LM K?RRCP FMU 5C?JGRW KMTCQ 7FC QCOSCLAC        GQ 6ICNRGAQ A?NGR?J NPMACQQ $JMLE UGRF ?JJ RFC MRFCP NPMRMAMJQ AMLQGBCPCB GL RFGQ ?PRGAJC RFGQ GQ ? NCPDCARGLDMPK?RGML QC OSCLRG?J NPMRMAMJ! KMTCQ ?PC K?BC GL RFC MPBCP JGQRCB ?LB C?AF NJ?WCP QCCQ RFC MRFCP NJ?WCPQ KMTCQ ?Q RFCW ?PC K?BC 1MRGAC RFC JGKGRCB PMJC NJ?WCB @W RFC NPM@?@GJGRW BGQRPG@SRGML 3 ,RQ MLJW PMJC GQ RM EGTCQ NPGACQ ?R UFGAF 6ICNRGA A?L @SW RGAICRQ :C A?L RFGLI MD GR ?Q ? QRP?RCEW DMP )MPCA?QRCP UFM BMCQ LMR ?NNC?P GL RFC E?KC @CA?SQC FGQ QRP?RCEW GQ VCB 5C?JGRW GQ LMR PCOSGPCB RM QCJCAR  Q T?JSC " P?LBMKJW ?AAMPBGLE RM RFC AMLBGRGML?J NPM@?@GJGRW 3 "  " "       " "  ))%.! $*,%4*) 4!,* +,*%'%.3 9CPQGMLQ MD 9GJJCQ RFCMPCK FMJB UFCLCTCP ? AMKNJCRC NPM@?@GJGRW BGQRPG@SRGML 3 GQ EGTCL DMP MSRAMKCQ "  "      @SR DMP QGKNJGAGRW JCR SQ AMLQGBCP MLJW 3PMRMAMJ  UFCPC RFC MSRAMKCQ ?PC @GL?PW 7FC RFCMPCK Q?WQ RF?R 6ICNRGAQ ECRRGLE GLLGRCJW PGAF GL RFGQ NPMRMAMJ GQ COSGT?JCLR RM ?L CTCLR MD XCPM NPM@?@GJGRW F?NNCLGLE GL RFC DMJJMUGLE QCLQC  :FCL 6ICNRGA DMJJMUQ ? QRP?RCEW RF?R EGTCQ  ?Q ? DSLARGML MD "      "  3         GQ SL@MSLBCB " 



 ,D GQ ? KC?QSP?@JC QS@QCR MD   UGRF 3  "  RFCL 6ICNRGA F?Q ? QRP?RCEW RF?R ES?P?LRCCQ JGK  "



UFCLCTCP "  "       

Defensive Forecasting

223

:C A?L QSKK?PGXC RFCQC RUM QR?RCKCLRQ @W Q?WGLE RF?R 6ICNRGAQ @CGLE ?@JC RM KSJRGNJW FGQ A?NGR?J @W ?L GLLGRC D?ARMP GQ COSGT?JCLR RM RFC F?NNCLGLE MD ?L CTCLR UGRF NPM@?@GJGRW XCPM ))%.! $*,%4*) -('' +,*%'%.3 3P?ARGA?J ?NNJGA?RGMLQ PCOSGPC RF?R UC AMLQGBCP QK?JJ NPM@?@GJGRGCQ P?RFCP RF?L XCPM NPM@?@GJGRGCQ ,L RFGQ A?QC UC A?L SQC RFC DMJJMUGLE LGR?PW TCPQGML MD 9GJJCQ RFCMPCK UFGAF GQ NPMTCL GL 6F?DCP ?LB 9MTI  AF?NRCP   :FCL 6ICNRGA DMJJMUQ ? QRP?RCEW RF?R EGTCQ  ?Q ? DSLARGML MD "      "  

 3 QSN    

 



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

224 

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Defensive Forecasting

225

PGEFR ,R GQ C?QW RM EGTC CV?KNJCQ ,L  UC UGJJ JMMI ?R ? QAMPGLE K?PRGLE?JC UFMQC @CGLE @MSLBCB ?Q NPCBGARCB UGRF NPM@?@GJGRW MLC @W  GKNJGCQ RFC QRPMLE J?U MD J?PEC LSK@CPQ ,L  UC UGJJ JMMI ?R ? QAMPGLE K?PRGLE?JC UFMQC @CGLE @MSLBCB @W   ?Q NPCBGARCB UGRF NPM@?@GJGRW  @W  GKNJGCQ RFC UC?I J?U MD J?PEC LSK@CPQ !-.%)# 7FC ?ARS?J T?JSCQ MD ? QAMPGLE K?PRGLE?JC RCQR RFC T?JGBGRW MD RFC NPM@?@GJGRW BGQRPG@SRGML 3 7FC J?PECP RFCQC =QAMPCQ ?PC RFC KMPC QRPMLEJW 3 GQ PCDSRCB 9GJJC SQCB QAMPGLE K?PRGLE?JCQ GL @MRF U?WQ (OS?RGML  RFC PSJC DMP SNB?RGLE RFC A?NGR?J GL 3PMRMAMJ  GKNJGCQ RF?R  GQ RFC CVNCARCB T?JSC ?R RGKC   MD RFC DSRSPC T?JSC MD  (  "      "  "  "      " 



%CA?SQC GRQ JCDRF?LB QGBC GQ COS?J RM  "      "  

3"      "   3"      "     "      "    3"      "  3"      " 

(OS?RGML  GQ COSGT?JCLR RM 4"      "    4"      "   " 4"      " 



UFCPC 4 GQ RFC DSLARGML ML LGRC QRPGLEQ MD Q ?LB Q BCLCB @W 4"      "  "  "      " 3"      "  (OS?RGML  GQ LCACQQ?PW ?LB QSAGCLR DMP ? LMLLCE?RGTC DSLARGML 4 ML LGRC QRPGLEQ MD Q ?LB Q RM BCLC ? NPM@?@GJGRW KC?QSPC ML    6M ? LMLLCE?RGTC K?PRGLE?JC UGRF PCQNCAR RM 3 GQ RFC Q?KC RFGLE ?Q RFC P?RGM MD ? NPM@?@GJGRW KC?QSPC 4 RM 3ZGC ? JGICJGFMMB P?RGM UGRF 3 ?Q BCLMKGL?RMP 'MM@ GKNMPRCB 9GJJCQ LMRGML MD ? K?PRGLE?JC GLRM KC?QSPCRFCMPCRGA NPM@?@GJGRW UFCPC RFC E?K@JGLE NGARSPC GQ LMR K?BC CVNJGAGR @W R?IGLE (OS?RGML  ?Q RFC BCLGRGML MD ? K?PRGLE?JC *?KCRFCMPCRGA NPM@?@GJGRW EMCQ GL ? BGCPCLR BGPCARGML ,R SQCQ 9GJJCQ BCLGRGML MD ? K?PRGLE?JC GL KMPC ECLCP?J DMPCA?QRGLE E?KCQ UFCPC RFC DMPCA?QRQ ?PC EGTCL @W ? NJ?WCP GL RFC E?KC LMR @W ? NPM@?@GJGRW BGQRPG@SRGML 3 DMP 5C?JGRWQ KMTCQ ,L RFCQC E?KCQ K?PRGLE?JCQ ?PC LMR JGICJGFMMB P?RGMQ 7FCW ?PC QGKNJW A?NGR?J NPMACQQCQ DMP 6ICNRGA 2LC TCPW GKNMPR?LR NPMNCPRW MD K?PRGLE?JCQ RF?R 9GJJC SQCB UFGAF BMCQ A?PPW MTCP RM RFC KMPC ECLCP?J E?KCQ QRSBGCB GL E?KCRFCMPCRGA NPM@?@GJGRW GQ RFGQ ?L ?TCP?EC MD QAMPGLE QRP?RCEGCQ NPMBSACQ RFC Q?KC ?TCP?EC MD RFC

226

G. Shafer

QAMPGLE K?PRGLE?JCQ ,L MPBCP RM ?TMGB ?LW AMLDSQGML JCR KC QNCJJ MSR UF?R RFGQ KC?LQ GL RFC A?QC UFCPC UC ?PC R?IGLE ? QGKNJC ?TCP?EC MD RUM QRP?RCEGCQ ?LB ?JQM GL RFC ECLCP?J A?QC UFCPC UC ?PC ?TCP?EGLE ? AJ?QQ MD QRP?RCEGCQ %(+'!-. -! 6SNNMQC RFC QAMPGLE QRP?RCEGCQ  ?LB  PCAMKKCLB KMTCQ  "      "  ?LB  "      "  PCQNCARGTCJW ML RFC RF PMSLB MD RFC E?KC 7FC ?TCP?EC MD  ?LB  ZA?JJ GR ZPCAMKKCLBQ RFC KMTC  "      "  "


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ML RFC RF PMSLB MD RFC E?KC ,D UC UPGRC     ?LB  DMP RFC AMPPCQNMLBGLE QAMPGLE K?PRGLE?JCQ RFCL  "      "  "


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DMP ?JJ  ?LB ?JJ "      "  !)!,' -! ,L ECLCP?J AMLQGBCP QRP?RCEGCQ
 UFCPC RFC GLBCV  P?LECQ MTCP ? QCR  :C K?W ?TCP?EC RFCQC QRP?RCEGCQ UGRF PCQNCAR RM ? NPM@ ?@GJGRW BGQRPG@SRGML  ML  M@R?GLGLE ? QRP?RCEW UGRF KMTCQ   "      "  " 
 "      " 

NPMTGBCB MLJW RF?R RFGQ GLRCEP?J ?JU?WQ AMLTCPECQ 7FC AMPPCQNMLBGLE QAMPGLE K?PRGLE?JCQ RFCL ?TCP?EC GL RFC Q?KC U?W   "      "  " 
"      " 

7FCQC ?QQCPRGMLQ ?PC RPSC @CA?SQC RFC GLAPCKCLR   GQ ?JU?WQ JGLC?P GL 6ICNRGAQ KMTC   7FGQ JGLC?PGRW GQ ? DC?RSPC MD ?JJ RFC NPMRMAMJQ AMLQGBCPCB GL 6F?DCP ?LB 9MTI  

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9GJJC QRSBGCB QMKC QRP?RCEGCQ DMP 6ICNRGA GL RFC QNCAG?J A?QC MD RFC NPCACB GLE NPMRMAMJ GL UFGAF 3 "  " "       " "  GQ ?JU?WQ COS?J RM  ,L MPBCP RM EGTC RFC PC?BCP ? EJGKNQC MD 9GJJCQ KCRFMBQ , UGJJ LMU PCTGCU RFC QAMPGLE K?PRGLE?JC FC SQCB RM NPMTC RFC QRPMLE J?U MD J?PEC LSK @CPQ DMP RFGQ QNCAG?J A?QC 6ICNRGA A?L @CAMKC GLLGRCJW PGAF SLJCQQ 5C?JGRW K?ICQ RFC PCJ?RGTC DPCOSCLAW MD Q AMLTCPEC RM 

Defensive Forecasting

227

8LBCP RFC ?QQSKNRGML RF?R 3 "  " "       " "  GQ ?JU?WQ COS?J RM  MSP NPMRMAMJ JMMIQ JGIC RFGQ '%)%%"  %'() $ + )  %$()$) &'% " )-  ,(!.!, PC?J LSK@CP  Q?RGQDWGLE     "  )25  "      6ICNRGA ?LLMSLACQ    5C?JGRW ?LLMSLACQ "     "    "  7FGQ NPMRMAMJ PCNPCQCLRQ UF?R AJ?QQGA?J NPM@?@GJGRW A?JJQ GLBCNCLBCLR %CPLMSJJG RPG?JQ QSAACQQGTC CTCLRQ RF?R ?PC GLBCNCLBCLR ?LB ?JJ F?TC RFC Q?KC NPM@?@GJGRW 7FC QAMPGLE K?PRGLE?JC 9GJJC SQCB RM NPMTC RFC QRPMLE J?U MD J?PEC LSK @CPQ DMP RFGQ NPMRMAMJ GQ  "      "  "

   
       



UFCPC  GQ COS?J RM   ?LB  GQ RFC LSK@CP MD Q ?KMLE "      "  "



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QCC 9GJJC  N ! 9GJJC  ?LB 5M@@GLQ  7M AMLPK RF?R  GQ ? QAMPGLE K?PRGLE?JC GR QSACQ RM LMRGAC RF?R GR GQ ? JGICJGFMMB P?RGM UGRF 3 ?Q RFC BCLMKGL?RMP ,L D?AR    " 3"      "  ?LB

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UFCPC 4 GQ RFC NPM@?@GJGRW BGQRPG@SRGML M@R?GLCB @W ?TCP?EGLE NPM@?@GJGRW BGQRPG@SRGMLQ SLBCP UFGAF RFC CTCLRQ " "  ?PC GLBCNCLBCLR ?LB ?JJ F?TC ? NPM@?@GJGRW  NMQQG@JW BGCPCLR DPMK  ,*+*-%.%*)          
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228

G. Shafer

&!.$ *" ,**" %CA?SQC  GQ ? QAMPGLE K?PRGLE?JC 9GJJCQ RFCMPCK (OS?RGML  Q?WQ RF?R UGRF NPM@?@GJGRW MLC RFCPC GQ ? AMLQR?LR QSAF RF?R    
        DMP  "       2LC A?L BCBSAC  DPMK  SQGLE 6RGPJGLEQ DMPKSJ? 7FC PC?BCP K?W TCPGDW RF?R  GQ 6ICNRGAQ A?NGR?J NPMACQQ UFCL FC DMJ JMUQ RFC QRP?RCEW RF?R NPCQAPG@CQ RFC KMTCQ        "      "  "   "      "     7FC P?RGM       GQ AJMQC RM RFC PCJ?RGTC DPCOSCLAW MD Q QM D?P    5MSEFJW QNC?IGLE RFC QRP?RCEW  @CRQ RF?R " UGJJ @C  UFCL RFC PCJ?RGTC DPCOSCLAW GQ KMPC RF?L   UFCL RFC PCJ?RGTC DPCOSCLAW GQ JCQQ RF?L  6M UFCLCTCP RFC PCJ?RGTC DPCOSCLAW BGTCPECQ DPMK  5C?JGRW KSQR KMTC GR @?AI RMU?PBQ  RM ICCN 6ICNRGA DPMK GLAPC?QGLE FGQ A?NGR?J 7FC P?RC MD AMLTCPECLAC MD   RM XCPM GKNJGCB @W  GQ JME   9GJJC  8QGLE MRFCP QAMPGLE K?PRGLE?JCQ M@R?GLCB @W ?TCP?EGLE    ?Q GL  @SR UGRF PCQNCAR RM BGQRPG@SRGMLQ AMLACLRP?RCB ?PMSLB  P?RFCP RF?L RFC SLGDMPK BGQRPG@SRGML 9GJJC ?JQM BCPGTCB RFC D?QRCP AMLTCPECLAC ?QQCPRCB @W RFC J?U MD RFC GRCP?RCB JME?PGRFK 6F?DCP ?LB 9MTI  AF?NRCP  :C A?L ?JQM SQC K?PRGLE?JC KCRFMBQ RM NPMTC MRFCP AJ?QQGA?J RFCMPCKQ GLAJSBGLE RFC UC?I J?U MD J?PEC LSK@CPQ 6F?DCP ?LB 9MTI  AF?NRCP  ?LB RFC ACLRP?J JGKGR RFCMPCK 6F?DCP ?LB 9MTI  AF?NRCP  5?RFCP RF?L NSPQSC RFGQ N?RF FCPC , UGJJ LMU QRCN MSRQGBC AJ?QQGA?J NPM@?@GJGRW RFCMPW GLRM KMPC ECLCP?J NPMRMAMJQ JGIC RFC NPM@?@GJGRW DMPCA?QRGLE NPMRMAMJ MD  ,L RFCQC NPMRMAMJQ K?PRGLE?JC KCRFMBQ ?JJMU SQ RM NPMTC RFCMPCKQ ?L?JMEMSQ RM RFC AJ?QQGA?J RFCMPCKQ @SR UC KSQR CVNPCQQ RFCK BGPCARJW GL E?KCRFCMPCRGA RCPKQ ,LQRC?B MD Q?WGLE RF?R QMKCRFGLE F?NNCLQ UGRF NPM@?@GJGRW MLC UC Q?W RF?R GD GR BMCQ LMR F?NNCL 6ICNRGA UGJJ KSJRGNJW RFC A?NGR?J FC PGQIQ @W ?L GLLGRC D?ARMP ,LQRC?B MD Q?WGLE RF?R QMKCRFGLE F?NNCLQ UGRF RFC FGEF NPM@?@GJGRW   UC Q?W RF?R GD GR BMCQ LMR F?NNCL 6ICNRGA UGJJ KSJRGNJW RFC A?NGR?J FC PGQIQ @W RFC J?PEC D?ARMP   

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Defensive Forecasting

229

T?PGCRW JCR SQ AMLQGBCP RFC UC?I J?U MD J?PEC LSK@CPQ P?RFCP RF?L RFC QRPMLE J?U $AAMPBGLEJW , LMU ?QQSKC RF?R RFC E?KC F?Q ? VCB LGRC FMPGXML GR CLBQ ?DRCP ? J?PEC @SR LGRC LSK@CP MD PMSLBQ   '%)%%"   $ )%' .%$ &'% " )- %'() $ ,(!.!, L?RSP?J LSK@CP   "  )25  "       )MPCA?QRCP ?LLMSLACQ   < > 6ICNRGA ?LLMSLACQ    5C?JGRW ?LLMSLACQ "     "    "   2SP QRSBW MD RFGQ NPMRMAMJ UGJJ DMASQ ML  "



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7FC UC?I J?U MD J?PEC LSK@CPQ Q?WQ RF?R GD  GQ QSAGCLRJW J?PEC RFC PCJ?RGTC  DPCOSCLAW MD Q     "  QFMSJB @C AJMQC RM RFC ?TCP?EC NPM@?@GJGRW  DMPCA?QR       7FGQ KC?LQ   QFMSJB @C AJMQC RM XCPM 7FC UC?I J?U A?L @C DMPKSJ?RCB ?LB NPMTCL DMP 3PMRMAMJ  ?Q DMJJMUQ ,*+*-%.%*)          
"-*7* *=.898 & 8(47.3, 2&79.3,&1* 9-&9 <.11 *=(**)   :31*88                 744+ &MLQGBCP RFC QRP?RCEW DMP 6ICNRGA RF?R NPCQAPG@CQ RFC KMTC  "

  

7FGQ NPMBSACQ RFC A?NGR?J NPMACQQ   

    "     "     UFGAF GQ ? QAMPGLE K?PRGLE?JC    DMP  "       @CA?SQC "     DMP C?AF  )MP RFC Q?KC PC?QML RFC FWNMRFCQGQ     GKNJGCQ 

230

G. Shafer

:FCL  ?LB  ?PC QK?JJ NMQGRGTC LSK@CPQ ?LB  GQ CVRPCKCJW J?PEC QM RF?R       3PMNMQGRGML  GKNJGCQ RF?R UC A?L @C ?Q AMLBCLR RF?R      ?Q UC ?PC RF?R 6ICNRGA UGJJ LMR KSJRGNJW FGQ A?NGR?J @W RFC J?PEC D?ARMP   7FC K?PRGLE?JC GL RFC NPMMD ?NNC?PQ GL KC?QSPCRFCMPCRGA DMPK GL .MJKMEMPMTQ  NPMMD MD RFC UC?I J?U .MJKMEMPMT  $ NMQGRGTC NPM@?@GJGRW BGQRPG@SRGML 3 ML   A?L QCPTC ?Q ? QRP?RCEW DMP )MPCA?QRCP GL 3PMRMAMJ  NPCQAPG@GLE  " 3 "  " "       " "  :FCL QSAF ? QRP?RCEW GQ GKNMQCB 3PMRMAMJ  PCBSACQ RM ? LGRCFMPGXML TCPQGML MD 9GJJCQ @GL?PW NPMRMAMJ 3PMNMQGRGML  Q?WQ RF?R RFCPC GQ ? QAMPGLE K?PRGLE?JC GL RFGQ LGRCFMPGXML NPMRMAMJ RF?R UGJJ CVACCB   SLJCQQ    

    " 3 "  " "       " "           ?LB @W 9GJJCQ RFCMPCK RFGQ GKNJGCQ RF?R      

    " 3 "  " "       " "      3       :FCL 3 "  " "       " "  GQ ?JU?WQ COS?J RM  UC ?PC GL ? LGRCFMPGXML TCPQGML MD 3PMRMAMJ  ?LB UC ECR      

    "      3       RFC CJCKCLR?PW TCPQGML MD RFC AJ?QQGA?J UC?I J?U SQS?JJW BCPGTCB SQGLE &FC@W QFCTQ GLCOS?JGRW 3PMRMAMJ  BMCQ LMR PCOSGPC )MPCA?QRCP RM DMJJMU ? QRP?RCEW BCLCB @W ? NPM@?@GJGRW BGQRPG@SRGML DMP "      "  +C A?L SQC GLDMPK?RGML DPMK MSR QGBC RFC E?KC RM BCAGBC FMU RM KMTC MP FC A?L QGKNJW DMJJMU FGQ UFGKQ 6M GLQRC?B MD F?TGLE ? AJ?QQGA?J NPM@?@GJGRW BGQRPG@SRGML UC ?PC GL UF?R $ 3 '?UGB A?JJCB RFC 57*6:*39.&1 +7&2*<470 '?UGB  ,L RFGQ DP?KC UMPI UC A?LLMR Q?W RF?R  F?Q NPM@?@GJGRW ?R JC?QR    @CA?SQC UC F?TC LM NPM@?@GJGRW BGQRPG@SRGML @SR 3PMNMQGRGML  F?Q RFC Q?KC GLRSGRGTC KC?LGLE ?Q QSAF ? NPM@?@GJGRW QR?RCKCLR :C AMLQGBCP 6ICNRGAQ KSJRGNJWGLE

Defensive Forecasting

231

FGQ A?NGR?J @W   MP KMPC ?E?GLQR ? EMMB )MPCA?QRCP SLJGICJW GL RFC Q?KC U?W ?Q UC AMLQGBCP RFC F?NNCLGLE MD ?L CTCLR SLJGICJW UFCL GR GQ EGTCL NPM@?@GJGRW  MP JCQQ @W ? T?JGB NPM@?@GJGRW BGQRPG@SRGML $JRFMSEF , F?TC DMASQCB ML RFC UC?I J?U FCPC GR GQ ?JQM QRP?GEFRDMPU?PB RM NPMTC RFC QRPMLE J?U DMP RFC NPM@?@GJGRW DMPCA?QRGLE NPMRMAMJ 3PMRMAMJ  @SR UGRF RFC E?KC NJ?WCB DMP ?L GLLGRC LSK@CP MD PMSLBQ GLQRC?B MD MLJW  PMSLBQ .SKML ?LB 7?ICKSP? F?TC EGTCL ? NPMMD SQGLE ? KMBGCB DMPK MD 9GJJCQ K?PRGLE?JC  .SKML ?LB 7?ICKSP?  ,L 6F?DCP ?LB 9MTI  AF?NRCP  9MTI ?LB , E?TC ?L ?JRCPL?RGTC NPMMD SQGLE ? K?PRGLE?JC CVRP?ARCB DPMK .MJKMEMPMTQ  NPMMD MD RFC QRPMLE J?U .MJKMEMPMT @ 

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,L MPBCP RM GJJSQRP?RC RFC NMGLR RF?R E?KCRFCMPCRGA NPM@?@GJGRW ECLCP?J GXCQ AJ?QQGA?J NPM@?@GJGRW , LMU ?QI RFC PC?BCP RM AMLQGBCP ? NPMRMAMJ UFCPC )MPCA?QRCPQ R?QI ML C?AF PMSLB GQ RM EGTC LMR ? NPM@?@GJGRW DMP ?L CTCLR @SR ? NPCBGARGML MD ? @MSLBCB OS?LRGRW 7M V GBC?Q , ?QQSKC RF?R RFC OS?LRGRW ?LB RFC NPCBGARGML ?PC ?JU?WQ GL RFC GLRCPT?J DPMK  RM  '%)%%"   $ )%' .%$ %*$ %'() $ ,(!.!, L?RSP?J LSK@CP   "  )25  "       )MPCA?QRCP ?LLMSLACQ   < > 6ICNRGA ?LLMSLACQ    5C?JGRW ?LLMSLACQ "  < >  "    "   $Q GR RSPLQ MSR RFC E?KCRFCMPCRGA RPC?RKCLR MD 3PMRMAMJ  GQ QA?PACJW BGCPCLR DPMK RF?R MD 3PMRMAMJ  6CRRGLE  "



"  



UC PC?QML ?JKMQR ?Q @CDMPC ,*+*-%.%*)       
"-*7* *=.898 & 8(47.3, 2&79.3,&1* 9-&9 <.11 *=(**)   :31*88                

232

G. Shafer

744+ &MLQGBCP RFC QRP?RCEW DMP 6ICNRGA RF?R NPCQAPG@CQ RFC KMTC  "

  

7FGQ NPMBSACQ    "   







 

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UFGAF GQ LMLLCE?RGTC @CA?SQC "     DMP C?AF  )MP RFC Q?KC PC?QML RFC FWNMRFCQGQ     GKNJGCQ  7FC NPMMDQ MD RFC QRPMLE J?U DMP RFC GLLGRCFMPGXML TCPQGML MD 3PMRMAMJ  RF?R UC HSQR AGRCB GL .SKML ?LB 7?ICKSP?  ?LB 6F?DCP ?LB 9MTI  AF?NRCP  ?JQM ?NNJW RM RFC GLLGRCFMPGXML TCPQGML MD 3PMRMAMJ 



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, LMU RSPL DPMK QRP?RCEGCQ DMP 6ICNRGA UFGAF RCQR )MPCA?QRCP RM QRP?RCEGCQ DMP )MPCA?QRCP UFGAF QCCI RM UGRFQR?LB QSAF RCQRQ )MPCA?QRCP CLE?ECQ GL )*+*38.;* +47*(&89.3, UFCL FC NJ?WQ GL MPBCP RM BCDC?R ? N?PRGASJ?P QRP?RCEW DMP 6ICNRGA $Q GR RSPLQ MSR )MPCA?QRCP A?L BCDC?R ?LW N?PRGASJ?P QRP?RCEW DMP 6ICNRGA NPMTGBCB MLJW RF?R C?AF KMTC NPCQAPG@CB @W RFC QRP?RCEW T?PGCQ AMLRGLSMSQJW UGRF PCQNCAR RM )MPCA?QRCPQ NPCTGMSQ KMTC )MPCA?QRCP U?LRQ RM BCDC?R KMPC RF?L ? QGLEJC QRP?RCEW DMP 6ICNRGA +C U?LRQ RM BCDC?R QGKSJR?LCMSQJW ?JJ RFC QAMPGLE QRP?RCEGCQ 6ICNRGA KGEFR SQC %SR ?Q UC UGJJ QCC )MPCA?QRCP A?L MDRCL ?K?JE?K?RC RFC QAMPGLE QRP?RCEGCQ FC LCCBQ RM BCDC?R @W ?TCP?EGLE RFCK ?LB RFCL FC A?L NJ?W ?E?GLQR RFC ?TCP?EC 'CDC?RGLE RFC ?TCP?EC K?W @C EMMB CLMSEF @CA?SQC UFCL ?LW MLC MD RFC QAMPGLE QRP?RCEGCQ PCHCARQ )MPCA?QRCPQ T?JGBGRW RFC ?TCP?EC UGJJ ECLCP?JJW PCHCAR ?Q UCJJ ?J@CGR JCQQ QRPMLEJW , @CEGL RFGQ QCARGML GL  UGRF ?L MTCPTGCU MD PCJ?RCB UMPI ML NPCBGA RGML 7FCL , CVNJ?GL BCDCLQGTC DMPCA?QRGLE GL RFC A?QC MD @GL?PW DMPCA?QRGLE JC?TGLE RFC PC?BCP RM AMLQSJR MRFCP CVNMQGRGMLQ DMP MRFCP NPMRMAMJQ QCC 9MTI CR ?J ?! 9MTI @ ,L  , PCTGCU RFC QGKNJC NPMMD RF?R )MPC A?QRCP A?L BCDC?R ?LW AMLRGLSMSQ QRP?RCEW DMP 6ICNRGA GL RFC @GL?PW A?QC ,L  , CVNJ?GL FMU RM ?TCP?EC QRP?RCEGCQ DMP 6ICNRGA RM RCQR DMP RFC A?JG @P?RGML MD @GL?PW NPM@?@GJGRW DMPCA?QRQ ?LB GL  , CVNJ?GL FMU PCQMJSRGML A?L ?JQM @C ?AFGCTCB 7FCPC GQ KSAF KMPC RM Q?W ?@MSR BCDCLQGTC DMPCA?QRGLE KSAF MD GR GL PCACLR N?NCPQ @W 9MTI  ?@AB ?@A , A?LLMR @CEGL RM

Defensive Forecasting

233

QSKK?PGXC RFGQ UMPI FCPC @SR , QFMSJB KCLRGML RF?R NPM@?@GJGRGCQ NPMBSACB @W BCDCLQGTC DMPCA?QRGLE A?L MDRCL @C SQCB GL BCAGQGML NPM@JCKQ UFCPC A?JG @P?RGML ?LB PCQMJSRGML MDRCL QSAC RM ES?P?LRCC RF?R BCAGQGMLQ BCPGTCB DPMK RFCK ?PC MNRGK?J GL RFC Q?KC QCLQC ?Q BCAGQGMLQ BCPGTCB DPMK AJ?QQGA?J NPM@ ?@GJGRGCQ 

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

234

G. Shafer ?LB 6KGRF  ?LB NPCBGARGML UGRF CVNCPR ?BTGAC &CQ?%G?LAFG ?LB /SEMQG ! 9MTI  $JJ RFGQ UMPI QCCIQ RM GKNPMTC QRP?RCEGCQ DMP )MPCA?QRCP NPM@?@GJGRW BGQRPG@SRGMLQ DMP 5C?JGRWQ KMTCQ "  "      @W ?TCP?EGLE MP MRFCPUGQC ?K?JE?K?RGLE RFCK $TCP?EGLE QRP?RCEGCQ DMP )MPCA?QRCP A?L @C CCARGTC ?QWKNRMRGA?JJW @CA?SQC RFC ?TCP?EC UGJJ CTCLRS?JJW @CF?TC JGIC RFC QRP?RCEW GL RFC ?TCP?EC RF?R @CQR K?RAFCQ RFC @CF?TGMP MD 5C?JGRW :FCRFCP RFGQ F?NNCLQ GL NP?ARGAC BCNCLBQ ML FMU K?LW PMSLBQ RFC NPCBGARGML E?KC PSLQ ?LB FMU AJMQC 5C?JGRWQ @CF?TGMP GQ RM ?LW MD RFC QRP?RCEGCQ @CGLE ?TCP?ECB

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

!"!.%)#  *).%)/*/- .,.!#3 "*, &!+.%

,L RFGQ QCARGML , PCNC?R ? QGKNJC NPMMD PQR EGTCL GL 9MTI CR ?J @ QFMUGLE RF?R )MPCA?QRCP A?L BCDC?R ?LW N?PRGASJ?P VCB QRP?RCEW DMP 6ICNRGA

Defensive Forecasting

235

NPMTGBCB RF?R C?AF KMTC NPCQAPG@CB @W RFC QRP?RCEW T?PGCQ AMLRGLSMSQJW UGRF PCQNCAR RM )MPCA?QRCPQ NPCTGMSQ KMTC &MLQGBCP ? QRP?RCEW DMP 6ICNRGA GL 3PMRMAMJ  RFC @GL?PW DMPCA?QRGLE NPMRMAMJ UC QRSBGCB GL  :PGRC

  "        "   



DMP RFC KMTC RF?R NPCQAPG@CQ ML RFC RF PMSLB MD RFC E?KC $R RFC @CEGLLGLE MD RFC RF PMSLB HSQR @CDMPC )MPCA?QRCP K?ICQ FGQ KMTC   RFC C?PJGCP KMTCQ   "          " ?PC ILMUL ?LB RFCPCDMPC VCB ?LB QM  @CAMKCQ ? DSLARGML MD  MLJW Q?W

   7FGQ BCLCQ ? DSLARGML  ML RFC GLRCPT?J < > ,D  GQ AMLRGLSMSQ DMP ?JJ  ?LB ?JJ   "        "  JCR SQ Q?W RF?R GQ +47*(&89(439.3:4:8 :FCL UC V RFC QRP?RCEW  6ICNRGA LM JMLECP F?Q ? PMJC RM NJ?W GL RFC E?KC ?LB UC A?L MKGR FGK DPMK RFC NPMRMAMJ '%)%%"   $'- &'% " )- %'() $  $()   , )() ,(!.!, 6RP?RCEW DMP 6ICNRGA  "  )25  "      )MPCA?QRCP ?LLMSLACQ   < > 5C?JGRW ?LLMSLACQ "     "     "  

,*+*-%.%*)  + 9-* 897&9*,> .8 +47*(&89(439.3:4:8 47*(&89*7 -&8 & 897&9*,> 9-&9 *38:7*8      


 744+ %W RFC GLRCPKCBG?RCT?JSC RFCMPCK RFC AMLRGLSMSQ DSLARGML  GQ ?JU?WQ NMQGRGTC ?JU?WQ LCE?RGTC MP CJQC Q?RGQCQ   "  DMP QMKC   < > 6M )MPCA?QRCP A?L SQC RFC DMJJMUGLE QRP?RCEW  GD  GQ ?JU?WQ NMQGRGTC R?IC  " !  GD  GQ ?JU?WQ LCE?RGTC R?IC  " !  MRFCPUGQC AFMMQC  QM RF?R    "  7FGQ ES?P?LRCCQ RF?R   "     QM RF?R    

236

G. Shafer

,D RFC PC?BCP LBQ GR AMLDSQGLE RF?R RFC LMR?RGML    JC?TCQ RFC BCNCL BCLAC ML RFC C?PJGCP KMTCQ   "          " GKNJGAGR FC MP QFC K?W UGQF RM RFGLI ?@MSR RFC DMJJMUGLE ?JRCPL?RGTC NPMRMAMJ UFGAF JC?TCQ 6ICNRGA GL RFC E?KC ?LB F?Q FGK ?LLMSLAC RFC DSLARGML  HSQR @CDMPC )MPCA?QRCP K?ICQ FGQ KMTC  '%)%%"  !&)  %%((  ()')- %$  '%*$  "  )25  "      6ICNRGA ?LLMSLACQ AMLRGLSMSQ  < >  )MPCA?QRCP ?LLMSLACQ   < > 5C?JGRW ?LLMSLACQ "     "     "   3PMRMAMJ  EGTCQ 6ICNRGA ? JGRRJC KMPC CVG@GJGRW RF?L 3PMRMAMJ  BMCQ! GR ?JJMUQ 6ICNRGA RM R?IC GLRM ?AAMSLR GLDMPK?RGML AMKGLE DPMK MSRQGBC RFC E?KC ?Q UCJJ ?Q RFC NPCTGMSQ KMTCQ   "        " UFCL FC BCAGBCQ ML   %SR GR QRGJJ PCOSGPCQ RF?R    BCNCLB ML  AMLRGLSMSQJW ?LB GR QRGJJ K?ICQ QSPC RF?R )MPCA?QRCP ILMUQ  @CDMPC FC K?ICQ FGQ KMTC   /GIC ?JJ NPMRMAMJQ GL RFGQ ?PRGAJC 3PMRMAMJ  GQ ? NCPDCARGLDMPK?RGML NPMRMAMJ! RFC NJ?WCPQ KMTC GL QCOSCLAC ?LB QCC C?AF MRFCPQ KMTCQ ?Q RFCW ?PC K?BC 6M RFC NPMMD ?LB AMLAJSQGML MD 3PMNMQGRGML  QRGJJ FMJB )MPCA?QRCP F?Q ? QRP?RCEW RF?R CLQSPCQ      


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

'%,.%*)

, LMU CVFG@GR ? DMPCA?QRAMLRGLSMSQ QRP?RCEW DMP 6ICNRGA GL 3PMRMAMJ  RF?R KSJRGNJGCQ 6ICNRGAQ A?NGR?J @W ? J?PEC D?ARMP UFCLCTCP )MPCA?QRCPQ NPM@?@GJGRGCQ D?GJ RM @C UCJJ A?JG@P?RCB $AAMPBGLE RM 3PMNMQGRGML  )MPC A?QRCP A?L EGTC NPM@?@GJGRGCQ RF?R ?PC UCJJ A?JG@P?RCB @W NJ?WGLE ?E?GLQR 

Defensive Forecasting

237

7FGQ QS@QCARGML DMJJMUQ AJMQCJW RFC KMPC ECLCP?J CVNMQGRGML GL 9J?BGKGP 9MTIQ =1ML?QWKNRMRGA A?JG@P?RGML ?LB PCQMJSRGML 9MTI ? *' /C?TGLE ?QGBC 6ICNRGA DMP ? KMKCLR AMLQGBCP @GL?PW DMPCA?QRGLE UGRF )MPCA?QRCP ?LB 5C?JGRW ?JMLC '%)%%"  %'()' " ))25  "       )MPCA?QRCP ?LLMSLACQ   < > 5C?JGRW ?LLMSLACQ "    ,D ? DMPCA?QRCP GQ BMGLE ? EMMB HM@ UC CVNCAR ?@MSR  MD RFC CTCLRQ RM UFGAF FC EGTCQ NPM@?@GJGRGCQ LC?P  RM F?NNCL ,D RFGQ GQ RPSC LMR MLJW DMP  @SR ?JQM DMP ?JJ MRFCP NPM@?@GJGRW T?JSCQ RFCL UC Q?W RFC DMPCA?QRCP GQ <*11 (&1.'7&9*) +CPC ?PC RUM U?WQ RM ?NNJW RFGQ QMKCUF?R DSXXW AMLACNR RM RFC NPM@?@GJGRGCQ        ?LB MSRAMKCQ "      "   )MP ? EGTCL   < > BGTGBC RFC  GLRM RFMQC UC AMLQGBCP LC?P  ?LB RFMQC UC AMLQGBCP LMR LC?P :PGRC    DMP RFC LSK@CP GL RFC EPMSN UC AMLQGBCP LC?P  'GTGBC RFGQ EPMSN DSPRFCP GLRM RFMQC DMP UFGAF " GQ COS?J RM  ?LB RFMQC DMP UFGAF " GQ COS?J RM  :PGRC   " DMP RFC LSK@CP GL RFC J?RRCP EPMSN 7FSQ RFC DP?ARGML MD Q ML RFMQC PMSLBQ UFCPC  GQ LC?P  GQ "        :C Q?W RFC DMPCA?QRQ ?PC UCJJ A?JG@P?RCB ?R  GD  GQ ?NNPMVGK?RCJW COS?J RM  :C Q?W RFC DMPCA?QRQ ?PC UCJJ A?JG@P?RCB MTCP?JJ GD RFCW ?PC UCJJ A?JG@P?RCB ?R  DMP CTCPW  DMP UFGAF RFC BCLMKGL?RMP MD  RFC LSK@CP MD  LC?P  GQ J?PEC  $JRCPL?RGTCJW GLQRC?B MD BGTGBGLE RFC  QF?PNJW GLRM RFMQC UC AMLQGBCP LC?P  ?LB RFMQC UC AMLQGBCP LMR LC?P UC A?L GLRPMBSAC ? AMLRGLSMSQ KC?QSPC    MD QGKGJ?PGRW ? DSLARGML QSAF ?Q RFC &:88.&3 0*73*1    " 






UGRF
  UFGAF R?ICQ RFC T?JSC  UFCL  GQ CV?ARJW COS?J RM  ? T?JSC AJMQC RM  UFCL  GQ LC?P  ?LB ? T?JSC AJMQC RM  UFCL  GQ D?P DPMK  7FCL UC Q?W RFC DMPCA?QRQ ?PC UCJJ A?JG@P?RCB GD     "       

238

G. Shafer DMP CTCPW  DMP UFGAF RFC BCLMKGL?RMP GQ J?PEC

7FC QCAMLB ?NNPM?AF K?IGLE RFC AJMQCLCQQ MD  ?LB  AMLRGLSMSQ RQ FCPC @CA?SQC 6ICNRGA A?L CLDMPAC  SQGLE ? DMPCA?QRAMLRGLSMSQ QRP?RCEW 5CA?JJ RF?R ? 0*73*1 ML ? QCR ML  GQ ? DSLARGML    RF?R Q?RGQCQ RUM AMLBGRGMLQ  ,R GQ QWKKCRPGA    "    DMP ?JJ        ,R GQ NMQGRGTC BCLGRC     #  #    DMP ?JJ PC?J LSK @CPQ        ?LB ?JJ CJCKCLRQ #      # MD  7FC *?SQQG?L ICPLCJ  GQ ? ICPLCJ ML < > GL RFGQ QCLQC ,R ?JQM Q?RGQCQ        DMP ?JJ     $LB ?Q , F?TC ?JPC?BW LMRCB    GQ  UFCL  "   AJMQC RM  UFCL  GQ LC?P  ?LB AJMQC RM  UFCL  GQ D?P DPMK  2SP PC?QMLGLE GL RFGQ QCARGML UMPIQ UFCL  GQ ?LW ICPLCJ ML < > Q?RGQDWGLE RFCQC AMLBGRGMLQ  7FC EM?J RF?R  FMJB DMP ?JJ  DMP UFGAF     GQ J?PEC GQ DSXXW ML RUM AMSLRQ =  GQ DSXXW ?LB =J?PEC GQ DSXXW %SR PC?QML?@JC GLRCPNPCR?RGMLQ MD RFCQC RUM DSXXW NPCBGA?RCQ JC?B RM RFC DMJJMUGLE NPCAGQC EM?J %"

   

      "       





!/,%-.% ,*+*-%.%*)  8LBCP PC?QML?@JC GLRCPNPCR?RGMLQ  MD =  ?LB =J?PEC  GKNJGCQ RF?R  FMJBQ DMP ?JJ  DMP UFGAF     GQ J?PEC 2+').%*) &MLBGRGML  GQ COSGT?JCLR RM 

  "        





:C ?PC ?QQSKGLE RF?R    GQ QK?JJ UFCL  GQ D?P DPMK   7FGQ GKNJGCQ RF?R            

Defensive Forecasting

239

  UFCL     GQ J?PEC 6M  FMJBGLE UFCL     GQ J?PEC GQ COSGT?JCLR RM 

  "        



FMJBGLE UFCL RM KC?L RF?R

 

   GQ J?PEC ,D UC R?IC 

   

 



   @CGLE J?PEC

 



RFCL RFGQ AMLBGRGML RMECRFCP UGRF  GKNJGCQ          "          UFGAF GQ ? PC?QML?@JC GLRCPNPCR?RGML MD  !..%)# -.,.!#3 ) "*,!-.%)# '#*,%.$( ,L MPBCP RM BCPGTC ? QRP?RCEW DMP )MPCA?QRCP RF?R ES?P?LRCCQ RFC EM?J  GL 3PMRMAMJ  JCR SQ GK?EGLC RF?R 6ICNRGA GQ ?JJMUCB RM CLRCP RFC E?KC ?LB @CR ?Q GL 3PMRMAMJ  2SP QRP?RCEW DMP )MPCA?QRCP UGJJ @C RM NJ?W ?E?GLQR RFC DMJJMUGLE DMPCA?QR AMLRGLSMSQ QRP?RCEW DMP 6ICNRGA

 )) $ )')   "



  "  



:C A?JJ RFGQ QRP?RCEW DMP 6ICNRGA RFC . @CRRGLE QRP?RCEW @CA?SQC GR GQ KMBCJCB ML RFC K?PRGLE?JC RF?R GQ SQCB GL .MJKMEMPMT  RM NPMTC RFC UC?I J?U MD J?PEC LSK@CPQ QCC  $AAMPBGLE RM RFC NPMMD MD 3PMNMQGRGML  NJ?WGLE ?E?GLQR RFC . @CRRGLE QRP?RCEW KC?LQ SQGLE RFC DMJJMUGLE ?JEMPGRFK RM AFMMQC  DMP  "       

240

G. Shafer

 %'() $ "%' )#  ,D RFC COS?RGML 

  "   " 





F?Q ?R JC?QR MLC QMJSRGML  GL RFC GLRCPT?J < > QCR  COS?J RM QSAF ? QMJSRGML   ,D    "     DMP ?JJ   < > QCR  COS?J RM    ,D    "    DMP ?JJ   < > QCR  COS?J RM  $3 .$! "*,!-.%)# '#*,%.$( 1*,&- :FW BMCQ RFC . DMPCA?QRGLE ?JEMPGRFK UMPI# , UGJJ EGTC ? DMPK?J NPMMD RF?R GR UMPIQ ?LB RFCL BGQASQQ GRQ QSAACQQ @PGCW DPMK ?L GLRSGRGTC TGCUNMGLR

,*+*-%.%*)  !:5548* 47*(&89*7 51&>8 .3 7494(41  :8.3, 9-*  &1 ,47.9-2 <.9- & 0*73*1  8&9.8+>.3,     +47 &11   < > "-*3   <.11 -41) 34 2&99*7 -4< *&1.9> (-448*8 "      " 

744+ 0CPACPQ RFCMPCK Q?WQ RF?R DMP ?LW ICPLCJ  ML < > RFCPC GQ ? K?NNGLE
A?JJCB ? +*&9:7* 2&55.3, MD < > GLRM ? +GJ@CPR QN?AC  QSAF RF?R    "


 



MJIMND DMP ?JJ   GL < > UFCPC
GQ RFC BMR NPMBSAR GL  0CPACP ! 6AF[ ?LB 6KMJ?  8LBCP RFC . @CRRGLE QRP?RCEW 6ICNRGAQ KMTCQ ?PC

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   "  



3PMNMQGRGML  Q?WQ RF?R UFCL )MPCA?QRCP NJ?WQ RFC . DMPCA?QRGLE ?JEMPGRFK

Defensive Forecasting

241

?E?GLQR RFGQ QRP?RCEW 6ICNRGAQ A?NGR?J BMCQ LMR GLAPC?QC 6M     "



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+CLAC



    

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)PMK  ?LB RFC D?AR RF?R 
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8QGLE  RFC &?SAFW6AFU?PRX GLCOS?JGRW ?LB RFCL  UC M@R?GL        

 

      "   " 
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$BBGRGML?J GLQGEFR ?@MSR RFC QSAACQQ MD RFC . DMPCA?QRGLE ?JEMPGRFK A?L @C EJC?LCB DPMK (OS?RGML  

  "   " 



,L NP?ARGAC RFGQ COS?RGML SQS?JJW F?Q ? SLGOSC QMJSRGML GL < > ?LB QM RFGQ QMJSRGML GQ )MPCA?QRCPQ AFMGAC DMP   %CA?SQC    GQ QK?JJ UFCL  ?LB

242

G. Shafer

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

Defensive Forecasting

243

KCPCJW UGRF FGEF NPM@?@GJGRW  )MPCA?QRCPQ QRP?RCEW ML C?AF PMSLB  BMCQ LMR BCNCLB ML RFC FMPGXML   6M GD RFC E?KC GQ NJ?WCB GLBCLGRCJW RFC EM?J A?JG@P?RGML GL RFC A?QC MD RFC . DMPCA?QRGLE ?JEMPGRFK GQ ?AFGCTCB DMP CTCPW   6CC 9MTI ? DMP ? KMPC AMKNPCFCLQGTC BGQASQQGML 

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Defensive Forecasting

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246

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Defensive Forecasting

247

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A Snapshot on Reasoning with Qualitative Preference Statements in AI Carmel Domshlak Faculty of Industrial Engineering and Management, Technion, Israel

Abstract Preference elicitation is a well-known bottleneck in decision analysis and decision automation tasks, especially in applications targeting lay users that cannot be assisted by a professional decision analyst. Focusing on the ordinal preferences of the users, here we discuss the principles that appear to underly various frameworks developed in the AI research for interpretation and formal reasoning about sets of qualitative preference statements.

1. Introduction The ability to make decisions and to assess potential courses of action is a corner-stone of numerous AI applications, including expert systems, autonomous agents, decision-support systems, recommender systems, configuration software, and constrained optimization applications (e.g, see [13, 14, 32, 37]). To make good decisions, we must be able to assess and compare different alternatives. Sometimes, this comparison is performed implicitly, as it is done in some recommender systems. However, explicit information about decision-maker preferences should be communicated to the decision-support system. At first view, the conceptual foundations to support such decision-aid tools should be abound—the field of decision theory and its companion methodology of decision analysis deal with making decisions and evaluating their quality. As developed over some 50 years, these disciplines have provided many powerful ideas and techniques, which exert major influences over the biological, cognitive, and social sciences. Unfortunately, in spite of these remarkable achievements, the tools of traditional decision theory have not proven fully adequate for supporting recent attempts in artificial intelligence (AI) to automate the process of decision making. While there are several reasons for such a gap between theory and practice (e.g., see an excellent discussion of this issue by Luce and Raiffa [35] back in 1957), we believe that the Achilles heel of the traditional decision theory is the amount and

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form of the information that the decision maker needs to communicate regarding her preferences. In classical decision theory and decision analysis a utility function is used to represent the decision-maker’s preferences. Utility functions are a powerful form of knowledge representation. They provide a quantitative measure of the desirability of different outcomes, capture attitude toward risk, and support decision making under uncertainty. However, the process of obtaining the type of information required to generate a good utility function is time-consuming and requires considerable effort on the part of the user. In some applications, this effort is necessary and/or possible, e.g., when uncertainty plays a key role, and when the decision-maker and the decision analyst are able and willing to engage in the required preference elicitation process. One would expect to see such effort invested when medical or important business decisions are involved. However, there are many applications where either uncertainty is not a crucial factor, or the user cannot be engaged for a lengthy period of time (e.g., in on-line product recommendation systems), or the preference elicitation process cannot be supported by a human decision analyst and must be performed by a software system (e.g., due to replicability or mass marketing aims). In many such cases, direct elicitation of a good (even ordinal) utility function is not a realistic option; the traditional approach provides little help for decision makers who (for these or other reasons) exhibit discomfort with numerical evaluation of lotteries and/or outcomes. This shortcoming has been precisely formulated by Doyle and Thomason [21]: ”Traditional decision theory provides an account of the information that, in principle, suffices for making a rational decision. In practice, however, the decision maker might have never considered the type of choice in question and so might not happen to possess this information.” But if direct eliciting of a good utility function is not a realistic option in our application, then what is? When a utility function cannot be or need not be obtained, we may resort to other, more qualitative forms of information communicating user preferences. Ideally, this qualitative information should be easily obtainable from the user by non-intrusive means. That is, we should be able to generate it from natural and relatively simple statements about preferences obtained from the user, and this elicitation process should be amenable to automation. In addition, automated reasoning with this information should be both semantically reasonable and computationally efficient. In this paper we aim at sketching a general scheme for reasoning about user preferences that unifies treating this cognitive paradigm with the way other cognitive and computational paradigms are treated in the AI research. We then try to identify the limitations of such a general scheme, pointing on some of its shortcomings that unavoidably should be addressed in practice. Both the general scheme, as well as bypasses of its shortcomings, are then illustrated on two principle methodologies and their instances.

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Figure 1: The AI approach to reasoning about preferences. 2. The AI approach: Models + Languages + Algorithms In many senses, reasoning about user preferences bare similarity to many other problems faced by the AI research, and thus we may try dealing with user preferences in a way AI aims at treating other cognitive and computational paradigms. The general scheme underlying the AI approach has three major components: 1. mathematical models to concretize, understand, and classify various cognitive paradigms, 2. languages for describing models conveniently, 3. algorithms for answering queries about these models as efficiently as possible. To allow a system to reason about user preferences, the user should realize to herself her preferences as an instance of the model, and communicate this model instance in the (agreed-upon) language to the system. Figure 1 illustrates this scheme along with some examples for models, language(s), and queries about user preferences the system may have to answer. On the positive side, the foundations of decision theory suggest that we can restrict ourselves to models constituting (possibly partial, possibly weak) orderings over the space of alternatives1. While the concept of model in preferential reasoning is rather clear, the choice of language is much more open, and this due to numerous reasons. Suppose we adopt a language lying closest possible to our model of user preferences, that is, the language of pairwise comparisons between possible alternatives. The good news are that this type of information poses no ambiguity on its interpretation 1 When reasoning under certainty, the alternatives constitute concrete outcomes of the decisions, while in reasoning under uncertainty, the alternatives constitute, e.g., lotteries over such outcomes. In this paper we restrict ourselves to discussing reasoning under certainty, and thus the former perspective suffices.

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(and later we’ll show that these are good news). Thus, if the number of generally acceptable alternatives is small, then adopting such a language appears appealing. This language, however, do not “scale” well to applications in whose the number of acceptable alternatives is larger than trivial. As an example, imagine yourself sending your web-agent to look for a used car for you on the web. Even if there are only some 100 car configurations that you are ready to accept, specifying an ordering over 100 alternatives is not a walk in the park. Typically, however, we have no such strict “threshold” on acceptability, and thus we’ll have to specify an ordering over a sea of manufactured, used, and sold car configurations. So what are other alternatives for qualitative preference specification languages? While explicit ordering specification as above might be infeasible due to both time and cognitive effort required from the user2 , our everyday practice shows that users are capable of providing generalizing statements of preference such as - “I like ecologically friendly cars”, - “For a sport car, I prefer red color over black color”, - “This car would be better in red”. Assuming the preferences of the users are not entirely unsystematic, one would expect a relatively compact set of guidelines structuring (at least the core of) the user’s preferences. Given that, it is only natural to assume that by such generalizing statements of preference the user will aim at communicating to the system exactly these guidelines. Note also that generalizing statements refer not (only) to alternatives, but to the properties of the alternatives. To do anything with statements over properties, both the user and the system should consider the alternatives  not as monolithic entities, but as compound objects drawn from some combinatorial space. This property-based perspective is typically natural to the users—when choosing a used car to buy, we implicitly consider each car as a composition of its model, age, color, number of previous owners, accidents history, etc. And targeting this perception of the users, the systems also represent and keep track of the objects of users’ interest in terms of this or another attribution         
, by that effectively abstracting the alternatives’ space  to
 Dom . Taking now a closer look at the general scheme depicted in Figure 1, it appears that the scheme complicates when facing the realm of real users. The lessons learnt in the areas of psychology, philosophical logic, AI, and information systems indicate at least four major obstacles to operationalising this scheme in practice as is: 2 To illustrate the potential cognitive issues with ordering realization, recall yourself choosing a car from a lot of say 20 cars standing for sale.

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1. Incompleteness and noise in model specification. 2. System’s uncertainty about the semantics the user puts into her statements. 3. Language constraints stemming from system’s design decisions. 4. Potential intractability of reasoning about the model on the basis of the user statements. The first problem is not at all specific to generalizing-statements-based languages, and we have already discussed it above3. In contrast, the next three problems are inherited from the necessity to adopt this or another compact preference specification language. Let us first illustrate the issue of uncertainty in statements’ interpretation. Here and in what follows, we assume the user provides the system with a qualitative preference expression            


 

 


 

(1)

consisting of a set of preference statements   
  , where    are logical formulas over ,
    , and , , and have the standard semantics of strong preference, weak preference, and preferential equivalence, respectively. First, consider an “instance comparison” statement “ is better than  ”, where   
. The interpretation of this statement poses no serious difficulties because it explicitly compares between complete descriptions of two alternatives. However, this is the exception, rather than the rule. Most of the preference statements that we use in our everyday activities (e.g., “I prefer compact cars to SUVs”) have this or another generalizing nature. As such, these statements typically mention only a subset of attributes. This creates an ambiguity with respect to the actual referents of these statements. Several proposals on how to interpret generalizing preference statements have been made both in philosophy and AI. However, there is no agreed-upon solution to this problem (e.g., see [5, 22, 27]), and the inherent uncertainty about the message that such a statement aims to communicate makes developing such a solution very questionable. Having said that, different existing proposals are not entirely tangential. Specifically, all these proposals suggest to interpret generalizing preference statements as indirectly comparing between sets of alternatives from , while possibly disagreeing on what what sets of alternatives are actually compared by each statement separately, and/or by a multi-statement preference expression as a whole.4 Next, as if interpreting preference statements over  is not complicated enough already, note that 3 A very thorough overview of lessons learnt on this matter in psychology and philosophical logic can be found, e.g., in [28]. 4 For an excellent survey of this topic, we refer the reader to [28].

270

C. Domshlak  The attributes  and their domains effectively constitute the maximal alphabet for any language that can be used to provide the system with an information about . In particular, the users have to communicate their preferences to the system also only in terms of .  The attribution  is typically fixed by the system to be used by all its users. Moreover, this attribution typically serves various components of the system, and often the attribution is selected to optimize system components other than this dealing with reasoning about user preferences. (This situation is typical to the choice of schema attributes of most database systems.)  The attribution  is just one out of many possible attributions of the physical alternatives , and as such it does not necessarily correspond to the criteria affecting preferences of a user over .

To stress the latter point, suppose that a used-cars catalog maintains in its records the model and the color of the cars as a pair of attributes     . Suppose that a user of the system likes Porsche Cayenne, likes green color, and yet strongly dislikes green Porsche Cayenne, providing a statement capturing the latter piece of information. In this case, the user articulates her preferences over a combination of a particular pair of values of  and   , and no single attribute directly maintained by the system captures this preference-related criterion. On the positive side, however, if the user does articulate some preference information in terms of , then the implicit preference-related criteria behind this information obviously have some encoding in terms of  (as it in fact happens in this “green Porsche Cayenne” example). 3. Independence-based Preference Modeling Probably one of the best-known instantiations of the general scheme as in Figure 1 these days is this of CP-nets [7], along with its various extensions and derivatives [11, 12, 6, 19, 18, 38, 33, 44, 43]. Here we use this tool for representing and reasoning about preferences to illustrate both the different elements of the general scheme, as well as how the problematic issues discussed in the previous section may pop up in its concrete instantiations. In terms of Eq. 1, the language underlying CP-nets corresponds to sets of (conditional) preference statements for values of variables ; each statement expresses user’s preference over a single variable. For instance, in the statement ”if the car is a sports car, I prefer black to red as its color”, the addressed variable corresponds to the exterior color of the car. Formally, the expressions supported by CP-nets are  
                 Dom 
 Dom  (2)

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Reasoning about the ordering induced by such an expression on  requires a commitment to a concrete logical interpretation of these natural language statements. While different interpretations are feasible, the CP-nets adopt the ceteris paribus (all else equal) semantics for statement interpretation [28]. In this conservative semantics, a statement “I prefer    to    ” means that given any two alternatives that are identical except for the value of , the user prefers the one assigning  to  to the one assigning  . If these two alternatives differ on some other attribute as well, then they cannot be compared based on this preference statement alone. Conditional statements have the same semantics, except that they are restricted to comparisons between elements that satisfy the condition. Thus, “I prefer    to    given that  
 ” is interpreted exactly as above, but only for objects that satisfy  
 . The model underlying the CP-nets language is this of strict partial orders. Clearly, each preference statement as in Eq. 2 induces such a preference ordering over . The “global” preference (ordering) relation specified by a collection of such statements (that is, by a CP-net) corresponds to the transitive closure of the union of these “local” preference relations. If the user provides us with consistent information about her preferences, then the binary relation induced by the TCPnet on  is a strict partial order. Note that this order is rarely complete, and thus typically not all pairs of alternatives are comparable with respect to a CP-net. Having specified the model and the language, we shall now proceed with discussing reasoning algorithms for various queries with respect to CP-nets. All these algorithms exploit an intermediate graphical representation of preference expressions. Given an expression  as in Eq. 1, it is represented by an annotated graph. The nodes of the graph correspond to the attributes  and the edges provide information about direct preferential dependencies between the variables; there is an edge from  to   implies that user preference over values of   vary with values of . Each node  in a CP-net is annotated with a conditional preference table (CPT) describing the user’s preference order over Dom for every possible value assignment to the immediate predecessors of  (denoted  ). Figure 2 depicts a CP-net over four variables—binary , ,  and ternary  . The graph shows that the preference over values of  depends on  ’s value, while the preference over values of  depends on the value of both  and . Actual preferences are provided in CPTs. For example, when    , we prefer  to be false. The structure of the CP-net graph plays an important role in determining the consistency of preference specification and in reasoning about preference, although the user need not be aware of this structure [7]. While not all preference expressions representable as CP-nets are consistent [15], the latter is the case, e.g., for acyclic CP-nets [7]. Note that the latter property of the CP-nets can be recognized in time linear in —this is probably the simplest example for queries that can be efficiently answered, at least for some sub-classes of statements as in Eq. 1.

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Figure 2: A schematic example of a TCP-net. A more striking illustration of algorithmic exploiting of the structure induced by the preference expressions comes in processing three other important queries:  optimization—finding a preferentially-optimal alternative from , that is, finding an alternative  such that      for any   ,  dominance testing—for a pair of given alternatives    , determining whether preference expression  entails    (denoted as      ), and  ordering—given a subset   , order  in a non-increasing order of preference communicated by  (that is, if some  comes before  on this ordering, then we have     ). In preferential reasoning, all these three queries typically constitute NP-hard problems [33]. Surprisingly, optimization for acyclic CP-nets can be solved in time linear in  by a simple, top-down traversal of the graph [7]. The situation with dominance testing is not as sharp—while NP-hard in general even for acyclic CP-nets, this query still can be answered efficiently for boolean attributes  and certain topologies of the CP-net’s graph [7]. Finally, the situation is most intriguing with the ordering query. As this query corresponds to sorting a given set of alternative, and pairwise comparison between alternative is a basic operation of any sorting procedure, at first view, it seems that ordering is at least as hard as dominance testing. This, however, is not precisely so. To order a pair of alternatives     consistently with  it is sufficient to know only that    or   . Note that this information is weaker than knowing the precise preference relationship between  and  with respect to . While obtaining the latter knowledge would unavoidably require answering dominance queries, turns out that (i) answering its weaker counterpart can be done in time linear in  for any acyclic CP-net over any finite-domain attributes  [7], and (ii) this procedure can be used to sort any set of alternative    in time   
   [8]. At the big picture, the CP-nets framework constitutes an instance of direct logical reasoning about , and, of course, other proposals for such frameworks can

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be found in the literature. More specifically, the CP-nets framework corresponds to what we call here independence-based methodology, roughly characterized by the following three steps. 1. One defines certain preferential independence conditions on , and provides a “representation theorem” stating that under these conditions the preference model can be compactly specified using certain (hopefully, intuitive) language. Some of such foundational representation theorems come from the classical works on measurement and multi-criteria decision theories [31, 29, 23], while some other (e.g., those underlying the CP-nets tool) come from the research in AI (e.g., see [7, 44, 2]). 2. Next, one possibly defines some additional preferential independence conditions on  under which the model is not only compactly specifiable, but can also be reasoned about efficiently (at least for some queries of interest). 3. Finally, the system poses restrictions on the form of allowable preference expressions, so that they are constrained to a sufficiently simple language for which the conditions from above are fulfilled. Unfortunately, all independence-based methodologies for direct logical reasoning about preferences are limited by severe trade-offs between computational efficiency and semantic expressiveness (i.e., the richness of the supported preference expressions) [7, 33, 24, 44]. In attempt to escape these trade-offs as much as possible at least for optimization and ordering queries, several works in AI (e.g., see [4, 10, 9, 25, 34, 36]) suggested compiling the information carried by  into an ordinal utility function  

(3) consistent with (what we believe  tells us about) , that is requiring   
           

(4)

The consistency requirement posed on  by Eq. 4 says that ordering a set of alternatives from
in a non-increasing order of values provided to these alternatives by  will never put alternative  before alternative  if our interpretation of the user’s expression  implies that the user strictly prefers  to . The task of constructing such a utility function  from  is called ordinal utility revelation. Observe that specifying a utility function  as in Eq. 3 can be expensive due to the fact that 
  
. Thus, previous works on ordinal utility revelation suggested to stick here as well to the independence-based methodology (e.g., see [1, 6, 9, 25, 26, 32, 36]); in case of ordinal utility revelation, the first two steps of the above methodology are as follows.

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1. One defines certain preferential independence conditions on , and provides a “representation theorem” stating that under these conditions  can be compactly represented. 2. Next, one possibly defines some additional preferential independence conditions on  under which a utility function  consistent with  is not only compactly representable, but also can be efficiently generated from  [25, 36, 9]. Considering the first step, note that any function  as in Eq. 3 can be represented in a generalized additive form [23, 1]:   

 

 

(5)



where each sub-function  depends on some subset of variables   , and  is the restriction of a complete assignment  
to the attributes in . Note that  can be selected to be any function that maps a (partial) assignment to a real value. The generality of Eq. 5 is immediate since we may have    and   , which means that one can effectively select a function   that assigns an arbitrary utility to each possible complete assignment  
. While by itself this generality of Eq. 5 has no practical implications, Eq. 5 allows us to generalize the core assumptions of previous works on ordinal utility revelation — each concrete instance of the independence-based methodology corresponds to some independence conditions on  that guarantee existence of  (i) satisfying Eq. 4 and (ii) having a compact decomposition in terms of Eq. 5 (that is, defined by a small number  of small attribute subsets). An additional key property of all previous works is that attribute subsets       decomposing  are assumed to be known to the system.Finally, to our knowledge, all the works on independence-based ordinal utility revelation (except for the approach suggested in [26]) assume that the user’s preference expression is consistent, that is, the user makes no mistakes in specifying her preferences. In practice, however, this assumption is not necessarily reasonable. 4. Unstructured Preference Modeling In short, the independence-based methodologies for multi-attribute preference revelation are parametrized by the structure that user preferences induce on , and thus are applicable only when such compact structure exists and is known to the system. In Section 2, however, we have discussed already the limited power of  to provide the “ultimate alphabet” for preference specification. The attributes  do not necessarily correspond to the criteria affecting preference of each individual

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user over , and thus, even if user preferences are compactly structured, they should not necessarily be compactly structured over . Having in mind the “lessons learnt” from exploring the independence-based methodology for a few decades, let us try sketching the “universal instance” of the general information-processing scheme as in Figure 1. The vision here is threefold. 1. The user should be able to provide preference expressions  while being as little constrained in her language as possible. 2. The preference-model revelation machinery should be completely non-parametric, that is free of any explicit assumptions about the structure of the user preference ordering. 3. Both the process of model revelation and reasoning with the revealed model should be computationally efficient, including the case where user preferences pose no significant independence structure on  whatsoever. While possibly not formalized precisely this way, this vision is obviously not new. For instance, while listing open problems in the area, Doyle [20] addresses the problem of language constraints and fixed attribution, and writes: “Can one recast the underlying set [of attributed alternatives] in terms of a different span of dimensions such that the utility function becomes linear? If so, can one find new linearizing dimensions that also mean something to human interpreters?” The same questions were considered by Shoham in the scope of his work on utility distributions [39, 40]. In an attempt to provide a unified view on probabilities and utilities, Shoham showed that, in principle, such a set of linearizing dimensions (called, in [39, 40], factors) exist for any utility function, and that this set of dimensions may have to be exponentially larger than the original set of attributes. However, the result of Shoham is more foundational than operational. The connection between the attributes and the particular set of factors proposed in [39, 40] is not generally natural, and thus it is rather unclear how to perform preference elicitation with respect to this set of factors. Likewise, no efficient computational scheme for reasoning about this, potentially very large, set of factors has been suggested until these days. It is possible, of course, that the “universal methodology” as above is simply utopia. Some recent results, however, provide a positive evidence that, at least getting close to, this utopia is possible. In particular, a recently suggested by Domshlak and Joachims [16, 17] framework for ordinal utility revelation appears to satisfy to a large degree all the desiderata of the “universal methodology” as above. Connecting some ideas from knowledge representation, machine learning, and philosophical logic, this frameworks is based on a high-dimensional preference decomposition, and a specific adaptation of certain standard techniques for high-dimensional continuous optimization. Here we briefly outline the basic

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knowledge-representation idea underlying this proposal, referring the reader for details and computational solutions to [17]. Suppose the user provides the system with a preference expression , and the task of the system to suggest an ordinal utility function  satisfying Eq. 3-4. In general case, the system is not provided with any information about the independence structure the user preferences induce on . In fact, the system cannot even assume that a significant such independence structure on  exists. The basic idea underlying the framework in [17] is simple: If no useful preferential independence information in the original representation space
is provided, the only escape is to switch to a different space in which no such independence information is required. But what this “different space” can be? Assuming here that the attributes  are all binary-valued, let us schematically map the alternatives
into a new, higher dimensional space  using a certain mapping
 
  
  (6) Specifically, the mapping  connections between the dimensions of
and  as follows. Let      
 be a labeling of the dimensions of  , and





       
 
 



be the union of attribute domains in . Let     
be a bijective mapping from the dimensions of  onto the power set of
, uniquely associating each dimension  with a subset  
, and vice versa. Let 
   denote the subset of attributes “instantiated” by  . For example, if  

     , then 
       . Given that, for each  
and   , we set:
        (7)    
  That is, geometrically,  maps each -dimensional vector  
to the 
dimensional vector in  that uniquely encodes the set of all projections of  onto the subspaces of
. Considering the semantics of the transformation , recall the pitfalls of the system-level fixing of the attribution , along with the example of green Porsche discussed in Section 2. In that example, the user articulates her preferences over a combination of a particular pair of values of two attributes     , and no single attribute in  can directly capture the user’s utility alone. However, as we also already suggested in Section 2, (i) if such a complex preference is articulatable in terms of the attributes , then it has to correspond to this or another

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set of value assignments to a subset of , and (ii) the evaluation of any abstracted alternative  
with respect to such a complex preference corresponds to a single value assignment to a subset of , and that is, to a single dimension of  . This means that there are not only dimensions in  that capture the utilities of “Porsche” and “green”, but there is also a dimension that directly captures the utility of “green Porsche”. This correspondence between complex preferences and individual dimensions of  makes the semantics of  very attractive. In addition, it can be shown that any preference ordering over alternatives
can be captured by a linear real-valued function from  (to which the alternatives in
are transformed using .) Now, let us dedicate a few words to interpretation of preference expressions  suggested in [17]. Assuming here that the referents of the statements in  correspond to propositional logic formulas over , consider an arbitrary comparative statement   . Let    (and similarly  ) be the variables involved in , and    Dom be the set of all ’s models in the subspace of
defined by  . For instance, if          , and      , then      , and            . In [17], the statement   is compiled into a set of constraints on the space of (candidate) linear, real-valued functions from  . Specifically,   is compiled into a set of        constraints                (8)



 







 



where 
denotes the set of all non-empty value subsets of the local model . For example, statement      (e.g., “It is more important that the car is powerful or fast than not having had an accident”) is compiled into 
 
   
 
   
 
  

(9)

The constraint system
resulting from such compilation of a user expression  defines the space of solutions for the framework in [17]. (The reader is referred to [17] for a thorough discussion on why this approach constitutes a least committing interpretation of preference statement.) Note, however, that solving
poses numerous complexity issues. First, though this constraint system is linear,
it is linear in the exponential space
 . Second, the very description size of
, and, in fact, of each individual constraint in
, can be exponential in . Interestingly, Domshlak and Joachims show that these complexity complexity issues can be overcome by using some duality techniques from optimization theory [3] and Reproducing Kernel Hilbert Spaces (RKHS) [30, 42, 41]. At the very bottom

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line5 , these techniques allow for computing a solution for
(corresponding to the desired ordering function    
), and subsequently computing the values of
 on
without requiring any explicit computation in
 . 5. Summarizing Discussion The goal of this paper was mainly to provide a personal perspective of the author on the way user preferences are (or should be) modeled and reasoned about by general purpose systems such as these considered by the AI research. One important thing implied by this perspective is that the general model/language/algorithms scheme can be arguably instantiated in lots of ways, and each such proposal will unavoidably be open for (not less arguable) critique from this or another angle. If so, then the only real evidence for/against this or another concrete proposal will have to come from the experience with these proposals in practice. Unfortunately, here we observe a paradoxical deadlock situation suggesting the “chicken-andegg” metaphor. On the one hand, it is only natural to assume that reasoning about user’s preference expressions is useful in many applicative domains (e.g., in online catalog systems). On the other hand, to our knowledge, no application these days allows its users to express any but trivial (e.g., “bag-of-word”) preference expressions. It seems that the real-world players wait for the research community to come up with a concrete suggestion on how natural-language style preference expressions should be treated, while the research community waits for the real-world to provide it with the data essential to make the former decision. It is clear that this deadlock situation should somehow be resolved, and we believe that now this should be a primary goal for both sides. Acknowledgments My personal perspective on preferences in AI outlined here resulted (directly or indirectly) from numerous discussions I had on the topic with Ronen Brafman, Thorsten Joachims, Craig Boutilier, Hector Geffner, and many other colleagues and friends. This, however, does not imply that any of them necessarily agree with this perspective as a whole, or any of its parts. 5 Presenting the computational machinery here is simply infeasible, and thus the reader is referred to [17].

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