Why Least Squares and Maximum Entropy? An Axiomatic Approach to... Inverse Problems Author(s): Imre Csiszar

Why Least Squares and Maximum Entropy? An Axiomatic Approach to Inference for Linear
Inverse Problems
Author(s): Imre Csiszar
Source: The Annals of Statistics, Vol. 19, No. 4 (Dec., 1991), pp. 2032-2066
Published by: Institute of Mathematical Statistics
Stable URL: http://www.jstor.org/stable/2241918 .
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The Annals ofStatistics
1991,Vol. 19, No. 4, 2032-2066
WHY LEAST SQUARES AND MAXIMUM ENTROPY?
AN AXIOMATIC APPROACH TO INFERENCE FOR
LINEAR INVERSE PROBLEMS'
BY IMRECSISZAR
Institute
Mathematical
oftheHungarianAcademyofSciences
An attemptis made to determinethe logicallyconsistentrules for
when
selectinga vectorfromanyfeasibleset definedbylinearconstraints,
or thosewithpositivecomponents
or the probability
eitherall n-vectors
ifand onlyifthe
Somebasicpostulatesare satisfied
vectorsare permissible.
selectionruleis to minimizea certainfunction
which,ifa "priorguess" is
available,is a measureof distancefromthe priorguess. Two further
naturalpostulatesrestrictthe permissibledistancesto the author's fAs corollaries,axand Bregman'sdivergences,
divergences
respectively.
iomaticcharacterizations
of the methodsof least squares and minimum
thelatterare also
information
are arrivedat. Alternatively,
discrimination
As a specialcase,a
characterized
bya postulateofcomposition
consistency.
froma smallset ofnatural
derivation
ofthe methodofmaximumentropy
axiomsis obtained.
1. Introduction. A frequently
occurring
problemin statisticsand applied
frominsufficient
is thata function
has to be inferred
information
mathematics
that specifiesonlya feasibleset of functions.
Problemsofthiskindare often
ofa signalor
calledinverseproblems.Typicalexamplesare thereconstruction
and the assignmentof a
image fromthe resultsof certainmeasurements,
that is,
probability
densityor mass functionsubjectto momentconstraints,
thatspecify
theexpectations
ofcertainfunctions
oftheunderlying
constraints
randomvariable.Oftenthe practicalsolutionto such problemsis to selectan
elementofthefeasiblesetbya moreor less ad hocrule,usuallybyminimizing
somefunctional
such as the L2-normor negativeentropy.If somefunction
is
specifiedas a "priorguess," it is naturalto minimizea measureof distance
fromthelatter,mostoftenthe L2-distance,
densityor mass
or,forprobability
Kullback'sI-divergence
fordiscrimination
functions,
(also calledinformation
or cross-entropy).
In thispaperwe addressthe questionofwhatselectionrulesare "good" in
this framework
and, in particular,whetherthe mentionedstandardones are
indeedthe "best." Unfortunately,
meaningto
it is hardto givea mathematical
thisquestion.It does not seempossibleto definea generalcriterion
by which
December
Received
1990.
May1989;revised
ofTokyo,supported
by
'This researchwas startedduringtheauthor'svisitat theUniversity
forthePromotion
ofScience.
theJapanSociety
Its completion
wassupported
bytheHungarian
National
ScienceFoundation,
Scientific
Research
Grant1806.
AMS 1980subject
classifications.
Primary
62A99;secondary
68T01,94A17,92C55.
andphrases.
linearconstraints,
logically
consistent
inference,
Keywords
Imagereconstruction,
minimum
discrimination
nonlinear
distance,
selection
rule.
information,
projection,
nonsymmetric
2032
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AXIOMATICAPPROACHTO INFERENCE
2033
the goodness of selectionrules could be compared.Still, whereas people
apparently
feellittleneedforanyspecialjustification
ofleastsquares(L2-norm
variousreasonshave been put forwardto justifyI-divergence
minimization),
intostatisticsby Kullback(1959) as the methodof
minimization
[introduced
minimumdiscrimination
and entropymaximization.
The recent
information]
widespreadapplicationsof "maximumentropy"have been pioneeredto a
greatextentbyJaynes[forhis viewscf.Jaynes(1982)].The presentauthorhas
arguedelsewhere[Csisza6r
(1985)] that the conditionallimittheoremsofVan
Campenhoutand Cover(1981) and Csiszar (1984) suggestthe interpretation
thatthe minimumI-divergence
distribution
"updating"ofa priorprobability
is a limitingformofBayesianupdating.
to meetmomentconstraints
Here we adoptan axiomaticapproachand considerthoseselectionrulesas
in the sense of
"good" thatlead to a logicallyconsistentmethodofinference,
not
natural
The
term
inference
is
meant in a
some
satisfying
postulates.
even
statisticalsense. Indeed, our considerationswill be nonprobabilistic,
distributions.
The relation
thoughtheobjectsto be inferred
maybe probability
ofthisworkto previousaxiomaticapproacheswillbe discussedat the end of
thissection.
As a typicalexample,we briefly
sketcha modelofthe imagereconstruction
and in variousotherfields
problemthat occursin computerized
tomography
[cf.,e.g.,Hermanand Lent (1976) or Censor(1983)].An imageis represented
functionf definedon some domain.The availableinforby a positive-valued
mation consistsin the measuredvalues of some linear functionalsR if,
i = 1,.. ., k. In X-raytomography,
f is the unknownX-rayattenuationfunctionand Ri f is itsintegralalongthepathofthe i's ray.Now,thedomainof f
is partitioned
intoa finitenumberofpictureelements,calledpixels,numbered
in some way from1 to n. Assumingthat f is nearlyconstantwithineach
pixel,we can write
n
(1.1)
f=Ev
j=1
fi,
where fj is the indicatorfunctionof the jth pixel.Then,settingaij
and bi = Ri f, we have
(1.2)
=
Ri fj
n
, aijvj
j=1
=
bi,
i
=
1,...,k.
Thus the unknownfunctionf is represented
by the vectorv = (v1,.. ., vn)T
and the feasibleset is identified
withthe set of thosevectorsv withpositive
thatsatisfythe linearconstraints
components
(1.2). The reconstruction
problemis to selecta "suitable"elementofthisfeasibleset (possiblydependingon
a "priorguess" of f represented
by a vectoru).
In thispaper,we concentrate
on linearinverseproblemsofform(1.2). The
functions
extensionofour resultsto the continuouscase, thatis, to inferring
definedon some domainand not representable
by finite-dimensional
vectors,
should not be hard but will not be entered.The above example will be
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I. CSISZAR
2034
repeatedlyused as an illustration,but it should be emphasizedthat our
axiomaticapproachis by no means limitedto imagereconstruction.
On the
other hand, since a generalapproachinevitablyinvolvesidealizations,the
solutionsit leads to are not necessarily"best" forspecificpracticalproblems,
includingimagereconstruction.
Our goal is to determinethe "logicallyconsistent"rules forselectingan
elementfromanypossiblefeasibleset.We adopttheidealizedassumptionthat
all conceivablelinearconstraints
mayoccur;thusthepossiblefeasiblesetsare
all thosesubsetsofa basic set S ofpermissible
vectorsthatcan be definedby
constraintsof form(1.2). Threecases willbe consideredin a parallelmanner
namelywhen S consistsofall n-vectors,
or ofthosewithpositivecomponents
(as in imagereconstruction)
or ofthe probability
vectorswithpositivecomponents.The choiceofvectorswithpositiveratherthannonnegative
components
has been preferred
in orderto reducetechnicaldifficulties;
also, thisensures
that a nonnegativequantityis never inferredto be 0 when the available
information
permitsit to be positive,whichis generallyconsidereddesirable.
Our postulateswill be stated and intuitively
justifiedin Section2. The
resultswillbe statedin Section3 and provedin Section5, usingthelemmasin
Section4. The keyresultis Theorem1, namelythat the basic postulatesof
"regularity"and "locality"of a selectionrule are satisfiedif and onlyifthe
selectionis by minimizing
a certainfunction.
If a priorguess is available,this
functionis a measureof distance(nonsymmetric
in general)fromthe prior
guess. The subsequenttheoremsshow how certainadditionalpostulatesrestricttheclass offunctions
thatmaybe used. Perhapsthemoststriking
result
in Theorem5. It providesa parallelcharacterization
of the methodsof least
squares and minimumI-divergence
as the onlyones satisfying,
in additionto
and locality,a postulateof"composition
regularity
The intuitive
consistency."
meaningofthispostulateis thatifthe objectofinference
is composedoftwo
and the availableinformation
components,
says nothingabout theirinteraction,we shouldinferthatno interaction
is present.
It shouldbe mentionedthatin practice(1.2) is oftenrelaxedto
(1.3)
n
E aij + ei =bi,
j=1
i = ly,...,)k,
wheree = (el, ..., ek)' is an errorvector.For the reconstruction
problemof
positronemissiontomography,
Vardi,Sheppand Kaufman(1985) describeda
modelequivalentto (1.3) withdata bi that were Poisson randomvariables
(countsofdetectedemissionsin "tubes" determined
by pairsofdetectors).
A
vectore as in (1.3) did notexplicitly
entertheirmodel,but,definingei as the
ofthe actualand expectedcountsforthe ith tube,(1.3) wouldhold.
difference
The suggestedreconstruction
was the maximumlikelihoodestimateofv, and
forits computationthe EM algorithmof Dempster,Laird and Rubin(1977)
was used. This reconstruction
techniqueis sometimesconsideredas relatedto
"maximumentropy"[cf.,e.g.,Millerand Snyder(1987)],fortheformalrather
thanconceptualreasonthatthe EM algorithm
is equivalentto an alternating
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AXIOMATICAPPROACHTO INFERENCE
2035
I-divergence
minimization;
Vardi,Sheppand Kaufman(1985) haveshownthat
the convergence
of theiralgorithmis an instanceof a resultof Csiszar and
minimization.
Tusnady(1984) on alternatingI-divergence
Maximumlikelihoodestimationis not a generallyapplicablemethodfor
"solving"inverseproblemsof form(1.3) because (i) the "errors"ei maybe
nonrandomor else theirjointdistribution
maybe unknownand (ii) evenifthe
the maximumlikelihoodesti"errors"are randomwithknowndistribution,
mate is typicallynonuniqueif n > k. Vardi, Shepp and Kaufman(1985)
theobjectintofewerpixelsthanthe
avoidedthelatterdifficulty
bypartitioning
numberof "tubes" thatwas sufficiently
large.If boundson the magnitudeof
the errorsare known,it maybe convenientto interpret
(1.3) as a systemof
inequalitiesofform
(1.4)
bi
n
j=l1
aijvj < btzi=l,i ,
k,
or as an inequalityofform
k
(1.5)
k
i=l
(
n
j=l
2
aiivi - b)
< c.
Our axiomaticapproachcould be extendedto the problemof selectingan
elementfromanyset definedbyinequalityconstraints
suchas (1.4) or (1.5) or,
moregenerally,
fromanyconvexsubsetofthebasic set S. Alternatively,
(1.3)
could be interpreted
as determining
a "feasible set" of pairs (v,e), and a
selectionfromthis set could be made by any methodthat has been deemed
"good" for the problem(1.2). Indeed, this is oftendone in practice,for
somequadraticfunction
ofthe pair(v,e) [cf.Herman
example,by minimizing
and Lent (1976) or Censor(1983)]. It remainsto be seen whether"solutions"
of this kind to the problem(1.3) can be coveredby an extensionof our
axiomaticapproach;one of the difficulties
is that the possiblelinear constraintson pairs(v, e) are ofthe veryspecialform(1.3).
The approachin thispaperwas strongly
motivatedby Shore and Johnson
(1980), where-forthe problemofassigninga probability
distribution
subject
to momentconstraints-an intuitively
appealingaxiomaticderivationof the
methodsof maximumentropyand minimumI-divergencewas provided.
forinferring
Skilling(1988) gave a similarderivation
arbitrary
positive-valued
because of the greaterliberty,
this case turnedout to be considerfunctions;
ablysimpler.BothShoreand Johnson(1980) and Skilling(1988) startedfrom
the assumptionthat inferencewas based on minimizingsome functionor,
equivalently,on some transitiverankingthat could be describedby real
numbers.Afterhavingsubmitted
thispaper,theauthorlearnedthatParis and
ofmaximumentropy"from
Vencovska(1990) had arrivedat "the inevitability
axiomsthat-like ours-did not a prioriassume the minimization
of some
fromtheirs,
function;in otherrespects,our approachappearsquite different
in the axiomsdo exist.The authoris indebtedto Profesalthoughsimilarities
sor Paris forsendinghimmanuscripts
ofthisand relatedworks.
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2036
I. CSISZAR
Our resultsalso provideaxiomaticcharacterizations
ofmeasuresofdistance
whoseminimization
leads to "good" methodsofinference.
Whereasthereis an
of entropy,I-divergence
extensiveliteratureon axiomaticcharacterizations
and theirgeneralizations
[cf.,e.g.,Aczeland Dar6czy(1975)],ourcharacterizadiffer
fromthe usual ones: Our postulatesinvolvenot the
tionssubstantially
but ratherthe inferencemethodit leads to. In
measureto be characterized
Theorem2(ii), a class of distancesintroduced
by Csisza6r
(1963) [and independentlybyAli and Silvey(1966)] is characterized;
relatedresultsalso appearin
Shoreand Johnson(1980) and Skilling(1988). Theorem3 characterizes
a class
ofdistancesintroduced
by Bregman(1967); sincethispaperhad been submitted,a different
axiomaticcharacterization
ofthisclass (in thecontinuouscase)
was givenbyJonesand Byrne(1990). Finally,we commenton theone-parameterfamilyofdistancescharacterized
in Theorem4(ii). In theoriginalversionof
this paper,the previouslynot consideredmembersof that familyhad been
mentionedas candidatesforbecomingpractically
useful.Recently,
Jonesand
Trutzer(1989) reportedapplicationsof (continuousversionsof) these distancesthatseemto confirm
thatprediction.
2. Preliminaries, postulates. The reallineand thepositivehalf-line
are
denotedbyR and R+, respectively.
We emphasizethat R+ does notcontain0.
The vectorsin Rn whosecomponents
are all 0 or all 1 are denotedby0 or 1.
All vectorsare columnvectors.
The set of n-dimensional
vectorswith positivecomponentsof sum 1 is
denotedby An
nthat is,
(2.1)
A = (v: vE R, Tv= 1}.
The familyofaffinesubspacesof Rn, thatis, the familyofnonvoidsubsets
of Rn definedby linearconstraints,
will be denotedby .Zn. In otherwords
#
LE
iff
L
0
and
n
L = {v: vE Rn,Av = b}
forsomeA E Mk xn (k x n matrix)and b E Rk. We denoteby Yn' thefamily
of all nonvoidsubsetsof R+ of form(2.2), withv E Rn replacedby v E R+.
The familyof those L E Yn+ whichare subsets of An will be denotedby
Y4+(1). For any fixeddimensiond < n, the set of d-dimensional
elementsof
4n (or 4+k) is endowedwith the natural topology,that is, the quotient
topologyderivedfromthe Euclideantopology
oftheset ofthosepairs(A,b) E
elementof .4n (or Yn+).
M(n-d)xn x Rn-d thatdefinea d-dimensional
Throughoutthis paper,our basic set S willbe eitherof Rn, Rn and An
The set ofcomponents
ofvectorsin S willbe denotedby V, thatis, V stands
for R, R + or the open interval(0, 1), accordingto the choiceof S. Unless
statedotherwise,u, v and w willalwaysdenoteelementsof V, whereasu, v
and w denotevectorsin Vn. Further, we denote by ./ the familyof nonvoid
subsetsof S determined
by linearconstraints.Thus, accordingto the three
cases, ? equals .n, .+
or .n'(1). For convenience,we will call the
elementsof Y subspacesof S also if S $ Rn.
(2.2)
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AXIOMATICAPPROACHTO INFERENCE
2037
Noticethat S itselfis an elementof ..f; amongthe propersubsetsof S the
maximalsubspacesare thoseofdimensionn - 1 (if S = Rn or R+) or n - 2
(if S = An). The subfamily
of Y consisting
ofthesemaximalsubspaceswillbe
denotedby X4/.
Thus X/ consistsofthe (nonvoid)sets
(2.3)
_
({v:a av
a
=b
0,
if S
=
R~ or R+,
{v: aTv = b, ITv = 1}, a A1, if S = Azn
As mentionedpreviously,
the conditionv E Vn is implicitin the notationin
(2.3).
Having in mindinferenceproblemsas in Section 1, we are interestedin
rulesthatspecify
foreach L E Y (and "priorguess" u) an elementof L, to be
selectedif L is in the feasibleset (and the priorguess was u). The vector
selectedfromL whena priorguessu was availableis regardedas a nonlinear
projectionon L ofu, denotedby fl(Llu).
(2
L
DEFINITION
1. A selectionrule (withbasic set S) is a mapping[l: Yt-*S
suchthat 11(L) E L foreveryL E .. A projectionrule is a familyofselection
rules fIl( Iu), u e S, such thatu E L impliesfl(Liu) = u. A selectionrule is
generatedbya functionF(v), v E S, if foreveryL E Y, fl(L) is the unique
elementof L whereF(v) is minimizedsubjectto v E L. A projectionrule is
generatedbya function
F(vlu), u E S, v E S, if its componentselectionrules
fl( Iu) are generatedbythe functionsF( Iu).
REMARK.A necessaryconditionforF(viu) to generatea projectionrule is
thatforanyfixedu E S theuniqueglobalminimumof F on S be attainedat
v = u. Attention
maybe restricted
to functionsF forwhichthisminimum
is 0
[because a projectionrule generatedby some F(vlu) is also generatedby
F(vIu) = F(vIu) - F(uIu)I. A functionF(vIu), u E S, v E S, withtheproperty
F(vlu) 2 0, withequalityiffv = u, willbe calleda measureofdistanceon S.
EXAMPLE1. In the case S
=
Rn, the Euclidean distance F(vlu) = ilu - vil
generatesa projectionrule that gives rise to the ordinaryprojectionin
Euclideangeometry.
It willbe calledthe least squaresprojectionrule and its
componentselectionrules will be called least squares selectionrules. In
particular,the selectionrule generatedby F(v) = livilis the standardleast
squares selectionrule. A projectionrule generatedby a weightedL2-distance
(E n 1ai(vi - ui)2)1/2 willbe calleda weightedleastsquaresprojectionrule.Of
course,"least squares" is a standardmethodof a verylong history.Notice,
however,
thatit is notsuitablein thecases S = R or An; thenF(v) = liv- uil
does not generatea selectionrule forany u E S (neitherdoes any weighted
L2-distance)becauseit does notattaina minimumon somesubspacesL E .Y.
EXAMPLE2.
definedby
(2.4)
Let S
=
R or An. The I-divergence
ofv E S fromu E S is
I(viiu)
=
f (vilog
- vi+ ui.
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2038
I. CSISZAR
This generalization
ofKullback'sformula
I(v||u)
n
v
vilog,U
u E- Anx
V E- A\n
retainsthe propertyI(vllu) ? 0, withequalityif and onlyif u = v. Clearly,
F(vlu) = I(vllu) generatesa projectionrule (forany fixedu E S, it attainsa
unique minimumon any L E /, and if u E L, this minimumis attainedat
v = u); this will be called the I-divergence
projectionrule,and eitherof its
componentselectionrulesis an I-divergence
selectionrule.The selectionrule
generatedby the negativeentropyF(v) = E n 1vilogVi will be called the
maximumentropy
selectionrule.This is a specialcase of I-divergence
selection
rulescorresponding
to u = (1/n)1 if S = A\ and u = (1/e)1 if S = Rn. As a
comparisonof the naturalityof these choicesof u indicates,the maximum
role mainlyin the case S = An,
entropyselectionrule plays a distinguished
thatis, forinferring
probability
distributions.
We emphasizethat we do not a priorirestrictattentionto selectionrules
2 and 3
generatedby some function.However,the postulatesin Definitions
belowwillsuffice
to provesuch generatedness.
Givena selectionrule II withbasic set S, we willdesignateHI(S) byv?(H).
Then the componentselectionrules H( Iu) of a projectionrule satisfy,by
Definition1,
(2.5)
vo(H(. Iu)) = u.
DEFINITION2. A selectionrule H:Y-+ S is regular if it satisfiesthe
axioms:
following
if L' c L and H(L) e L', then E(L')=l(L);
(i) (consistency)
(ii) (distinctness)
if L # L' are bothin X#,then H(L)
M
H(L') unlessboth
L and L' containv?(HI);
(iii) (continuity)
the restriction
of Hl to the subspacesof any fixeddimension is continuous.
A projectionrule is regularifits componentselectionrulesare such.
The consistencyaxiom formalizesthe intuitiveidea that if v = H(L) selected on the basis of constraintsspecifyingL also satisfiesthe stronger
constraintsspecifying
L', thenthe additionalconstraints
provideno reasonto
changethe selectionof v. The case forthis axiomappears quite strong,for
one wouldhardlywantto use a selectionrule
example,in imagereconstruction
not satisfying
it. Nevertheless,
forcertain
this axiom may be inappropriate
problems.Whenthe selectedelementoughtto resemblethe otherelementsof
thefeasibleset L as muchas possible,it is reasonableto selectthev E L that
minimizes-forsome given measure of distance d-either sup,, L d(v,w)
of d(v,w)
["barycenter
expectation
method,"Perez (1984)] or the conditional
on the conditionw E L (Bayesianrule,requiresa priordistribution
on S).
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AXIOMATICAPPROACHTO INFERENCE
2039
These selectionrulesdo not satisfythe consistency
axiomand are outsidethe
scopeofthispaper.
in a
The distinctness
axiomsaysthatdifferent
bothconsisting
information,
singlelinear constraint,must lead to different
conclusions,unless both are
This
consistent
withtheselectionthatwouldbe madewithoutanyconstraints.
is a technicalpostulateand it wouldbe desirableifit couldbe dispensedwith.
Notice that the distinctnessaxiom is certainlysatisfiedfor selectionrules
functionsF.
generatedby differentiable
is an obviousregularity
Continuity
hypothesis.Notice,however,that (for
projectionrules) we did not postulatethe continuousdependenceof H(LIu)
on u.
For any set of indices J = Ul, ... , jk, 1 < il <
< Jk < n, and any
vectora E Rn, we denoteby aj the vectorin Rk definedby
(2.6)
aj
= (ajlX
...
, aik)
For a selectionrule [I, we willdenote(II(L))j briefly
by rlj(L).
S is local ifforeveryJ c {1, ... , n}
DEFINITION3. A selectionrule II: Y
ofarbitrary
size k, and any L' and L" in Y ofform
(2.7)
L'= {v: vj e L, vjcEL'},
L" = {v: vjE L, vjcEL"),
L" E4k
whereLo E..4, L' E Yk,
(if S = Rn) or Lo E
E4yn+-k (if S = Rn orAn), we have
L"
-k
XL'
E/A-k,
(2.8)
H( L') = 11JW) .
A projectionruleis local ifits componentselectionrules IL( Iu) are local and,
in addition,forL' and L' as abovewe have
(2.9)
HIj(L'u') = llJ(L"'Iu") ifu', = U'.
= +,then anyL' and L" of
REMARK.If S=Rn, )Y=n
orS=R+,
form(2.7) necessarily
belongto ..?. On theotherhand,if S = An, ..= 4+ (1),
the sets L' and L" in (2.7) belongto Y iffthe sum of componentsof each
vectorin Lo is equal to the same 0 < c < 1, and the sum of componentsof
each vectorin L' and L" is equal to 1 - c.
Localitymeans,in otherwords,thatforL' E Y as in (2.7), [Ij(L') depends
onlyon LO) and [IH(L'Iu') dependsonlyon Lo and u'j.
the localityaxiomsays thatwheneverthe availableinformation
Intuitively,
consistsof two piecesthat involvecomplementary
sets of componentsof the
vectorto be inferred,
each component
ofthevectorselectedon thebasis ofthis
whichinvolvesthe
willbe determined
information
bythatpieceofinformation
componentin question.In the reconstruction
problemof X-raytomography,
this means that if two sets of ray paths coveringdisjointsets of pixelswere
wouldbe determined
used,thenforeach pixelthereconstruction
bythoserays
whose paths are in the set coveringthat pixel. This axiom appears very
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2040
I. CSISZAIR
natural,but strictadherenceto it mayperhapsbe criticized
in thetomography
exampleon thebasis ofsmoothnessproperties
oftheunknownX-rayattenuationfunction.For the case ofinferring
probability
mass functions,
Shoreand
Johnson(1980) used a similarbut strongerpostulatecalled "subset independence."
In this paper, regularityand localitywill be the basic postulates.The
selectionand projection
rulessatisfying
themwillbe characterized
in Theorem
1. In the restofthissectionwe formulate
someotherdesirableproperties
that
suggestthemselvesas additionalpostulates,ifwe wantto arriveat methodsof
practicalinterest.
An important
rolewillbe playedbythe specialsubspaces
({v: vj + Vj=t},
(2.10)
Lij(t)
|
v: + vi = t,
E
if S =R
v= 1-t},
orR,
if S = 'A
lo,j
Given a selectionrule fl, we will writev ij v' to designatethat v and v'
equal the ith and jth componentsof il(Lij(t)) forsome t; of course,then
necessarilyt = v + v'. Similarly,givena local projectionrule,we will write
(vlu) -ij (v'u') to designate that v and v' equal the ith and jth components
of [l(Lij(t)Iu) ifthe ith and jth component
ofu are u and u' (and t = v + v').
Clearly,v <ij v' means the same as v' *ji v. Further,we will show
(corollaryof Lemma 3) that if v -ij V' and v' jk V" for a regular,local
selectionrule II, then also v *>ik V", providedin the case S = An that v +
v' + v" < 1. Ofcourse,similarstatements
holdfortherelations(v u) i*- (v'Iu')
associatedwitha regular,local projectionrule.
DEFINITION4. (i) A local projectionrule is semisymmetric
if for every
Lij(t) as in (2.10), the ith and jth componentsof Hl(Lij(t)u) are equal
wheneverui = uj, that is, (vyu) <-ij (v'lu) iffv = v' (providing,
in the case
S = An, that v < 1/2, u < 1/2).
(ii) A projectionrule withbasic set R+ or An is statisticalifit is regular,
local,and the relation(ylu) *-ii (v'lu') is equivalentto v/u = v'/u',withthe
additionalconditionsv + v' < 1, u + u' < 1 if S = An.
The term"semisymmetric"
refersto the factthatthisweak and plausible
postulateoftenimpliesthe apparentlymuchstrongerpropertyof symmetry
(permutation
invariance);see Theorem2(i). It wouldnot be unreasonableto
use symmetry
as a postulate,as did Shore and Johnson(1980), but by not
doingso we will obtain strongermathematicalresultswithlittleadditional
effort.
Also the strongerpostulatein (ii), applicablewhen S = R n or An, appears
in thelattercase. Namely,ifa priorguess
natural;it is particularly
compelling
about a probabilitymass functionhas to be updated subject to a single
ofa givenset,it is standardto assign
constraintthatspecifiesthe probability
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AXIOMATICAPPROACHTO INFERENCE
2041
to the prior ones.
probabilitiesto the elementsof this set proportionally
4(u)
Definition
requires this for two-elementsets only. The proportional
the probabilitiesof several
updating,in general for constraintsspecifying
pairwisedisjointsets ["Jeffrey's
rule,"cf.Diaconisand Zabell(1982)] appears
uncontroversial.
It was also partof the axiomsof Shore and Johnson(1980).
5. (i) A projectionrule withbasic set S = R' or R+ is scaleDEFINITION
invariantifforeveryA > 0, L E Y and u E S we have [l(ALJAu) Afl(LIu).
(ii) A projectionrule withbasic set S = Rn is translation-invariant
if for
everyL E Y, u E S and ,uE R, fl(L + ,1Iu + 1) = fl(LIu) + 41.
Here AL and L + ,i1 denotethe set ofvectorsAvand v + At1,respectively,
such thatvE L.
The intuitivemeaningof these invariancepropertiesis obvious,and they
are clearlydesirable.The characterization
ofall (regularand local) projection
rules satisfying
eitheror both of thesepostulatesis not difficult
but will be
omittedbecause of its limitedpracticalinterest.Rather,thesepostulateswill
be used in connection
withthe nextone only.
DEFINITION
6. (i) A projectionrule is subspace-transitive
if forarbitrary
L' c L in Y and anyu E S we have
fH(L'Ju)= fl(L'jfl(L1u)).
(ii) A projectionrule is parallel-transitive
if (2.11) holds forsubspaces L
and L' that can be representedin the form{v: Av = b} [cf.(2.2)] withthe
same matrixA.
(2.11)
Subspacetransitivity
meansthatifupdatinga "priorguess" u based upon
information
thefeasibleset L resultsin v = Hl(LJu),and additional
specifying
information
leads to L' as the actual feasibleset,thenupdatingthe "present
guess" v on the basis of all availableinformation
leads to the same resultas
wouldthe directupdatingof the "priorguess" u. For example,if an image
reconstruction
has beenobtainedfrommeasurements
Ri f, i = 1,. . ., k, using
some priorguess,and thenfurther
measurementsRi f, i = k + 1,. . ., k + 1,
are made,thereconstruction
fromall themeasurements
Ri f,i = 1,..., k + 1,
willbe the same no matterwhetherthe originalpriorguess or the previous
reconstruction
is used as " priorguess." Parallel transitivity
has a similar
intuitivemeaningforthecase whenafterhavingobtainedthefirstreconstruction,the same measurements
are repeatedwithresultsdifferent
fromthose
before;the second reconstruction
is now based on the new results,the
previouscontradicting
ones beingdiscarded.
Subspace transitivity
is a highlydesirableproperty
also because it implies
(and is actuallyequivalentto) the commutativity
of two-stageupdatings.
Indeed,letus be giventwosetsoflinearconstraints
determining
subspacesL1
and L2 withL1 n L2 = 0. Then (2.11) impliesthatupdatinga priorguess u
based upon the firstset of constraints
and thenupdatingthe resultfl(L1Iu)
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2042
I. CSISZAR
based uponbothsets ofconstraints,
theresultwillbe H(L1 n L21u); thesame
resultwouldbe obtainedifthe firstupdatingwerebased on the secondset of
constraints.
Ofcourse,thiscommutativity
holdsfor"proper"two-stage
updatingsonly,thatis, whentheconstraints
used in thefirststageare notdiscarded
in the secondstage.
For regularprojectionrules,paralleltransitivity
impliessubspacetransitivity,and more generally,also that (2.11) holds wheneverL = {v: Av = b},
L {v: Av = b'}, wherethe matrixA containsall rowsof A. In fact,then
L = {v: A'v = A'v*} c L, wherev* = fl(Llu), and thus fl(Llu) = v* by the
axiom.Then paralleltransitivity
consistency
impliesthat
= I(L'Il(LIu)),
[(L'ju)
and (2.11) holdsas claimed.
We will show that for regular,local projectionrules the two kinds of
definedabove are actuallyequivalent.The familyof all regular,
transitivity
local and transitive
in Theorem3.
projectionruleswillbe characterized
Combiningresultscharacterizing
projectionruleswithpropertiesstatedin
Definitions4-6 will lead us to the least squares and I-divergence
projection
rules as the onlyones that simultaneously
satisfysome intuitively
appealing
postulates.On theotherhand,a singlepostulate(in additionto regularity
and
locality)willalso suffice
to uniquelycharacterize
theseprojection
rules,as well
as the least squares and I-divergenceselectionrules. To formulatethis
postulate(Definition7 below),it is necessaryto considervectorsindexedby
pairsofintegers(i, j), 1 ? i < m, 1 < j < n, ratherthanbyintegers1 < i < n.
This is the natural representation
for vectorsdescribingtwo-dimensional
objects,such as images.
The marginalsof v = fvij: 1 < i < m, 1 <j < n} E Rmn are the vectors
E Rm, v = (v,..
=(i1,.. . ., v
(2.12)
v=
n
j=1
..,
vn
vij,
E
R, where
m
V
E
i=l
vij.
As before,we considerthreechoicesof our basic set S, namely,S
Rmn or Amn
=
Rmnor
For any v = {vij} E S, we denoteby L, the subspaceof S consistingof
thosevectorsthathave the same marginalsas v, thatis,
(2.13)
We say thatv
(2.14)
Lv
=
=
(w: w = v; w = v}.
{vij} is of sum or productformif
vij = si + tj or vij = s t,
respectively.
DEFINITION 7. A selection rule H with basic set S as above
is composition-consistent,
more precisely,sum- or product-consistent,
if
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AXIOMATICAPPROACHTO INFERENCE
2043
A projecHI(L,) = v wheneverv E S is of sum or productform,respectively.
if its compotion rule is composition-consistent
(sum-or product-consistent)
wheneveru itselfis of sum or
nentselectionrules fl( Iu) have thisproperty
productform,respectively.
the vectorsv = {vij, 1 < i < m, 1 < j < n} are interpreted
Intuitively,
as
oftwointeracting
compositions
the individualcomponents
components,
being
represented
by the marginalsv and v. Then Lv beingthe feasibleset means
thattheavailableinformation
but nothing
specifiesthe individualcomponents
else. The postulateofcomposition
formalizes
theintuitiverequireconsistency
mentthaton the basis ofsuch information
we shouldinfer"no interaction,"
unless a prior guess is available that impliesinteraction.Implicitin this
interpretation
is that "no interaction"is represented
by the sum or product
formof v. In particular,for inferring
probabilitymass functions,product
is a hardlyavoidablepostulate.It could be argued,thoughwith
consistency
less weight,that productconsistency
is an appropriateaxiomalso forimage
that is, that if the marginalsof an image were knownand
reconstruction,
nothingelse (an unlikelysituationin practice),the "best" reconstruction
wouldbe ofproductform.In inverseproblemswithouta positivity
constraint,
the sum formofv appearsto be the naturaldescription
of "no interaction,"
suggestingthat in this case the postulateof sum consistencyshould be
adopted.
REMARKS. (i) Product-consistentselection rules cannot exist for S = Rmn
because in that case different
elementsof S, each of productform,can have
thesame marginals.On theotherhand,sum-consistent
selectionrules,satisfyingthecontinuity
axiomin Definition
2, cannotexistforS = R + n or S = Amn
To see this,considera sequencev(k) of elementsof S of sum formv(*)-*t
wees1t=
s(k) >O>
suhtaS(k)
S (k) +t (k)
> jh0,, such
t5(k)O,
-_>
that SI -*+( SSi' t(k)
i
tj, wheres, = t, = 0
I(
*j Ij)Si*
and si > 0 fori > 1, tj > 0 forj > 1. Then v(k) - v and L(k) - Lv, where
selectionrulewe shouldhave
vij = si + tj. Thus fora sum-consistent
fl(Lv) = limfl(Lv(k))
k-
= lim v(k) = v
k-
a contradiction
becausev i S.
(ii) In the case S = Rmn,everyLv as in (2.13) containsan elementofsum
form;hence fora sum-consistent
selectionrule II, fl(Lv) is alwaysof sum
form.It follows,subjectto regularity,
thatv? = v0(01)is ofsumform,because
=
v? II(Lvo) by the consistency
axiom. Similarly,in the cases S = Rmn or
if
a
is
HI
product-consistent
(regular)selectionrule,then v0(HI) is of
Amn
productform.
(iii) For thecase S = Amn, thepostulateofproductconsistency
is similarto
but weakerthan Shore and Johnson's(1980) "systemindependence"postulate.
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2044
I. CSISZAR
3. Statement of results. Recallthatourbasic set S is eitherRn, R n or
An,the familyofaffinesubspacesof S is denotedby /, and V denotesR, R +
or theinterval(0, 1), accordingto thechoiceof S. Throughout
thissection,we
assumethat n 2 3 (if S = Rn or R+) or n ? 5 (if S = An).
The following
willfacilitatethe statementofour results.
terminology
DEFINITION 8. A n-tupleoffunctionsf1,. . . ,fn definedon V willbe called
a standardn-tuplewith0 at vo = (vI, .. ., v? E S if:
(i) fi is continuously
differentiable
and vanishestogetherwithits derivativeat v ,i = 1, ..., n,
(ii) in the cases S = Rn or A ,lim oif(V) = -0i =L.
n
(iii) the function
n
(3.1)
F(v)=
E
fi(vi)
i=l
is nonnegativeand strictlyquasiconvexon S, that is, forany v and v' in S
and any 0 < a < 1 we have
(3.2)
F(av + (1 - a)v') < max(F(v), F(v')).
REMARK. The functionsfi in a standardn-tuplemustbe convexif S = Rn
or R n but not necessarilyif S = A\. The proofofthisis omittedin orderto
save space.
(i) For anyregular,local selectionrule [: s-- 5, thereexists
a standardn-tuplef1,.. ., fn with0 at v0 = v0(l) such thatr1 is generated
bythefunctionF in (3.1). Conversely,
any F as in (3.1) generatesa regular,
local selectionrule 1l withv?(JII)= v?.
(ii) For any regular,local projectionrule, thereexistfunctionsfi(vlu),
u e V, v E V, such thatforeveryfixedU = (U1,..., Un)T E S, thefunctions
fi(vIu) forma standardn-tuplewith0 at u and thegivenprojectionruleis
generatedby
THEOREM 1.
(3.3)
n
F(vlu) = E fi(vilui).
i=1
Conversely,
anysuch F generatesa regular,localprojectionrule.
(iii) Two functions F(v)=
E= if(vi),
F(v)=
En1fi(vi)
or F(vlu)=
E Z=lfi(vIui) F(VIU) = E1 fi(viIui) as in (i) or (ii) generatethesame selection or projectionrule, respectively,
if and onlyif fi= cfi,i = 1,... , n, for
someconstantc > 0.
REMARK. In part (iii),the assumptionthat F and F be ofthe statedform
is essential.Otherwise,F(v) and F = 4(F(v)) generatethesame selectionrule
for any strictlyincreasingfunctionFD,and F(vlu) and F = F(F(v1u),u)
generatethe same projectionrule whenever(?(* Iu) is strictlyincreasingfor
everyfixedu E S.
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AXIOMATICAPPROACHTO INFERENCE
2045
THEOREM
2. (i) A projectionrule withbasic set S = Rn or Rn is regular,
local and semisymmetric
(Definition4) ifand onlyifit is generatedbyF(vlu)
as in Theorem1(i) withfi notdependingon i.
(ii) A projectionrule withbasic set S = R or An is statistical(Definition
4) ifand onlyif it is generatedby
(3.4)
F(vlu)
=
E ui fI-'-I)
for some continuously
strictlyconvexfunctionf on R+, with
differentiable,
=
=
=
f(l) f'(1) 0 and limt 0 f'(t)
oo.
Functionsofprobability
ofform(3.4), as measuresofdistance
distributions
motivated
byinformation
theory,wereintroduced
by Csiszar(1963) underthe
name f-divergences
[cf. also Csiszar (1967)] and independently
by Ali and
Silvey(1966). For theirapplicationsin statistics,see, forexample,Liese and
Vajda (1987). The conditionson thederivative
of f are notpartoftheoriginal
definition
of f-divergences.
Notice that the conditionf'(1) = 0 is essential
onlyin the case S = R+ because if S = AnX
thenany f(t) and f(t) = f(t) +
c(t - 1) definethe same F in (3.4). The conditionlimt 0 f'(t)= -o is
necessaryforF in (3.4) to generatea projection
rule,thatis, to ensurethatits
minimumin v be attainedon everyL E Y.
THEOREM
3. (i) For any regular,local and subspace-transitive
projection
rule (Definition6), thereexists a standard n-tuple(P1, , fpnsuch that
?(V)= E=p1,i(vi) is strictlyconvexon S and the givenprojectionrule is
generatedby
F(vlu)
(3.5)
=
?(v)
n
E
i=l
- ?(u)
((v)
-
- (grad 1(u))T
i(Ui)
-
(v
-
u)
0(Up(iu)(V-ui)).
(ii) AnyF as in part (i) generatesa regular,local and parallel-transitive
projectionrule.
On accountofthistheorem,in the sequel we neednotdistinguish
between
the twokindsoftransitivity.
COROLLARY.The only transitivestatisticalprojectionrule (with S
An)
is theI-divergence
projectionrule (cf. Example2).
=
Rn or
The class of measuresof distanceassociatedas in Theorem3 withstrictly
convex functions'F (not necessarilyof a sum form)was introducedby
'F under
Bregman(1967). He developedan iterativealgorithm
forminimizing
linear(and, moregenerally,
underlinearinequality)constraints,
whosesteps
involvedprojectionsin the sense of the distancecorresponding
to 'P [cf.also
Censorand Lent (1981)].
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I. CSISZAR
2046
Bregman'sdivergencesinclude squared Euclidean distanceand I-divergence,and sharewiththesethe property
(3.6)
F(vlu) + F(wlv) = F(w1u) if [l(Llu) = v,
w E L.
Notice that for squared Euclidean distance,(3.6) is just the Pythagorean
also has thisPythagorean
property
playsa
theorem.The factthat I-divergence
keyrolein its applicationsin statistics[cf.Kullback(1959)].
A resultrelatedto but notdirectly
comparablewithTheorem3 was recently
thepreviousresultsofJones
obtainedbyJonesand Byrne(1990),generalizing
(1989). They showedthat amongthe continuousanalogs of the distancesof
form(3.3), onlythe analogsofthosein Theorem3 satisfieda postulatecalled
projectivity.
That postulateis stronglymotivatedfromthe pointof view of
inference,
but it also involvesthe distanceitself,as opposedto transitivity
which is a propertyof the projectionrule alone. Jones and Byrne(1990)
pointedout that theirdistancessatisfied(3.6), and, in fact,that (3.6) was
equivalentto the projectivity
postulate.
THEOREM4. (i) A projectionrule withbasic set S = Rn is regular,local,
transitive
and bothlocationand scale-invariant
ifand onlyifit is a weighted
least squares projectionrule (cf. Example 1). The above propertiesand
semisymmetry
uniquelycharacterize
theleastsquaresprojectionrule.
(ii) A projectionrulewithbasic setS = Rn+is regular,local, transitive
and
scale-invariantif and only if it is generatedby F(vlu)= ELz alha(vilhu4
wherea1, ... , an are positive constants,a < 1, and
v
v log-u
(3.7)
ha(VIU)
v + u,
if = 1,
u
v
logg-+---1,
1
_(ua
a
v
if a =0,
u
-
va)
+ Ua
l(v
-
u),
if 0 < a < 1 or a < 0.
A projectionrulewiththeaboveproperties
ifand onlyif
is also semisymmetric
it is generatedby
(3.8)
Fa(vlu) =
n
E
i=1
ha(vilui),
a < 1.
REMARK. The one-parameter
family(3.8) containstwo well-knowndistances: I-divergence (a = 1) and Itakura and Saito (1968) distance (a = 0).
RecentresultsofJonesand Trutzer(1989) indicatethat(continuousversions
of) the membersof this familywith a = 1/m (m some integer)may be
a tradipreferable
to the Itakura-Saitodistancein spectrumreconstruction,
tionalfieldofapplicationofthe latter.
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AXIOMATICAPPROACHTO INFERENCE
2047
ofthe least squaresand I-diverOur last resultprovidesa characterization
genceselectionand projectionrulesthat,insteadofinvarianceor transitivity,
relyon composition
consistency
(Definition
7). We emphasizethatthischaracterizationalso appliesto individualselectionrules,ratherthan to projection
rules only.On the otherhand,we have to assume thatthe basic set S is as
the inferenceproblemin question
describedbeforeDefinition7. Intuitively,
shouldrelateto objectswith(at least) twocomponents.
THEOREM5.
Let S be as in Definition7, S = Rmnor R+n or Ain' with
m > 2, n > 2, and if S = Amn, m + n ? 5.
(i) In the case S = Rmnn,the regular, local, sum-consistentselection and
projectionrulesare exactlythoseleastsquaresselectionrulesforwhichv0 is of
sum form,and theleast squaresprojectionrule,respectively
(cf. Example 1).
(ii) In the cases S = R mnor A
the regular, local and product-consistent
selectionrulesfor
selectionand projectionrulesare exactlythoseI-divergence
whichv? is ofproductform,and theI-divergence
projectionrule,respectively
(cf. Example 2).
(i) For S = Rmnn,thestandardleast squares selectionrule is
theuniqueregular,local, sum-consistent
selectionruleforwhichv0(H) = 0.
(ii) For S = Almn'themaximumentropy
selectionrule is theunique regular, local, product-consistent
selectionruleforwhichv0(Lt) is the "uniform
distribution"
(1/mn)l. The same holdsalso forS = R+ n ifthelast condition
is replacedby vo(H) = (1/e)1.
COROLLARY.
4. Basic lemmas.
LEMMA1. For a regularselectionrule H: S
, for everyL' E Y of
dimension less than n - 1 (if S = Rn or Rn) or less than n - 2 (if S = A
thereexists L E X/ [cf. (2.3)] such that L' c L and H(L') = [1(L).
-
COROLLARY
. For a regularH and L' E .,
[1(L) impliesL' c L unless H(L') = v0(H).
L E X#,theequality[I(L')
=
PROOF. Clearly, it sufficesto prove that if dim L' = d with 0 < d < n - 1
(or 0 < d < n - 2), then thereexistsan L e .. such that L D L', dimL =
d + 1, H(L') = H(L). Further,insteadofthe last equality,it suffices
to show
that [(L) E L' becausethis,bytheconsistency
axiomin Definition
2, already
implies (L') = [(L).
L1 D L' and supposethat [1(Ll) t L'.
Now, pick any (d + 1)-dimensional
Then we will "rotate" L1 to obtaina family{Lt: 0 < t < 2} and show that
1(Ld) E L' forsome t. To this end, pick any vo 0 L1 in S, set v, = [M(L,) and
let v2 be such that some interiorpointof the segment[vO,v2] is in L'. Set
vt = (1 - t)vo + tv1 if 0 < t < 1 and vt = (2 - t)v1 + (t - 1)v2 if 1 < t < 2,
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2048
I. CSISZAR
and let Lt denotethe subspaceof S spannedby L' and vt.Then Lo = L2 and
Ltn Lt2= L' if0< t1<t2 <2.
By the continuity
axiom,{I1(Lt): 0 < t < 21 is a continuousclosedcurvein
the subspaceL spannedby L1 and vo (t = 0 and t = 2 representing
the same
point).For ? > 0 sufficiently
small,fl(L1) and fl(Ll+E)-which are arbitrarily close to 11(L1)= vl-are separatedby L1 withinL; hence thereexists
some t with It- 11> E forwhich fl(Lt) E L1. Then fl(Lt) E Lt n L, = L'
and theproofofLemma 1 is complete.
The corollaryis immediate,because for L E X containingL' such that
axiomimpliesL = L. El
ME(L')= 11(L),the distinctness
LEMMA2. The restriction
ofa regularselectionrule Vt:.-* S to X'\ Y0
is a homeomorphism
ontoS \ {v?}, wherevo = vo(H) and Y0 - {L: vo E LI.
PROOF. ApplyingLemma 1 to the zero-dimensional
subspace L' = {v}, it
followsthatforeach v E S thereexistsL EX-#suchthat11(L) = v. Ifv 0 v?,
thenL 4 ?, bytheconsistency
axiom.Thus 11maps X#\f ? ontoS \ {v0}.
This mappingis one-to-one
and continuousbythe distinctness
and continuity
axioms.It remainsto provethe continuity
of the inverse.In otherwords,we
have to showthat U(Lk) --+H(L) #v? impliesLk-- L.
We do this by showingthat everysubsequenceof {Lk* containsa subseto L. Write
quenceconverging
L k J{v: akv =bk
{v: aTv = b; 1Tv1if=
ifS =Rn orS =R+,
S =A
[cf.(2.3)]. Here we may supposethat Ilakll= 1 and in the case S = An also
thatak ? 1.
Now, any subsequenceof {L*} containsa subsequence{Lkj such that
by asa, say. Writefl(Ld) = Vk, fl(L) = v*, aTv* = b. As Vkv*
aki
= ak.Vk
that
b.
This
a
means
that
if
a
aTv=
sumption,
implies
aki
bki
the set
-*
LJ[{v:aTv-b},
ifS=R
orRn,
(v: &Tv= b, 1Tv = 1}, if S = An,
is in
we have Lk.
L. But L EX,# holds because (i) L ? 0 (namely,
= 1 and,if S = An, also a ? 1.
We have provedthat everysubsequenceof {L*) containsa convergent
subsequenceLk. -+ L. Bythecontinuity
axiom,here1l(L) = limi lH(Lk.)=
11(L) # v?, and hence necessarilyL = L by the distinctnessaxiom. This
completestheproofofLemma2. oJ
[email protected],
-*
v*~E L bythe definition
of b) and (ii) IlaI
We will need the following
notation,fork < n: The vectorsin Rk whose
are all 0 or all 1, willbe denotedby Ok and lk, respectively
components
(thus
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o=
on, 1 = In). Two vectorsin
a a, iffforsome A # 0,
2049
will be called equivalent,denotedby
Rk
if S =Rn orS =RRn+
if S = An-
{Aa+
XAa+ ,k, AE R,
Observethatin the representation
L
(4.1)
)
-ftv:av
=
t{v: aTv blTV
ifS=R
orR+,
fS = A
= 1}
ofa subspaceL E X1,thevectora E Rn is determined
up to equivalence,and
it is notequivalentto 0.
LEMMA3. Let H: ..t-* S be a regular, local selectionrule, and let
therelationintroducedin thepassage containing(2.10).
*-*ij
be
(i) Let L fe, [(L) = v* * v? = vo(H). Then vi* ij vj* if and only if
(4.1) of L; further,in the cases S = Rn or
ai - aj in the representation
S =R+n, v = v ifand onlyifa =0.
(ii) Let L and L be both in Xt' with [(L) # v?, [(L) 0 vo, and let
= Hj(L) impliesaj
J c{1, ..., n}. Then[j(L)
iaj fora and -a representL
ing and L as in (4.1). In otherwords,aj is determined
by jII(L) up to
equivalence.
COROLLARY.If vi +>ij
also vi +ik
Vj
and vj *->jk Vk for some {i, j, k) c (1, .. ., n}, then
= An thatvi + Vj + Vk < 1.
Vk, providedin thecase S
PROOF. We will show that for L as in (i), arbitraryJ = {j1,
{1, . . ., n}, a E Rk, and for
*, iv=
a
-avTv
(4.2)
L
a={
aTvj*
i
J
l
lTvVjc),
iv
ifS =Rn
,ik} C
orR+,
=
if S = 'An)
the following
holds:aj _a is a sufficient
conditionfor
(4.3)
[j(L)
=
[rj(L')
and thisconditionis also necessaryunlessaj To provethis,set
(4.4)
L"
=
L n {V: vjc
?k.
= VJc.
Then,by locality,Hj(L') = r1i(L"). Since,of course,[ljc(L") = V5c, it follows
that(4.3) holdsifand onlyif [(L") = v*.
Now, if aj
a, then L" c L. By the consistency
axiom,the latterimplies
H(L") = v*. Thus aj
a is sufficient
for(4.3), as claimed.
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I. CSISZAR
2050
if(4.3) and therefore
Conversely,
[(L") = v* hold,the corollaryofLemma
1 yieldsL" c L. Comparing(4.1), (4.3) and (4.4), L" c L meansthat aTvi =
= aivj
(subjectto the additionalconstraintE jeJVj =
aTvj impliesaiv
v if S = An). Clearly,this implicationholds iffthe two conditions
E iO
representthe same constraint,
exceptforthe case a1 0?k. This provesthat
for
indeed
unlessajk.?
a"
is
(4.3)
necessary
aj,
To proveassertion(i) of the lemma,supposefirstthat S = R' or S = R+
and applythe resultjust provedto J = {i}, a = 0. Then L' in (4.2) equals S
and (4.3) meansthat vi = vo.Thus we obtainthat ai = 0 impliesvi = v,0and
if ai # 0, then vi*= vO cannot hold. Next, apply our resultto
conversely,
J = {i, j}, a = (1, 1)T. Then L' in (4.2) equals the specialsubspaceLij(t) (with
t = V* + VJ) thatdefines
therelation+-*ij [cf.(2.10)].Hence(4.3) is equivaa = (1,1)T, except when S = R'
lent to v*
v Now if ai = aj, then a
VJ.
or S = R+ and ai = a =O. Thus we have that ai = aj impliesvO*-*jV,
because the mentionedexceptionalcase has alreadybeen covered.
a; hence v*
VJ
if ai # aj, thenneitheraj - 02 nor ai
Conversely,
i
cannothold.
To proveassertion(ii), supposethat IIj(L) = lJ4(L)= vj and considerL'
as in (4.2) with a = &j. Then ljl(L)
= Hj(L)
= HII(L'), by assumption and
the sufficient
conditionfor(4.3). This implies,bythenecessarycondition,
that
when
L
L
Since
the
roles
of
and
except
perhaps
0k'
are
aj ~aj,
aj
we similarlyhave aj
when
is
symmetric,
not
equivalent
to
?k'
aj
aj
whereasthe same trivially
holdswhenbothaj ?k and aJOk.
To provethe corollary,
pick any v* E S with vi*= vi, Vj*= V, vk = Vk. If
#
v* v?, the hypothesesvi ij vj and vj **jk Vk implyby Lemmas2 and 3(i)
that v* = 11(L) for some L as in (4.1) with ai = aj = ak. This, in turn,
implies[again,byLemma3(i)] that Vi jik Vk whereasthelatteris obvious(by
the consistency
axiom)ifv* = v?.
LEMMA4. (i) Let Fij(u, v), i s j, {i, j) c {1, . . ., n), n > 3, be real-valued
functionsdefinedforu E Ai, v c Aj, whereA, ..., An are arbitrary
sets. If
theseFij satisfythefunctionalequations
(4.5)
Fi(u,
v)Fjk(V, w) = Fik(u, w),
Fij(u, v)Fji(v, u) = 1,
thenthereexistfunctions
gi definedon Ai, i = 1,... , n, such thatforeveryi
andj,
(4.6)
Fij1u,v)
=
gi(u)
(ii) Let n 2 4 and letBij c Ai x Aj, Cijkc Ai x Ai x Ak be setssuch that
(1) (u, v)E Bij if and onlyif (v,u) E Bji; (2) (u, v,w) C Cijk impliesthat
(u, v) E Bij, (u, w) E Bik, (v, w) E B3k; (3) forany distincti, j, k,I and any
uEAi, veAj, wEAk thereexists s e Al with (u, s) e Bil, (v, s) e Bjl,
(w, s) E Bkl; and (4) for any distincti, j, k,I and any u, v,s, s' with
(u,s),(u,s') in BiI and (v,s),(v,s') in Bjl, there exists w EAk with
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2051
(u, w,s), (u, w,s') in Cikl and (s, w, v),(s', w,v) in Clki. Thenifthefunctions
Fij are definedon thesetsB i and theequations(4.5) holdfor(u, v,w) E Cijk
theconclusionofpart (i) stillholds.
PROOF. (i) Fix some k and w~E Ak, and writegi(u) = FW(u, W) fori # k.
Then gi(u) # 0 bythesecondpartof(4.5), and thefirstpartof(4.5) gives(4.6)
if i, j are bothdifferent
fromk. In addition,(4.5) impliesthat
gi (u)
g,(v)
(V
Fjk
, W )
Fi k (U,
W )
foreveryi * k, j * k, u E A,, vE Ai. Denotingthe commonvalue of these
quotientsby gk(W),we obtain(4.6) also forj = k. Finally,the validityof(4.6)
forthe remainingcase i = k followsfromthe secondequationin (4.5).
(ii) On account of (i) it sufficesto provethat the functionsFij can be
extendedfromBij to Ai x Ai so thattheequations(4.5) remainvalid.To this
end,givenu E Ai, v E Aj, pickarbitrarily
1 = k (bothdifferent
fromi, j) and
s and s' in Al such as in hypothesis(4); choose w E Ak accordingto that
hypothesis.
Then,applyingthe firstand thenthe secondpartof(4.5), we get
Fil(U,
s)F1j(s,
v)
= Fik(U,
W)Fkl(W,
S)Flk(S,
= Fik(U,
W)Fkj(w,
v)
V)
W)Fkj(W,
and similarly
Fil(U,
s
( S%v)
)Flj
=
Fik(U,
v)
W)Fkj(W,
This meansthat
Fij( u, v) = Fil( u, s) F,j(s, v)
(4.7)
is welldefinedbecausetheright-hand
sidedoes notdependon I and s [subject
to (u, s) E Bii, (v, s) E Bjl, thelatterbeingequivalentto (s, v) E B,j]. Clearly,
(4.7) definesan extensionof Fij to Ai x Ai. To see thattheseextensionsFj
satisfythe functional
equations(4.5) foreveryu E Ai, v E Aj, wE Ak, let l
be different
fromi, j, k and pick s E Al accordingto hypothesis
(3). Then by
(4.7) and the secondpartof(4.5)
Fij(u,
V)Fjk(V,
w)
=
=
Fij(u,
v) Fji(u
Fil(U,
s)F,j(s,
v)Fjl(v,
FEA(u,
S)Flk(s,
W)
s) Flj(s,
v) Fjl(v,
I v) = Fil(u,
= Fil(u, s)F,i(s, u)
s)Flk(s,
= Fk(U,
=
w)
W),
s) Fli(s,
u)
1.
5. Proof of the main results.
PROOFOF THEOREM 1. (i) Let H:
S be a regular,local selectionrule,
and let v? = (v9,... , v2)T= v'(H1). By Lemma 2, foreveryv / v? there exists
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I. CSISZAR
2052
L(v) dependscontinua unique L = L(v) E X suchthatfl(L) = v; moreover,
on
v.
Write
L
=
L(v)
as
ously
(5.1)
(5.1)
~L
w aT
{w:
S= R'or R',
=aTV,if
aTw
= aTv; lTW = 1},
if S = A".
up to equivalence(cf.thepassagebefore
Here thevectora E R' is determined
Lemma3).
Our firstclaimis that thereexistcontinuousfunctionsgi(v), v e V, with
(5.1) of L = L(v) (for
gi(v) = 0, i = 1,...,n, such thatin the representation
arbitrary
v #v?)
a - (91(V J
(5.2)
gn(Vn))
For laterreference,
observethat(5.2) impliesby Lemma3(i) that
Vi ij vj ifand onlyif gi(vi) =gj(vj)
(5.3)
in the case S = An that vi + vj < 1.
providing
To proveour firstclaim,we startwiththe simplercases S = Rn or Rn. By
up to
Lemma3(i), appliedto J = {i, j}, it followsthat(a1, aj )T is determined
equivalenceby vi and vj. Thus
(5.4)
Fij(vi,vj)
=
of vi and vj wheneverv; 'vj9[which,by
is a well-defined
continuousfunction
(5.4)
Lemma3(i), is necessaryand sufficient
foraj # 0]. Clearly,thefunctions
equations
satisfythe functional
Fij(Vi
Vj)Fjk(Vj,
Vk) = Fik(Vi, Uk),
Fij(vi,vj)Fji(v,
vi) = 1
forvi E V\ {v9), i = 1,. . ., n (recallthatV = R or R+ accordingas S = Rn
or Rn). It followsby Lemma4 that
(5.5)
Fij(vi,vj) = gi(vi)
if vi #v10,vj #vjo,for suitable functionsdefinedand not equal to 0 on
V \ {v(0.Lettinggi(vo)= 0, (5.5) also holdsforvi = vo?wheneverFij(vi,vj) is
thatis, vj 0 vu?.
defined,
Comparing(5.4) and (5.5), we obtain(5.2). The functionsgi are continuous
becausethe Fij are such.
case S = An, applyLemma 3(i) with J=
Turningto the more difficult
{i, j, l} to obtainfora E Rn in (5.1) that(ai, aj, ai)T is uniquelydetermined,
up to equivalence,by vi,vj,vI. Hence
a
(5.6)
Fijl(vi, vj,v1) = a, -- a,
a,
is a well-defined
continuousfunctionof (vi,vj,v,) subjectto vi + vj+ v, < 1
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AXIOMATICAPPROACHTO INFERENCE
2053
foraj = a,, by Lemma
exceptforvj *j v1[whichis necessaryand sufficient
3(i)].
Clearly,the functions
(5.6) satisfythe functional
equations
(5.7)
if VI + V
(5.8)
(5.9)
(5.10)
Fijl(Vi
+ Vk + Vl <1,
Vk) VI) = Fikl(Vi) Vk, Vl)
vI
vjlV)Fjkl(vjy
as wellas
Fijl(vi, Vj,vl)Fjil(vj, vi, vl)
Fijl(vi v,
vj)Fjli(
,
vjv, vvi)Fij( vl,Vij)
+ Fij(vi,vl,vj)
Fij(vi,vj,vl)
=
1,
= -1,
= 1
if vi + vj + v1< 1, assumingin each case thatall functions
are defined.
These functional
equationscan be solvedapplyingLemma4(i) threetimes.
First,we use (5.7) and (5.8) fixingI and v1and restricting
the domainofthe
Fij1's-as functionsof vi and vj-by excludingvi *il vl, that is, Fil = 0.
Thenthehypotheses
ofLemma4(i) are easilychecked,takingintoaccountfor
hypotheses
(3) and (4) that(forfixedvl) the relationv >il vl neverholdsif v
is sufficiently
small;the latterfollowsfromthe continuity
axiomin Definition
2. Lemma4 gives
(5.11)
G-1(vi, vl)
Fijl(vi vj,vl) = G1('v)
for suitable functionsGil defined(and nonzero) for ui + v, < 1, unless
Vi *4il V1
Substituting
(5.11) into(5.9), we obtain,afterrearranging,
Gji(vj, vi) Glj(vl, vj)
Gij(vi, vj) Gjl(vj,vl)
Gli(vl vi)
'Gil(vi, vl)
NowweapplyLemma4(i) to thefunctions
-Gl1(vl,
sG
V(v)-vl
Hil(
vg VI)
Hi1(vi,
vi)
(definedfor vi + v, < 1, unless vi il vl), yielding
h vi)
((
.Hi1(vi,v1) = h1v1
forsuitablefunctionshi defined(and nonzero)in V= (0, 1). This meansthat
the functions
G 1(v ,v1)
Gil(v)
vl)
=
Gjl(vi, v/)
h (v1)
satisfy
(5.12)
Gil(vi, vl)
=
Gli(vl vi)
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2054
I. CSISZAR
and,by(5.11),
(5.13)
Fijl(vi,vj,vj)
=
G"(vi
)
At this point we removethe temporaryexclusionof vi ji, v1 fromthe
ofthe domainof Fijl; defining
definition
Gil(vi,vl) = 0 if vi 'a v1,(5.13) will
alwaysholdwheneverFijl in (5.6) is defined.
Finally,substituting
(5.13) into (5.10) gives,afterrearranging,
using also
(5.12),
(5.14)
vj) = Gij(vi,vj),
G6i(vi,
vi) + G1j(v1,
whenevervj + v + v1< 1. Moreexactly,the givenderivation
of(5.14) is valid
unless Vj -jl v1and a similarderivation
from(5.10), withtherolesof i and j
interchanged,
is validunlessvi*->il vl; ifbothvi*->il v1 and v3 >l vl,then
also vi ij vj and (5.14) holdstrivially.
(5.14) and (5.12) mean that Lemma 4(i) is applicableto the functions
Fij = expGij. It followsthat
G7ij(vi,vi) = gi(vi) - gi(vi),
forsuitablefunctionsgi definedon V = (0, 1).
SinceGij(vi,vj) = 0 if vi +->ijvj andthus,in particular,
if vi = v0,vj =
here gi(v ) is independentof i, and we may assume that gj(v0) = 0, i =
1,... I n.
Substituting
(5.15) into(5.13), we obtain
(5.15)
(5.16)
Fijl(vi, vjvl) = g(vi) -- g1(v1)
g1(v3) g1(v1)
whenevertheleft-hand
sideis defined.
As thefunctionsFijl are continuous,
so
are the gi's, too.
Comparing(5.16) with(5.6), we obtain(5.2).
Havingestablishedour firstclaim,we define
v
(5.17)
fi(v)
=
f g(t) dt,
v?
n
F(v)
=
E
i=l
fi(vi).
We willshowthat f1,..., fn forma standardn-tuple[noticethatproperty
(i)
in Definition
8 obviouslyholds]and that F(v) generatesII.
With(5.17), (5.2) becomesa - gradF(v). As the vectora in (5.1) is determinedup to equivalenceonly,it followsthat(forarbitrary
v 0 vo)L(v) is the
set ofall w E S satisfying
(5.18)
(gradF(v) )T(W
- V) =
0.
Observenextthat forany L E -t we have by the consistency
axiomand
Lemma1 thatfl(L) = v ifand onlyifv e L c L(v), providing
v # v?, whereas
ifv? E L, thenalways[1(L) = v?. Thus we mayassertwithoutanyrestriction
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2055
on v and forarbitraryL E t, that [1(L) = v if and onlyifv E L and (5.18)
fulfilled
ifv = vo).
holdsforeveryw E L (the latterbeingautomatically
For any distinctv and w in S satisfying
(5.18), the result in the last
paragraphappliedto the line L' throughv and w givesthat
(grad F(w))T(w - v) # 0;
herewe used the factthatthe difference
ofanytwoelementsof L' is a scalar
multipleofw - v.
By continuity,
this nonzeroinnerproductmust be of constantsign for
w 0 v satisfying
(5.18),whenv E S is fixed.Further,againbycontinuity,
this
sign cannotactuallydependon v, either.Withoutrestricting
generality,
we
may assume that this constantsign is positive.Indeed,the functionsgi in
(5.17) maybe multiplied
by(-1) ifnecessary,withoutchangingtheirproperties assertedin our firstclaim.
We have obtainedthat(5.18) withw 0 v alwaysimplies
(grad F(w))T(w - v) > 0.
(5.19)
It followsthat forany line L' in S, F(w) is strictlyincreasingas w moves
away fromv = 11(L') in eitherdirection.This immediately
givesthat F has
property
(iii) in Definition
8.
Further,foranyLEYE, thefactthat F(w) is strictly
increasingas w moves
awayfromv = 11(L) on anyline L' c L provesthat H(L) is the uniquepoint
whereF is minimizedoverL. Thus fl is generatedby F.
To provethat (ii) in Definition8 also holds,noticefirstthat as v = v?
trivially
satisfies(5.18) foreveryw e S, (5.19) gives
(grad F(w)j)T(W- v?) > 0 forallw # v' in S.
(5.20)
IfS = R' or A, let v = ( vl,... , Qn)" be a boundarypointof S such that
=
s 0 if j 0 i, further,fj'(Vj) f/(01)
v forsome j and 1 different
vi 0 and VQ
fromi. (5.20) impliesthat
liminf(grad F(w)"( v w
-r
)>0;
V
hence f'(v) is bounded from above as v -- 0. Thus, if the assertion
= -mowere false, fi' would have a finitelimiton a suitable
limv 0/f7'(v)
sequence of positivenumbersapproaching0. Then therewould exist a sequence vk -e v (withvk e S) such that gradF(vk) -- a, say, wherea is not
equivalentto 0. This providesthe desiredcontradiction
becausethe sequence
ofsubspaces
Lk =
(w: (gradF(vk)j)T(w-vk)
=
0, wES}
Es X
convergesto
L-={w:
and the sequence[(Lk)
aT(w
=
-_
)
=
0, w e
S} E,
g
to 11(L), forit goesto v
Vk does notconverge
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S.
I. CSISZAR
2056
This completesthe proofof the assertionsmade in the passage containing
(5.17).
Finally,let fl,..., fnbe any standardn-tuplewith0 at v0 = (v?,.. .,
and set F(v) = E 1fi(vi). Notice firstthat in the cases S = Rn or Rn
property
(iii) in Definition8 impliesthat fi(v) is strictly
increasing(decreasthelatter,
ing)forv > v? (v < v) and also that fi(v) Xc as v -- oo.To verify
that fi(v) - c < ooas v xc, choosevi> vf', > v) with
supposeindirectly
fi(vi)+ fj(vj) = c, and apply(3.2) to v = (v,..., vn)T withv, = vI for1 # i, j
and v' = (vi, ... V,v)T withv' = v' for1 # i. It followsthat
(5.21)
fi(avi + (1 - a)v') + fj(avj + (1 - a)vJ) < f-(vi) + fj(vj),
foreveryvi > v? and 0 < a < 1. Lettingfirstvi -> and then a -* 0, (5.21)
resultsin thecontradiction
c < fi(vi).One verifies
in the samewaythatin the
case S = R , fi(v) > ooalso as v or An, F(v) has a limit
. If S =
(finiteor + cm)as v convergesto a boundarypointof S, becauseproperty
(ii) in
Definition8 impliesthe monotonicity
of each fi near 0. It followsthat F,
extendedby continuityto the closure of S if S = R+ or A attains its
minimumon any L e /, moreexactly,if S = R n or An,on the closureof L.
(ii) rules out the minimumbeingattainedon the boundary.By
But property
property
(iii),the pointwhereF attainsits minimumoverL mustbe unique;
hence F does generatea selectionrule [I. Property(i) impliesthatthis HI is
regularand it is obviouslylocal.
(ii) Let us be givena regular,local projectionrule withcomponentprojection rules fl( Iu), u E S. For arbitrary
v ? u, let L(vlu) denotethe unique
L E X#withrl(LIu) = v. We claimthatthe following
modified
versionofour
firstclaimin the proofofpart(i) is valid:
There existfunctionsgi(vIu), u, v E V, continuousin v and vanishingfor
v = u, i = 1,.. , n, such that in the representation(5.1) of L = L(vlu) we
have
T
U
a - (g(V1 UI 1),
(5.22)
gn(Vn I n))
In the proofofpart(i), the functionF generating
the selectionrule LI was
constructed
fromfunctionsgi withthe property
(5.2). If (5.22) is established,
the functionsgi(vlui) therecan be takenas functionsgi(v) corresponding
to
fl = H(I Iu). Then [cf.(5.17)] Hf(Iu) willbe generatedby
(5.23)
n
F(vlu) = E fi(vilui),
i=1
fi(vilui)=
f
v.
ui
gi(vlui)dv
as a functionofv. This means,by definition,
thatthe givenprojectionrule is
generatedby F(vlu).
Thus it suffices
to provethe claimabout(5.22). This can be donealongthe
linesofpart(i); hencewe onlysketchthe proof,forthe hardcase S = An .
We needan obviousmodification
ofLemma3, namelythatforL as in (4.1)
with rl(Llu) = v # u, (a) (vilui) ij (vjluj) iffai = aj and (b) the vectoraj
is determined
by uj and vj up to equivalence.
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Applying
thisto J = {i, j, 1),it followsthatthe functions
(5.24)
Fijl(vi,vj,vljui,uj, ul) = a,
-
a1
are well definedand continuousin vi,vj,v1 subjectto ui + u; + u < 1, vi +
vj + v, < 1, exceptwhen(vjIuj) -jl (vilu,).
The functions(5.24) satisfyfunctionalequations similarto (5.7)-(5.10),
each variablevi in thelatterbeingreplacedbya pairofvariablesvi,u i, where
witheach constrainton sums of variablesvi a similarconstrainton sums of
variables u is imposed.This systemof functionalequationscan be solved
similarlyto (5.7)-(5.10), again applyingLemma4(u)threetimes.It is convein Lemma4 werenot requiredto be
nientthatthe variablesofthe functions
reals; presentlywe have to let them stand forpairs of real numbersv,u.
of the functions(5.24) analogousto
Finally,we arriveat a representation
(5.16), namely,
(5.25)
gi(vilui) -- g1(v1ul)
riJ~U~UJ,U1tt~U1,It1
Fiji
(vi,ij, VIlgi,X j,Xal) =
g1(v1iu1)
=gj(vjtu1)
part
Comparing
(5.24) and (5.25) provesthedesiredrelation(5.22) and thereby
(ii) ofTheorem1.
(iii) Suppose that F(v) = E ,7 1fi(vi) and F(v) =E f(vi) generatethe
same selectionrule,where(f1,. , fn) and ( f,..., fn) are standardn-tuples.
to prove fi = cfi forthecase when( fl,. . ., fn) is arbitrary,
Clearly,it suffices
with0 at vo = (v?,.. ., v')T, say,and (
to the (regufn) is constructed
lar,local) selectionrule 11generatedby F as in theproofofpart(i) [cf.(5.17)].
Now,since F generatesII, its minimumon L = L(v) is achievedat the point
v. Hence foreveryv + v' we have (gradF(v))T(w - v) = 0 forall w E L(v).
Since L(v) is the set ofall w E S satisfying
(5.18), it followsthatforv + v',
(5.26)
(5.26)
gradF(v)
=A(v)gradF(v)
gTad
A(v)gTadF(v) ifS
ifS = Rn orRn+,
F(v) =
gradF(v) = A(v)gradF(v) + i(v)1 if S = An;
the same holdstrivially
also forv = v?, withj,(v?) = 0 in the case S = An.
As thecomponents
ofthegradientvectorsdependonlyon thecorresponding
componentsof v, the scalar functionsA and ,u in (5.26) and (5.27) mustbe
constantand, in particular,,u in (5.27) is identically
0. Thus we actuallyhave
(5.27)
gradF(v) = c gradF(v).
Since f1,..., fn and f1,..., f, have the property
(i) in Definition8, (5.28)
rules
impliesthat fi = cfi,as claimed.The proofofassertion(iii) forprojection
is similar.El
(5.28)
PROOF OF THEOREM 2. By theproofofTheorem1(i), anyregularand local
projectionrule is generatedby a functionas in (5.23), where gi(vlu) is a
continuousfunctionof v thatvanishesat v = u, i = 1,. . . , n; moreover,
for
anyu # v thesubspaceL definedby(5.1) withai = gi(viIu1)has theproperty
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I. CSISZAR
2058
of Lemma 3(i) used in that proof,it
that Il(Llu) = v. By the modification
followsthat
(vilui) ij (vjluj) ifandonlyif gi(vilui) =gj(vjluj)
(5.29)
providedin the case S = An that ui + uj < 1, vi + vj < 1.
ruleis semisymmetric,
Now,if S = Rn or Ri+ and thegivenprojection
that
is, (vlu) jj (vyu) foreveryu and v in V and everyi, j E {1, ... , n}, (5.29)
impliesthatthe functionsgi do not dependon i; hence,by (5.23), neitherdo
the functionsfi.
Further,if S = R n or Ai and thegivenprojection
ruleis statistical,
thatis,
(vyu) * i1(v'lu')ifand onlyif v/u = v'/u',(5.29) impliesthat
(5.30)
(5 30)
g~vJ~ = g(~u)if~~gi(vilui)
gi(vilui)
i
V.
Vi
providedin the case S = An that ui + uj < 1, vi + vj < 1. Actually,the last
constraintcan be dispensedwith, because for any given u1,1U
jv , vj [in
V = (O,1)] thereexist Uk, Vk such that ui + Uk < 1, vi + Vk < 1, uj + Uk < 1,
Vj + Vk < 1 and vj/ui = Vk/U. (5.30) means that gi(vlu) is a one-to-one
functionof v/u, notdependingon i, thatis,
gj(vlu) = g
(5.31)
).
The continuityof gi as a functionof v and the one-to-onepropertyof g
implies that g(t) is a continuous,strictlymonotonicfunctionof t, and
g(l) = gi(u lu) = 0. Substituting
(5.31) into(5.26) gives
(5.32)
= Uif
fi(vijui) = |dv
)
f (t)
=
fg(s) ds.
This completesthe proofofTheorem2. o
PROOFOF THEOREM3. (i) First,we showthatifa regularselectionruleis
thenfordistinctelementsu, v,w of S,
subspace-transitive,
(5.33)
L(vlu) n L(wlv) c L(wlu)
ifwEE L(vju).
Here,as in theproofofTheorem1(i), L(vlu) denotestheunique L E X. with
H(LIu)= v.
Indeed, let L' = L(vlu) n L(wiv). Then [I(L'iv) = w by the consistency
axiom,and the transitivity
postulateappliedto L' c L = L(vlu) gives
fl(L'u)
= [l(L'Iv)
= w.
But Hl(L'lu)= w impliesthat L' c L(wlu), by the corollaryof Lemma 1,
proving(5.33).
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2059
Now let us be givena regular,local projectionrule,generatedby F(vlu) as
in (5.23). In particular,
(5.34)
gi(vlu)
=
-fi(vju)
dv
is a continuousfunctionof v, vanishingat v = u. Then L(vlu) consistsof
thosew E S thatsatisfy
n
(5.35)
E
i-l
gi(vilui)(wi - vi) = 0.
Hence,ifthe givenprojection
ruleis subspace-transitive,
it followsfrom(5.33)
thatforany distinctu, v,w in S satisfying
there
existscalars a, ,l, y,
(5.35),
possiblydependingon u, v,w, withy = 0 unless S = An, suchthat
(5.36)
i - 1,..., n.
agi(uilui) + fgi(wi(vi) + y = gi(wilui),
We claimthatactually
(5 37)
gj(vlu) + gj(wlv)= gj(wlu),
foreveryu, v,w in V and i
1,.. . , n. Clearly,it sufficesto provethis for
=
i = 1.
Considerfirstthe simplercases S = Rn or R+ . Then forany u, v,w in V,
thereexistu,v,win S satisfying(5.35)
suchthat u1 = u, v1 = v, w1 = w and,
in addition,
=
(5.38)
U2 = V2 + W29
U3 ? V3 W3.
Withthese u,v,w, (5.38) impliesthat in (5.36), wherenow y- 0, we have
a = f = 1 [usingthat,by Lemma3, gi(vlu) # 0 if v + u]. This proves(5.37)
fori = 1.
If S = An then v # u does not necessarilyimplygi(vlu) / 0. It follows,
however,from(iii) in Definition
8, thatforat mostone index i can u < 1/2
and an intervalI c (0, 1/2) be foundsuch that fi(vlu)is constantforv E I.
This implies(assuming,withoutany loss ofgenerality,
thatthe exceptionali,
if any, is different
from2 and 3) that for any 8 < 1/2, the numbersE
satisfying
(5.39)
g2(Ej8) + 0,
g3(cJ)
#
0
are densein theinterval(0, 1/2). Usingthis,it is easyto see thatforanyfixed
v and w sufficiently
close to v, thereexist u, v, w in S = An satisfying
(5.35), such that u1 = u, v1= v, w1 = w and, withsome E and 8 satisfying
(5.39),
ui,
(5.40)
U2 = V2 = U3
W2 = V3 = W3 =?
and
(5.41)
U4 = V4 = W4.
Withtheseu, v,w, (5.41) impliesthatin (5.36) we have y = 0, then(5.40) and
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2060
I. CSISZAR
(5.39) implythat a = f3= 1. This proves(5.37) (fori = 1) if w is sufficiently
closeto v.
Observenext that givenany u and v in V = (0,1), the numbersw E V
satisfying
(5.37) fori = 1 forman openset. Indeed,bythe last paragraph,for
closeto w we have
w' sufficiently
gl(w'lw) + gl(wlv) = g1(w'v),
g1(w'Jw)+ gl(wlu) = gl(w'Iu);
it followsthatif w satisfies(5.37) fori = 1, thenso does w'.
of w, the(nonvoid)set
Since gl(wlu) and gl(wlv) are continuousfunctions
of those w E V that satisfy(5.37) fori = 1 can be open onlyif it equals the
wholeV = (0, 1). This completesthe proofof(5.37) in the case S = An
The functionalequations(5.37) implythat the functionsgi can be representedas
(5.42)
gi(VIU) = iV(V) - i(u),
whereit may be assumed that Ofi(s)= 0, i = 1,... , n, forsome s E V [set
i(v) = gi(slv), say]. Since gi(vlu) is a continuousfunctionof v, i mustbe
continuous,too.Writing
(5.43)
(Pi(v)=
v
i(t) dt,
the functions'Pi .. ., (Pn satisfy(i) in Definition
8, withvo = s 1, and (in the
cases S = R n or An) the validityof (ii) in Definition8 forthe functions'pi
followsfromthat forgif(lu), by (5.42). Property(iii) for (Pl,
Pn will,of
of C4(v)= E ' 1=
that
we
course,followfromthe strictconvexity
are going
li(vd)
to verify
immediately.
From(5.34), (5.42) and (5.43) we get
(5.44)
fi(vIu) =]
v
gi(tlu) dt = fi(v) - fi(u) - M(u)(v - u).
This provesthat F(vlu)
=
E=
fi(v(Iu ) has the claimedform
F(vlu) = ?F(v) - ?F(u) - (grad?(u) ) T (V - U).
Since F(vlu) 2 0, withequalityifand onlyifv = u, thisresultalso provesthe
strictconvexity
of CFon S.
(ii) Let (P1
'Pn be a standardn-tuplesuch that C(v) = E=>pi(vi) is
convexon S, and let fi(vlu)be definedby(5.43) where 0i((u)= f(u).
strictly
Then foranyfixedu E S, the functionsfif(Iun) forma standardn-tuplewith
0 at u; hencebyTheorem1(ii),F(vlu) = E. lfi(vilui) generates
a regular,
local projectionrule. To provethat the latteris parallel-transitive,
suppose
that v = H(LIu), w = H[('Iv), where L and L' are "parallel" subspaces.
Then gradF(vlu) and gradF(wlv) are both orthogonalto these subspaces
(wherethe gradientrefersto the firstvariable)and henceso is
gradF(wlu) = gradF(vlu) + gradF(wlv).
This meansthat l(L'Iu) = w, as claimed.
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2061
To provethe corollary,
recallthatbyTheorem2, everystatisticalprojection
ruleis generatedby an f-divergence
F(vlu)
z
f Ui
where f(t) is a continuouslydifferentiable
functionwith f(1) = f'(1) = 0.
Then the functionsgi(vlu) in (5.34) do not dependon i and are equal to
g(v/u), whereg(t) = f'(t). If thisprojectionrule is transitive,
we musthave
g()q
= 1(v) -q(u)
[cf.(5.42)] forsome continuousfunctionq. This impliesthat g satisfiesthe
functional
equation
g(ts) =g(t) +g(s),
t,s eR+,
whose onlycontinuoussolutionsare g(t) = c logt [cf.Aczel (1966), Section
2.1.2]. Hence f(t) = Jtg(t)dt = c(tlogt - t) and this means that F(vlu) =
cI(vIIu). o
(5.45)
PROOFOF THEOREM4. (i) First we show that a (regular, local) projection
rulewithbasic set S = R', generatedby
(5.46)
n
F(vlu)
=
E
i=l
fi(vilui)
as in Theorem1(i), is translation-invariant
ifand onlyif
(5.47)
fi(v + Alu + A) = c(u) fi(vlu),
foreveryu, v and ,Ain R, wherec(,u) is a suitablepositive-valued
function.
Indeed,sincev* = fl(L + ullu + Al) minimizesF(vlu + Al) subjectto v E
L + ,ul, v* - ,ul minimizesF,(vlu) = F(v + ,Illu + Al) subject to v E L.
Hence the givenprojectionrule is translation-invariant,
that is, Il(Llu) =
v* - 4l, if and onlyif this projectionrule is also generatedby Fu(vIu).The
latteris, by Theorem1(iii),equivalentto (5.47), as claimed.
By Theorem3, if(5.46) generatesa tranlsitive
projection
rule,then
(5.48)
fi(vIu) = gPi(v)-si(u)
- PJ(u)(v - u),
where(,P.. ., Spn) is a standardn-tuple.By the paragraphcontaining(5.43),
we mayassumewithoutanyloss ofgenerality
thatthisstandardn-tuplehas 0
at v? = 0. Our nextgoal is to determine
whatfunctionsfi(vlu) ofform(5.48)
satisfy(5.47).
Observethat (5.47) implies c(GL1+ tL2) = c(t1)c(A2) and that (5.47) and
(5.48) implythe continuity
of c(,u).It followsthat
c(,u) = el
[cf.Aczel(1966), Section2.1.2].
(5.49)
forsomef3e R
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I. CSISZAR
2062
From(5.47) (with,u = -u) and (5.49), we obtain
(5.50)
= ePufi(v
fi(vIu)
ulO);
-
thisand (5.48)-where, by assumption,(pi(O)= fp(O)= 0- resultin
(5.51)
fi(vlu)
=
e-
a).
-
i(V
by v, it followsthat
Comparing(5.48) and (5.51) and differentiating
(p'(v) - (p'(u) = epuSp(v - u).
(5.52)
This means, substitutingv = u + t, that ifr= < satisfiesthe functionalequa-
tion
(5.53)
f(u + t) = qf(u) + ePUif,(t).
This functional
equationis solvedeasily.First,
(5.54)
+lr(t)= at if,B= 0
[cf.Aczel (1966), Section 2.1.2]. For f3 0
0, observethat (5.53) impliesby
symmetry
+fi(u) + eiSulk(t) = +(t) + ept`O(u);
thus
ept - 1
+f(t)
q_(u) =eBU
-
1
This gives
1) if,B=kO.
Thus (p' mustbe eitherofform(5.54) or ofform(5.55), wheretheconstant
factora (but not ,B)maydependon i. Clearly,the positivity
or negativity
of
these factors,accordingas p 2 0 or p < 0, is necessaryand sufficient
for
gettinga standardn-tuple(Pi,. I n
fortranslationinvariFinally,just as (5.47) was necessaryand sufficient
if
ance,one sees thatthe projectionrulegeneratedby(5.46) is scale-invariant
and onlyif
(t
(5.55)
a(ept
-
(5.56)
fi(AvlAu) = c(A) fi(vIu),
foreveryA > 0. Now, fi(vlIu)definedby(5.48) with'p' ofform(5.55) does not
satisfy(5.56), whereaswith'p ofform(5.54) it does. In the lattercase (5.46)
becomes
n
(5.57)
F(vju)
=
E
i=l
ai(vi - ui)2,
ai > O,i
=
1,...,n,
thatis, onlythe weightedleast squaresprojectionrulesare transitiveas well
as location-and scale-invariant.
impliesthatall
By Theorem2(i), the additionalpostulateofsemisymmetry
in
coefficients
(5.57)
must
be
equal.
ai
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AXIOMATICAPPROACHTO INFERENCE
2063
(ii) Supposenowthat S = R+ and a projection
rulegeneratedbya function
and scale-invariant.
Then(5.48) and (5.56) hold;in the
as in (5.46) is transitive
former
we nowsupposethatthestandardn-tuple(pl, .. ., 9n)has 0 at vo = 1.
As (5.56) impliesthat c(AlA2)= c(A1)c(A2)and (5.48) and (5.56) implythe
of c(A), we have c(A) = Aa forsome a e R [Acz6l(1966), Section
continuity
2.1.2]. Hence from(5.56) (with A = l/u) and (5.48) [wherenow 'i(l)=
po(1) = 0], we obtain
(5.58)
fi(Vju) = Oaf - 1i) = U'9i
Comparing(5.48) and (5.58) and differentiating
by v, it followsthat
= u
q'i(U)
-V)
fi
u
v = tu,
or,substituting
(5.59)
p'i(tu)
=
0'i(u) + U-'p,(t).
Since (5.59) means that p(t) = '9(et) satisfiesthe functionalequation(5.53)
(with ( = a - 1), we obtain from(5.54) and (5.55)
(5.60)
'p4(t)
failog t,
a,=(ta-
if a =1,
- 1),
if a 0 1.
Clearly,(5.60) with (pi(l) = 0 definesa standardn-tuplefor S = Rn iff
a < 1 and, in addition,a > 0 in the case a = 1 or ai < 0 in the case a < 1.
Withthis choiceof pi,(5.48) becomes fi(vlu) = lailha(Vlu),with ha defined
by (3.7). On the otherhand,(5.46) withthese functionsfi does generatea
transitiveand scale-invariant
projectionrule. If this projectionrule is also
it followsby Theorem2(i) that it is generatedby (3.8). The
semisymmetric,
proofis complete.0
PROOFOF THEOREM5. Supposethat S Rmn,Rn or Ain the elements
of S beingrepresentedas v= {vij), i = 1,...,m, j = 1, . . ., n. Let H be a
regular,local selectionrulewithbasic set S.
By Theorem1, HIis generatedby a function
(5.61)
F(v) =
m
n
E E fii(Vii)
i=lj=1
wherethe functionsfij forma standardmn-tuplewith 0 at vo {v0} =
v?(11).In particular,
the functionsgij(t) = (d/dt) fj(t) are continuousand
-
(5.62)
fij(V&?)
=
gij
(v9j)
=
0.
Considerthe subspaces Lv definedby (2.13), that is, Lv= {w: w =v;
= v}. Then if HI(Lv) = v forsomev E S, thatis, ifthe minimum
of(5.61)
on Lv is attainedat v, we have
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I. CSISZAR
2064
Ai), . It followsfrom(5.63) that
forsuitable"Lagrangemultipliers"
(5.64)
+ gkl(Vkl)
gij(Vij)
=
+ gkj(vkj),
gil(vil)
forevery i,j,k,1.
then
Nowwe proceedto provepart(i) ofTheorem5. If fl is sum-consistent,
by Definition7, rl(L,) = v always holds ifv is of sum form,vij = si + tj. Thus
(5.64) givesthe systemoffunctional
equations
(5.65)
gij(si + tj) + gkl(sk + tl) = gil(si + tl) + gkj(Sk + tj).
of the (continuous)
Observefirstthat (5.65) impliesthe differentiability
bothsidesof(5.65)
can
for
this
be
functions
seen,
example,
by
integrating
gij;
withrespectto Sk*
bothsides of(5.64) by tj on the one hand and by Sk on the
Differentiating
other,we obtain
i'j(si
+ tj) = gkj(Sk
+ tj)
= gjl(Sk
+ tl).
This meansthat g' equals the same constantc foreveryi, j; hence
gij(v) = cv + dij.
(5.66)
Recalling(5.62), it followsthat
(5.67)
gjj(v)
= c(v - V?)g
=
fij(V)
_
(u
V)2.
it mustbe a least squares selection
This provesthat if 17is sum-consistent,
rule. Recall that,as remarkedafterDefinition7, vo mustbe of sum form.
it is easy to see thatthe least squares selectionruleswithvo of
Conversely,
sum formare sum-consistent.
The resultjust provedeasilyimpliesthatthe only(regular,local) sum-consistentprojectionrule is the least squares projectionrule. In fact,let this
projectionrulebe generatedby
m
F(vlu)
n
=
i=l j=1
f
fij(VijUij),
wherethe functionsfij( luij) forma standardn-tuplewith0 at u forevery
fixedu = (uijs E S. Then (5.67) givesthatif u is of sum form,the termsof
thisstandardmn-tuple
mustbe constantmultiplesof(v - u ij)2. Sinceforany
fixed u there exists u = {uij} of sum formwith uij = u, it followsthat
fij(vlu) alwaysequals a constanttimes(v - u)2.
The proofof part (ii) is similar.If fl generatedby (5.61) is product-consistent,thenby Definition
8, [l(L,) = v alwaysholdsifv is ofproductform,
=
Thus
(5.64)
gives
vij sitj.
+ gkl(Sktl)
+ 9kj(Skt;)(5.68)
=gil(Sitl)
9ij(Sitj)
Noticethatwhereasin thecase S = Amn theconditionE vij = 1 represents
on thepermissiblesi and t, in (5.68), the*equationmustcertainly
a constraint
bvai-oanfiei,
j
kg 1*i
si+S
1
t+
<
1.
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2065
to (5.65).
The systemof functionalequations(5.68) can be transformed
Namely,if the functionsgij(v) satisfy(5.68), then gij(v) = gij(ev) satisfy
(5.65). Hence,from(5.66),
gij(v) = gij(log v)
(5.69)
c log v + dij.
Recalling(5.62), it followsthat
gij(v) = clog V[oivlo
O,
fj( v) = cv log- - v + v9j.
(5.70)
This provesthata product-consistent
selectionrulemustbe an I-divergence
selectionrule(cf.Example2) withv? ofproductform.Conversely,
it is easyto
see that these I-divergence
selectionrules are product-consistent.
The assertionthattheonlyproduct-consistent
projection
ruleis the I-divergence
projectionrule followsin the same wayas did its analogin part(i). The corollaryis
immediate.o
REFERENCES
J.(1966). Lectureson FunctionalEquationsand TheirApplications.
ACZfL,
NewYork.
Academic,
ACZEL,J. and DAR6czY,Z. (1975). On Measuresof Information
and Their Characterizations.
NewYork.
Academic,
ALI,S. M. and SILVEY,
S. D. (1966). A generalclass ofcoefficients
ofdivergence
ofone distribution
fromanother.J. Roy.Statist.Soc. Ser. B 28 131-142.
BREGMAN,L. M. (1967). The relaxation
methodoffinding
thecommonpointofconvexsetsand its
applicationto the solutionof problemsin convexprogramming.
U.S.S.R. Comput.
Math. and Math. Phys.7 200-217.
VANCAMPENHOUT,J. M. and COVER,
T. M. (1981). Maximumentropy
and conditional
probability.
IEEE Trans. Inform.TheoryIT-27 483-489.
CENSOR, Y. (1983). Finiteseries-expansion
reconstruction
methods.Proceedings
oftheIEEE 71
409-419.
CENSOR, Y. and LENT, A. (1981). An iterative
row-action
methodforintervalconvexprogramming.
J. Optim.TheoryAppl.34 321-353.
CsiszAR,I. (1963). Eine Informationstheoretische
Ungleichungund ihre Anwendungauf den
Beweis der Ergodizitiit
von Markoffschen
Ketten.Publicationsof theMathematical
Institute
oftheHungarianAcademyofSciences8 85-108.
CsisZAiR,
I. (1967). Information-type
measuresofdifference
ofprobability
distributions
and indirectobservations.
Studia Sci. Math. Hungar.2 299-318.
I. (1984). Sanovproperty,
CsIsZAR,
generalizedI-projection
and a conditional
limittheorem.Ann.
Probab.12 768-793.
CsIsZAR,
I. (1985). An extendedmaximumentropyprincipleand a Bayesianjustification
(with
discussion).In BayesianStatistics2 (J.M. Bernardo,
M. H. DeGroot,D. V. Lindleyand
A. F. M. Smith,eds.) 83-89. North-Holland,
Amsterdam.
CSISZAR, I. and TUSNADY, G. (1984). Information
and alternating
geometry
minimization
procedures.Statist.DecisionsSuppl. 1 205-237.
DEMPSTER, A. D., LAIRD, N. M. and RUBIN, D. B. (1977). Maximum
likelihood
fromincomplete
data
via theEM algorithm.
J. Roy.Statist.Soc. Ser. B 39 1-37.
DIACONIS, P. and ZABELL, S. L. (1982). Updatingsubjective
probability.
J. Amer.Statist.Assoc.
77 831-834.
HERMAN, G.
T. and LENT, A. (1976). Iterativereconstruction
algorithms.
in Biology
Computers
and Medicine6 273-294.
This content downloaded from 152.2.104.17 on Wed, 3 Jul 2013 17:53:26 PM
All use subject to JSTOR Terms and Conditions
2066
I. CSISZAR
telephony
basedon themaximumlikelihood
F. and SAITO,S. (1968). Analysissynthesis
ITAKURA,
Congresson Acoustics(Y. Kohasi,ed.)
method.In ReportsoftheSixthInternational
17-20. Tokyo,Japan.
oftheIEEE 70
entropy
methods.Proceedings
JAYNES,E. T. (1982). On therationaleofmaximum
939-952.
entropyprinciplesfor
theoreticderivationof logarithmic
JONES, L. K. (1989). Approximation
methodto incorporate
ofthemaximum
entropy
inverseproblemsanduniqueextension
SIAM J. Appl. Math.49 650-661.
priorknowledge.
C. L. (1990). Generalentropycriteriafor inverseproblems,with
JONES, L. K. and BYRNE,
and clusteranalysis.IEEE
patternclassification
applicationsto data compression,
Trans.Inform.TheoryIT-36 23-30.
minimum-distance
feasiblehigh-resolution
JONES, L. and TRUTZER, V. (1989). Computationally
method.InverseProblems5 749-766.
whichextendthemaximum-entropy
procedures
Theoryand Statistics.Wiley,NewYork.
KULLBACK, S. (1959). Information
Distances.Teubner,Leipzig.
LIESE, F. and VAJDA,I. (1987). ConvexStatistical
and entropyin incomplete-data
MILLER, M. I. and SNYDER, D. L. (1987). The role of likelihood
and Toeplitzconstrained
intensities
point-process
to estimating
Applications
problems:
oftheIEEE 75 892-907.
covariances.Proceedings
Internaofmaximumentropy.
A. (1990). A noteon theinevitability
J. B. and VENcOvsKA,
PARIS,
tionalJournalofInexactReasoning4 183-223.
in statistical
measuresand its application
ofa set ofprobability
PEREZ, A. (1984). "Barycenter"
decision.Compstat
Lectures154-159.Physica,Heidelberg.
derivationof the principleof maximum
SHORE, J. E. and JOHNSON, R. W. (1980). Axiomatic
entropy.IEEE Trans. Inform.Theory
entropyand the principleof minimum-cross
IT-29 (1983) 942-943.]
IT-26 26-37. [Correction
SKILLING, J. (1988). The axioms of maximumentropy.In MaximumEntropyand Bayesian
Methodsin Scienceand Engineering1 173-187.Kluwer,Amsterdam.
VARDI, Y., SHEPP, L. A. and KAUFMAN, L. (1985). A statisticalmodel for positronemission
tomography
(withdiscussion).J. Amer.Statist.Assoc.80 8-35.
MATHEMATICALINSTITUTE OF THE
HUNGARIANACADEMYOF SCIENCES
BUDAPEST, P.O.B. 127
H-1364
HUNGARY
This content downloaded from 152.2.104.17 on Wed, 3 Jul 2013 17:53:26 PM
All use subject to JSTOR Terms and Conditions