Statistical Decision Theory: Concepts, Methods and Applications. (Special topics in Probabilistic Graphical Models)

Transcription

1 Statstcal Decso Theory: Cocepts, Methods ad Applcatos (Specal topcs Probablstc Graphcal Models) FIRST COMPLETE DRAFT November 30, 003 Supervsor: Professor J. Rosethal STA4000Y Aal Mazumder

2 Part I: Decso Theory Cocepts ad Methods Part I: DECISION THEORY - Cocepts ad Methods Decso theory as the ame would mply s cocered wth the process of makg decsos. The exteso to statstcal decso theory cludes decso makg the presece of statstcal kowledge whch provdes some formato where there s ucertaty. The elemets of decso theory are qute logcal ad eve perhaps tutve. The classcal approach to decso theory facltates the use of sample formato makg fereces about the ukow quattes. Other relevat formato cludes that of the possble cosequeces whch s quatfed by loss ad the pror formato whch arses from statstcal vestgato. The use of Bayesa aalyss statstcal decso theory s atural. Ther ufcato provdes a foudatoal framework for buldg ad solvg decso problems. The basc deas of decso theory ad of decso theoretc methods led themselves to a varety of applcatos ad computatoal ad aalytc advaces. Ths tal part of the report troduces the basc elemets (statstcal) decso theory ad revews some of the basc cocepts of both frequetst statstcs ad Bayesa aalyss. Ths provdes a foudatoal framework for developg the structure of decso problems. The secod secto presets the ma cocepts ad key methods volved decso theory. The last secto of Part I exteds ths to statstcal decso theory that s, decso problems wth some statstcal kowledge about the ukow quattes. Ths provdes a comprehesve overvew of the decso theoretc framework.

3 Part I: Decso Theory Cocepts ad Methods Secto : A Overvew of the Decso Framework: Cocepts & Prelmares Decso theory s cocered wth the problem of makg decsos. The term statstcal decso theory pertas to decso makg the presece of statstcal kowledge, by sheddg lght o some of the ucertates volved the problem. For most of ths report, uless otherwse stated, t may be assumed that these ucertates ca be cosdered to be ukow umercal quattes, deoted by. Decso makg uder ucertaty draws o probablty theory ad graphcal models. Ths report ad more partcularly ths Part focuses o the methodology ad mathematcal ad statstcal cocepts pertet to statstcal decso theory. Ths tal secto presets the decsoal framework ad troduces the otato used to model decso problems. Secto.: Ratoale A decso problem tself s ot complcated to comprehed or descrbe ad ca be smply summarzed wth a few basc elemets. However, before proceedg ay further, t s mportat to ote that ths report focuses o the ratoal decso or choce models based upo dvdual ratoalty. Models of strategc ratoalty (small-group behavor) or compettve ratoalty (market behavor) brach to areas of game theory ad asset prcg theory, respectvely. Thus for the purposes of ths report, these latter models have bee eglected as the terest of study s statstcal decso theory based o dvdual ratoalty. I a covetoal ratoal choce model, dvduals strve to satsfy ther prefereces for the cosequeces of ther actos gve ther belefs about evets, whch are represeted by utlty fuctos ad probablty dstrbutos, ad teractos amog dvduals are govered by equlbrum codtos (Nau, 00[]). Decso models led themselves to a decso makg process whch volves the cosderato of the set of possble actos from whch oe must choose, the crcumstaces that preval ad the cosequeces that result from takg ay gve acto. The optmal decso s to make a choce such a way as to make the cosequeces as favorable as possble. As metoed above, the ucertaty decso makg whch s defed as a ukow quatty,, descrbg the combato of prevalg crcumstaces ad goverg laws, s referred to as the state of ature (Ldgre, 97). If ths state s ukow, t s smple to select the acto accordg to the favorable degree of the cosequeces resultg from the varous actos ad the kow state. However, may real problems ad those most pertet to decso theory, the state of ature s ot completely kow. Sce these stuatos create ambguty ad ucertaty, the cosequeces ad subsequet results become complcated. Decso problems uder ucertaty volve may dverse gredets - loss or ga of moey, securty, satsfacto, etc., (Ldgre, 97). Some of these gredets ca be assessed whle some may be ukow. Nevertheless, order to costruct a mathematcal framework whch to model decso problems, whle provdg a ratoal

4 Part I: Decso Theory Cocepts ad Methods bass for makg decsos, a umercal scale s assumed to measure cosequeces. Because moetary ga s ofte ether a adequate or accurate measure of cosequeces, the oto of utlty s troduced to quatfy prefereces amog varous prospects whch a decso maker may be faced wth. Usually somethg s kow about the state of ature, allowg a cosderato of a set of states as beg admssble (or at least theoretcally so), ad thereby rulg out may that are ot. It s sometmes possble to take measuremets or coduct expermets order to ga more formato about the state. A decso process s referred to as statstcal whe expermets of chace related to the state of ature are performed. The results of such expermets are called data or observatos. These provde a bass for the selecto of a acto defed as a statstcal decso rule. To summarze, the gredets of a decso problem clude (a) a set of avalable actos, (b) a set of admssble states of ature, ad (c) a loss assocated wth each combato of a state f ature ad acto. Whe oly these make up the elemets of a decso problem, the decso problem s referred to as the o-data or wthout expermetato decso problem. However, f (d) observatos from a expermet defed by the state of ature are cluded wth (a) to (c), the the decso problem s kow as a statstcal decso problem. Ths tal overvew of the decso framework allows for a clear presetato of the mathematcal ad statstcal cocepts, otato ad structure volved decso modelg. Secto. The Basc Elemets The prevous secto summarzed the basc elemets of decso problems. For brevty purposes, ths secto wll ot repeat the descrpto of the two types of decso models ad smply state the mathematcal structure assocated wth each elemet. It s assumed that a decso maker ca specfy the followg basc elemets of a decso problem.. Acto Space: A = {a}. The sgle acto s deoted by a a, whle the set of all possble actos s deoted as A. It should be oted that the term actos s used decso lterature stead of decsos. However, they ca be used somewhat terchageably. Thus, a decso maker s to select a sgle acto a Afrom a space of all possble actos.. State Space: Θ = {}. (or Parameter Space) The decso process s affected by the ukow quatty Θ whch sgfes the state of ature. The set of all possble states of ature s deoted by Θ. Thus, a decso maker perceves that a partcular acto a results a correspodg state. 3. Cosequece: C = {c}. 3

5 Part I: Decso Theory Cocepts ad Methods The cosequece of choosg a possble acto ad ts state of ature may be multdmesoal ad ca be mathematcally stated as c( a, ) C. 4. Loss Fucto: l ( a, ) A Θ. The obectves of a decso maker are descrbed as a real-valued loss fucto l ( a, ), whch measures the loss (or egatve utlty) of the cosequece c ( a, ). 5. Famly of Expermets: E = {e}. Typcally expermets are performed to obta further formato about each Θ. A sgle expermet s deoted by a e, whle the set of all possble expermets s deoted as E. Thus, a decso maker may select a sgle expermet e from a famly of potetal expermets whch ca assst determg the mportace of possble actos or decsos. 6. Sample Space: X = {x}. A outcome of a potetal expermet e E s deoted as x X. The mportace of ths outcome was explaed (3) ad hece s ot repeated here. However, t should be oted that whe a statstcal vestgato (such as a expermet) s performed to obta formato about, the subsequet observed outcome x s a radom varable. The set of all possble outcomes s the sample space whle a partcular realzato of X s deoted as x. Notably, X s a subset ofr. 7. Decso Rule: δ ( x) A. If a decso maker s to observe a outcome X = x ad the choose a sutable acto δ ( x) A, the the result s to use the data to mmze the loss l ( δ ( x), ). Sectos ad 3 focus o dscussg the approprate measures of mmzato decso processes. 8. Utlty Evaluato: u(,,, ) o E X A Θ. The quatfcato of a decso maker s prefereces s descrbed by a utlty fucto u( e, x, a, ) whch s assged to a partcular coduct of e, a resultg observed x, choosg a partcular acto a, wth a correspodg. The evaluato of the utlty fucto u takes to accout costs of a expermet as well as cosequeces of the specfc acto whch may be moetary ad/or of other forms. Secto.3 Probablty Measures Statstcal decso theory s based o probablty theory ad utlty theory. Focusg o the former, ths sub-secto presets the elemetary probablty theory used decso processes. The probablty dstrbuto of a radom varable, such as X, whch s 4

6 Part I: Decso Theory Cocepts ad Methods depedet o, as stated above, s deoted as P (E) or P ( X E) where E s a evet. It should also be oted that the radom varable X ca be assumed to be ether cotuous or dscrete. Although, both cases are descrbed here, the maorty of ths report focuses o the dscrete case. Thus, f X s cotuous, the Smlarly, s X s dscrete, the X P ( E) = f ( x ) dx = df ( x ). E E P ) ( E) = f ( x. x E Although, E has bee used to descrbe a famly of expermets, a evet ad wll be used the ext sub-secto to deote expectatos, the meag of E wll be clear from the cotext. For e E (where e represets a expermet or eve a evet), a ot probablty measure P, x (, e) or smply deoted P, x e s assged ad more commoly referred to as the possblty space. Ths s used to determe other probablty measures (Raffa & Schlafer, 000): () The margal measure P ( ) o the state space Θ. Of course, here the assumpto s that P ( ) does o deped o e. () The codtoal measure P x ( e, ) o the sample space X for a gve e ad. () The margal measure P x ( e) o the sample space X for a gve e. (v) The codtoal measure P ( x) o the state space Θ for a gve e ad x. The codto e s ot stated as x s a result of the expermet ad hece the relevat formato s cotaed x. Before cocludg ths sub-secto, t s mportat to make two remarks. The frst s to summarze the three basc methods for assgg the above set of probablty measures. That s (a) f a ot probablty measure s assged toθ X, the the margal ad codtoal measures o Θ ad X ca be computed, (b) f a margal measure (probablty dstrbuto) s assged to Θ ad the codtoal for X, the the ot ca be foud ad smlarly (c) f a margal measure (probablty dstrbuto) s assged to X ad the codtoal for Θ, the the ot ca be determed. These elemetary methods or cocepts of probablty have a more practcal mportace. The secod remark s smply to clarfy that the prme o the probablty measure dcates a pror probablty where as the double prme dcates a posteror probablty. For the most part, these otatos are redudat but at certa tmes wll help to keep thgs clear. Whe t s obvous these superscrpts wll ot requred. A further dscusso of prors s provded Secto.5. 5

7 Part I: Decso Theory Cocepts ad Methods Secto.4 Radom Varables ad Expectatos I may staces, real umbers or -tuples (of umbers) descrbe the states of ature {) ad sample outcomes {x}. I the sectos of ths report a tlde sg may be used to dstgush a radom varable or fucto from a partcular value of the fucto. For example, the radom varables ~ x ad ~ ~ may be used to defe (, x) = ad ~ x (, x) = x, respectvely. Expectatos of radom varables are almost always cosdered ecessary whe dealg wth decso processes such as loss fuctos. The expectato for a gve value of, s defed to be E [ h( X ) = h X x X ( x) f ( x ), h( x) f ( x ), (cotuous) (dscrete) As before, the superscrpts ad subscrpts o the expectato operator wll perform much the same way as for the probablty measure. Whe ecessary, such scrpts wll be mmzed whe the cotext s clear. Thus, wth respect to pots ()-(v) the prevous secto, the otato for the followg expectatos s provded below: ~ () E or E ( ) s take wth respect to P. () E x s take wth respect to P x. () E z e, s take wth respect to P z e,. (v) E s take wth respect to P. x e x e Secto.5 Statstcal Iferece (Classcal versus Bayesa) Statstcal ferece s cosdered here wth the decso framework. Both classcal ad Bayesa perspectves are brefly preseted to show the varyg approaches. Classcal statstcs uses the sample formato to make fereces about the ukow quatty,. These fereces (wth decso theory) are combed wth other relevat formato order to choose the optmal decso. These other relevat formato/sources clude the kowledge of the possble cosequeces of the decsos ad pror formato whch was prevously metoed. The former o-sample formato, cosequeces, ca be quatfed by determg the possble loss curred for each possble decso. The latter, pror formato, s the formato about arsg from other relevat sources such as past expereces. Bayesa aalyss s the approach whch seeks to utlze pror formato (Berger, 985). Ths thrd type of formato s best descrbed terms of a probablty dstrbuto. The symbol π ( ) or smply p( ) wll be used to represet a pror desty of 6

8 Part I: Decso Theory Cocepts ad Methods. Smlar, to the other deftos, uder both the cotuous ad dscrete cases, the pror probablty dstrbutos ca be wrtte as P( E) = E df π ( ) = E π ( ) d( ), (cotuous) E π ( ), (dscrete) The uses of pror probabltes are dscussed the proceedg sectos. Both the o- Bayes ad Bayesa decso theory are dscussed ths report. Secto.6: Covex Sets Varous cocepts preseted throughout ths report make use of the cocepts of covexty ad cocavty; hece, the requred deftos ad propertes are summarzed below. Defto: A lear combato of the form αx + ( α) x wth 0 α s called a covex combato of xad x. (Ldgre, 97) If xad x Ω, the t ca be sad that Ω s covex f the le segmet betwee ay two pots Ω s a subset of Ω. (Berger, 985). Ths defto smply suggests that the set of all combatos of two gve pots s precsely the set of pots whch make up the le segmets otg these two pots. It may be coceved that the values α ad ( α) whch fall betwee 0 ad may be terpreted as probabltes. A covex combato of two pots the becomes a combato of probabltes. Defto: If {, x m x,...} s a sequece of pots R, ad 0 α are umbers such that α = =, the α x = (ad fte) s called a covex combato. The covex hull of a set Ω s the set of all pots whch are covex combatos of pots Ω. (Berger, 985). Fgure -: Ω Ω 7

9 Part I: Decso Theory Cocepts ad Methods The set Ωs covex whle the Ω s ot, for the fgure above. So a covex hull s a set wth o holes ts teror ad o detatos o ts exteror e.g. a egg s covex, whle a doughut or a baaa or a golf ball s ot. Formally, X s a covex set f every le segmet coectg two dstct pots X s wholly cotaed X, ad t s strctly covex f the teror pots of such a le segmet are the strct teror of X. Thus, a strctly covex set has o flat sdes or edges: ts exteror cossts oly of curved surfaces that are bowed outwards. A sphere s strctly covex, whle a cube s covex but ot strctly covex. (Nau, 000). Covex sets play a cetral role the geometrc represetato of prefereces (utlty theory) ad choces (statstcal decso theory), as wll be show. 8

10 Part I: Decso Theory Cocepts ad Methods Secto : No-statstcal Decso Processes Statstcal decso problems clude data or observatos o the ukow quatty,, (also referred to as the state of ature) whch may be used to choose a more optmal decso. Oe approach to hadlg such problems, s to cosder the o-statstcal approach. Smply, ths excludes the data avalable o the state of ature. Ths s reduces the problem to a smpler oe. The basc elemets or gredets of ths o-data problem were preseted the prevous secto. I smple decso problems, the most mportat elemets are the state space {}, the acto space {a} ad the loss fucto l(, a). These aspects form the theoretcal aalyss of decso makg. The state space Θ ad the acto space A are both fte.e. there are oly ftely may actos [states]. The loss curred s assumed to be measured egatve utlty uts where utlty s defed by a fucto defg the prospects of a decso maker. I ths stace, however urealstc, we assume that the loss fucto s kow. These cocepts wll be defed the proceedg secto wth the o-data decso problem costructs. Secto. The Set of Radomzed Actos A smple way to expla the geeral theory of decso makg s to cosder a typcal co toss. Ths troduces a extraeous radom devce whch s useful provdg decso rules that uder some crtera are better tha those that use oly the gve, oradom actos (Ldgre, 97). I geeral, a radom devce (such as a co (ordary or based)) s a expermet of chace havg as may possble outcomes as there are actos from whch to choose; each outcome s assocated wth a acto, ad whe a partcular outcome s observed, the correspodg acto s take. The use of a radom devce to select a acto from the set of possble actos s called a radomzed or mxed acto. Choosg a radomzed acto from amog all possble actos amouts to selectg a radom devce from amog all radom devces that could be used to determe the acto actually take; ad further selectg a set of probabltes for the varous actos. Ths leads to the followg defto of a radom acto. Defto: A radomzed acto, for a problem wth acto space cosstg of actos a, a,, a k, s a probablty vector (p, p,, p k ), that s a sequece of oegatve umbers p whose sum s. To summarze, cosder costructg a expermet of chace producg outcomes z, z,, z k wth probabltes p, p,, p k assged, respectvely. If the outcome of the expermet performed s z the acto a s take. To dstgush betwee pure (or orgal) actos a, a,, a k ad radomzed actos, probablty vectors are used to defe the latter type. A sgular probablty vector ca be used to determe a pure acto. For stace, the acto a ca be equvaletly wrtte as the probablty vector 9

11 Part I: Decso Theory Cocepts ad Methods (0,,0, 0) that assgs all of the probablty mass to acto a. Thus, pure actos costtute a subset of the radomzed actos. Secto. The Loss Fucto The use of a radomzed acto a decso problem wth a gve loss fucto evtably sets the loss to be a radom varable, for each state of ature. The most atural expected loss to cosder whe makg a decso volves the ucertaty. Thus, takg the expected value of the radom varable ( or l(, a) ) measures the cosequece of employg a gve radomzed acto (whe ature s a gve state). I partcular, f the loss of a fucto l(, a) ad the radomzed acto (p, p,, p k ) s used to choose amog the actos a, a,, a k, the expected loss s the followg weghted sum: l, a ) p + l(, a ) l(, a k ) p k. ( Notably, ths ca be wrtte as takg the tegral over all l(, a ) ad p, for the cotuous case. The above defto of expected loss s explaed detal Chapter of Berger s Statstcal Decso Theory ad Bayesa Aalyss (985). I a decso problem wth m states,,, m, there are m L s correspodg to each radomzed acto. For the case of k actos, these L s are the followg expected losses: L L M L = E[ l(, a)] = l(, a ) p = E[ l(, a)] = l(, a ) p = E[ l(, a)] = l(, a ) p m m l(, a ) p l(, a ) p l(, a ) p m k k k k k k defed by the radomzed acto (p, p,, p k ). These relatos ca be wrtte the matrx form L l(, a) l(, ak ) M = p M pk M, Lm l( m, a) l( m, ak ) whch suggests the terpretato of the vector of losses (L, L,, L m ) s a pot m- dmesoal space computed as a covex combato of the pots ( l(, a ),..., l( m, a )), for =,,k. The oto of covex sets was troduced subsecto.6. The latter pots are the loss vectors defed by the pure actos whle those defed by radomzed actos are covex combatos of those defed by the pure actos (Berger, 985) ad (Ldgre, 97). The closg example to ths sub-secto shows that the set of loss pots (L, L,, L m ) defed by all possble radomzed actos (p, p,, p k ) s a covex set m- 0

12 Part I: Decso Theory Cocepts ad Methods dmesoal space as descrbed above. It forms a covex polyhedro where the extreme pots are pure actos, although some pure actos may fall sde the set. The followg example demostrates a two dmesoal case.e. there are two states of ature. Example -: (Ldgre, 97) The followg table descrbes a problem wth fve possble actos, two states of ature ad the respectve loss fucto. a a a 3 a 4 a The pure actos defe the loss vectors (the colums the table) ad the radomzed actos defe the covex set geerated by these fve actos whch s foudg the fgure below. Note that oe acto, a 3 falls wth the covex polyhedro. Fgure -: L a5 a a3 a4 a L Recall from Secto.6 that a covex combato of a set of pots correspods to the ceter of gravty of a system of pot masses at those pots where these masses are proportoal to the weghts the covex combato. Thus, the radomzed acto (p, p,, p k ) yelds a pot (L, L,, L m ), whch s the ceter of gravty of a system of masses wth p uts at a, p uts at a,, ad p k uts at a k. Iterpretg radomzed actos as ceters of gravty helps geometrcally terpret the state. For stace, f oly p ad p are postve, wth o mass at a 3,, a k, the the ceter of gravty (ad therefore the pot (L, L )) must le o the le segmet og a ad a. If these actos are represeted by extreme pots, the such a mxture of ust these two actos les o the edge of the covex set of pots represetg all radomzatos (Ldgre, 97). Thus, for ths example, the acto a 3 results the same losses as a

13 Part I: Decso Theory Cocepts ad Methods certa mxture of a, a ad a 5, or as a mxture volvg oly a, a 4 ad a 5, as well as may other mxtures of a, a, a 4 ad a 5. However, t could ot be obtaed by mxg actos a, a 4 ad a 5. Secto.3 Regret If the state of ature s kow the the acto whch results mmal lost would be take. Thus, f t were kow that were the state of ature, the acto a for whch l(, a) s smallest should be take, ad the mmum loss m = m l(, a), A s a loss that could ot be avoded wth eve the best decso. Suppose, oe takes the acto a, whch does ot produce ths mmum, ad the dscovers that ature s deed state, the decso maker would regret ot havg chose the acto that produces the mmum; the amout of loss that could have saved by kowg that state of ature s called the regret. Regret s defed for each state ad acto a as follows: r(, a ) = l(, a ) m l(, a). So for each state of ature, subtract the mmum loss m from the losses volvg that state to obta the regret. Regret s ofte referred to as opportuty loss ad represets a thkg terms of ga rather tha loss (Ldgre, 97). The ga resultg from takg acto a whe ature s state s the egatve of the loss: g(, a) = l(, a). Hece, the mmum loss s the egatve of the maxmum ga: m l(, a) = max g(, a) ad the regret ca be re-expressed terms of ga as follows: A r, a ) = l(, a ) m l(, a) = max g(, a) g(, a ). ( A A Ths represets the maxmum that could have bee gaed f the state of ature had bee kow, mus the amout that actually was gaed by takg acto a. Example -: (cot d.) (Ldgre, 97) A A

14 Part I: Decso Theory Cocepts ad Methods Recall the example preseted the prevous sub-secto. The table has bee replcated below wth the addto of aother colum represetg the mmum loss over all fve actos. The regret table s provded below ad computed by subtractg the mmum loss from each of the losses ther respectve rows. a a a 3 a 4 a 5 m l(, a) A 0 a a a 3 a 4 a There s at least oe zero each state ad the remag etres are postve the regret table. Geometrcally, the obectve or effect s to traslate the set of pots so that at least oe acto s o each axs. Ths s demostrated the fgure below. Fgure -: L a5 a a3 a4 a L Notce that the whole covex set of radomzed actos shfts alog wth the fve pure actos. Ths s geerally the case because the amout subtracted from each loss s depedet of the acto a, upo whch a probablty dstrbuto s mposed a mxed acto (Ldgre, 97): R = E[ r(, a)] = E[ l(, a)] m l(, a) L m. A Questo: Would t make ay dfferece, studyg a decso problem, f oe used regret stead of loss? I some staces, the regret may be more paful the the losses, depedg o the role of the decso maker ad hs or her stakes. The classcal treatmet of statstcal problems statstcal decso theory, usually results assumg a loss 3

15 Part I: Decso Theory Cocepts ad Methods fucto that s already a regret fucto; however ths s ot always the case ad leads to a dscusso cases where t does make a dfferece. Secto.4 The Mmax Prcple Decso problems preset a dffculty determg the best decso because a acto that s best uder oe state of ature s ot ecessarly the best uder the other states of ature. Although, varous schemes have bee proposed - decso prcples that lead to the selecto of oe or more actos as best accordg to the prcple used oe s uversally accepted. (See Berger, ) By learly orderg the avalable actos, assgg values to each acto accordg to ts desrablty s a frequetst prcple. The mmax prcple places a value o each acto accordg to the worst that ca happe wth that acto. For each acto a, the maxmum loss over the varous possble states of ature: M ( a) = maxl(, a), Θ s determed ad provdes a orderg amog the possble actos (Ldgre, 97 ad Frech & Isua, 000). Takg the acto a for whch the maxmum loss M(a) s a mmum leds tself to the ame mmax. Berger states the same prcple wth the cotext of a decso rule. Ifδ s a radomzed rule the the quatty sup R(, δ ) represets the worst that could happe Θ the decso δ s used. Furthermore, the decso ruleδ s preferred to a ruleδ f sup R(, δ) < sup R(, δ ). Θ Smlarly, a mmax decso rule s a mmax decso rule f t mmzes sup R(, δ ) amog all radomzed rules. Θ Example -: (cot d.) (Ldgre, 97) Aga the loss table s repeated below wth the addto a row statg the maxmum loss for the varous actos. The smallest maxmum loss s determed by acto a to be. Θ maxl(, a) Θ a a a 3 a 4 a

16 Part I: Decso Theory Cocepts ad Methods The table of regrets s produced below ad shows dfferet results the that of the maxmum loss table. The table shows that the mmum maxmum regret s determed by acto a. maxl(, a) Θ a a a 3 a 4 a Recall the questo posed at the ed of the last sub-secto: Questo: Would t make ay dfferece, studyg a decso problem, f oe used regret stead of loss? A graphcal approach of determg the mmax pot-whch s feasble whe there are two states of ature s a useful approachg the prevously stated questo. For a gve acto a wth losses ( L, L ) uder (, ), respectvely, the maxmum of these losses s the frst co-ordate f the pot les below the bsector of the frst quadrat whle f the pot les above that bsector, or 45 le, the maxmum loss s the secod co-ordate L. If two pots le above the bsector, the lower oe has the smaller maxmum; ad f two pots both le below t, the left-most pot has the smaller maxmum. Ths approach ca be geeralzed to may states, but becomes more messy decpherg all the combatos. The graphcal approach to the mmax acto s smply that the mmax process s related to the locato of the org of the co-ordate system, ad that movg the acto pots relatve to the co-ordate system ca alter the process of fdg the mmax pot. Of course, some staces, the mmax loss acto ad the mmax regret acto wll ot dffer. To determe the mmax acto amog the set of all radomzed actos s geerally more complcated, because stead of choosg a acto from a fte set of actos oe must choose a probablty vector from a set of possble probablty vectors that s fte umber (eve f the set of actos s fte). There are two cases that ca be cosdered for brevty sake at ths pot - whe there are ust two states of ature, ad whe there are ust two actos. Whe there are ust two states of ature, a graphcal soluto to the problem of determg a mmax mxed acto ca be carred out by represetg the radomzed actos terms of ( L, L ) ad ( R, R ). Ths ca be show by returg to the example. Example -: (cot d.) (Ldgre, 97) Cotug wth the same example ad lookg at the fgure below, t s clear that ths pot (where the bsector meets the covex polyhedro) les o the segmet og the 5

17 Part I: Decso Theory Cocepts ad Methods pots represetg a ad a, ad so represets a mxture volvg oly those two actos. The pot (x,y) questo s a covex combato of a ad a : x = y 4 p + ( p). 3 0 Moreover, t s the pot o that le segmet wth equal co-ordates: p + 4( p) = x = y = 3p + 0( p), 4 ad equatg these two fuctos of p yelds p =. 5 Fgure -3: L a5 a a3 a4 a L The mmax acto therefore puts 5 4 of the total probablty () at a, 5 at a, ad oe 4 at ay other acto, producg the probablty vector (,,0,0,0). The mmax loss, that s, the smallest maxmum expected loss, s the value obtaed by settg p =, 5 amely, 3 p =. The pot o the graph represetg the mmax actos s thus 5 (, ). Notce that ths mmax expected loss s actually less that the mmum 5 5 maxmum loss acheved whe oly pure actos are admtted, 3, as determed the earler example preseted the prevous secto. The case of two acto s dscussed oly brefly as t ca be determed a smlar fasho as above. The radomzed actos are vectors of the form (p, - p), defed by a sgle varable p. The expected losses uder the varous states of ature ca be computed 6

18 Part I: Decso Theory Cocepts ad Methods as fuctos of p, ad from ths oe ca determe (at least graphcally) the fucto max E[ l(, a)]. The mmum pot of the latter fucto the defes the mmax Θ acto. See both Berger (985) ad Ldgre (97) for examples ad further dscusso. Secto.5 Bayes Solutos The frequetst approach has cosdered the states of ature to be fxed. I cotrast, cosderg cocept of radomess for the state of ature s the Bayes decso prcple. Ths stace cosders ature as radom or ot, evertheless, t s ratoal to corporate to the decso makg process oe s pror huches, covctos, or formato about the state of ature-ad how lkely (whatever that meas) the varous states of ature are to be goverg the stuato (Ldgre, 97). Ths s accomplshed by weghtg the states ad orderg the actos to permt the selecto of a best acto. Thus, f a large loss ca occur for a gve acto whe ature s a state that the decso maker feels s hghly ulkely, the extreme loss s mmzed slghtly by the state of ature that would have produced t. The role of pror probabltes was brefly troduced Secto ad wll be expaded upo here. I a geeral decso problem, a probablty weght g( ) s assged to each state of ature, where these assged weghts are oegatve ad add up to. Such a set of probabltes or weghts s called a pror dstrbuto for. Gve the dstrbuto g ( ), the loss curred for a gve acto a s a radom varable, wth expected value B ( a) = g( ) l(, a). Ths s referred to as the Bayes loss correspodg to acto a (Ldgre, 97). The Bayes acto s the defed to be the acto a that mmzes the Bayes loss B(a). That s, the computato of the expected loss accordg to a gve pror dstrbuto provdes a meas of arragg or orderg the avalable actos o a scale (amely, B(a)) such that the acto farthest to the left o that scale s the most desrable, ad s to be take. Whe radomzed actos are cosdered, a Bayes loss ca be defed as the expectato wth respect to a gve pror dstrbuto of the expected loss (wth respect to the radomzed acto): For a radomzed acto p = ( p, p,..., pk ), whch assgs probablty p to acto a, the expected loss for a gve state s E[ l(, a)] = l(, a ) p, ad the Bayes loss s obtaed by averagg these (for the varous s) wth respect to g ( ) : Ths otato s used so as to dffuse ay cofuso from the probablty vectors assged to each acto. 7

19 Part I: Decso Theory Cocepts ad Methods B ( p ) = g( ) E[ l(, a)] = g( ) l(, a ) p. Sce ths Bayes loss s a fucto of ( p, p,..., pk ), or k varables (p k s determed as soo as p, p,..., p k are specfed), the problem of determg the mmum Bayes loss s that of mmzg a fucto of (k ) varables. Example -: (Ldgre, 97) Cosder the followg loss table where there are ust two states of ature ad two actos. The pror probabltes are gve ad g ( ) = w, g ( ) = - w ad the expected losses are smply calculated as B( a) = 0 w + 6( w) = 6 6w B( a ) = w + 5( w) = 5 4w. B(a) a w a 5 5 4w g ( ) w - w The fgure below llustrates that for ay w to the left of 0.5, the value of w for whch the Bayes losses are equal, the smaller Bayes loss s curred by takg acto a. For ay w to the rght of 0.5, the Bayes acto s a as t yelds the smaller Bayes loss. Whe w = 0.5 the t s rrelevat whether acto a or a s take. Examato of that fgure shows that for a pror dstrbuto defed by a w the rage 0 w < w the Bayes acto s a ; for w < w < w t s a ; for w < w t s a 3. Fgure -4: 6 a a 0 w.5 w 8

20 Part I: Decso Theory Cocepts ad Methods The terpretato of small or large w s of course depedet o the problem; however, t ca be geeralzed that small w refers to a least favorable outcome where as w refers to a more favorable outcome. I geeral, the Bayes losses for the varous actos wll be lear fuctos of w, a problem wth two states of ature ad g ( ) = w, g ( ) = - w. These fuctos of w wll be represeted by straght les, ad for a gve value of w the acto correspodg to the le whose ordate at that w s smallest s the Bayes acto. Now whe cosderg radomzed actos (p, p,, p k ) for actos a, a k, respectvely, produces the expected losses B ( p ) = g( ) l(, a ) p = p g( ) l(, a ). Thus, the value of B(p) s a covex combato of the values of the Bayes losses for the varous pure actos, ad s at least as great as the smallest of those values (Ldgre, 97). Ths meas that there s o ga possble the use of radomzed actos; a pure acto ca always be foud whch yelds the mmum Bayes loss. The result that the Bayes actos are the same usg the regret as usg loss-s evdet for the problem volvg oly two states of ature; however, t s also true geeral. Substtuto of r (, a ) = l (, a ) m l (, a ) to the expresso for expected regret: yelds E[ r(, a = E[ l(, a E[ r(, a )] = r(, a ) g( ) )] = )] l(, a ) g( ) g( ) m l(, a). a g( ) m l(, a) a Thus, E[ r(, a )] dffers from E[ l(, a )] by a term that does ot volve a. The acto that mmzes oe must therefore mmze the other. Ths s llustrated the followg example. Example -3: (Ldgre, 97) Cosder the followg decso problem wth ts correspodg loss table. 9

21 Part I: Decso Theory Cocepts ad Methods a a a The regrets are r(, a) = l(, a) r(, a) = l(, a) The expected regrets, gve pror weghts g ) ad g ), are E[ r(, a E[ r(, a3 )] = 3g( ) + 5g( ) [g( ) + g( )] so that E r(, a )] = E[ l(, a)] [g( ) + g( )]. ( ( E[ r(, a )] = g( ) + 3g( ) [g( ) + g( )] [ )] = 5g( ) + g( ) [g( ) + g( )] Thus the acto that mmzes E[ r(, a )] also mmzes E[ l(, a )] by a dfferece of, rrespectve of the acto. Secto.6 Domace ad Admssblty The cocepts of domace ad admssblty are couterparts to decso rules. Some examples have bee ecoutered whch certa of the avalable actos would ever be used because there are others for whch losses are always less. It s mportat to state some deftos before dscussg ther mportace ay further. Defto: A acto a* (pure or radomzed) s sad to domate a acto a f the loss curred by usg acto a s always at least as great as that curred by usg a: Defto: l(, a) l(, a*), for all. A acto a* (pure or radomzed) s sad to domate strctly a acto a f t domates acto a ad f, addto, there s some state of ature for whch the loss equalty s strct: l(, a) > l(, a*), for some. Defto: 0

22 Part I: Decso Theory Cocepts ad Methods A acto (pure or radomzed) s sad to be admssble f o other acto domates t strctly. The proceedg paragraph was extracted from Ldgre (97). Returg to the case of two states of ature, a acto s represeted as a pot the (L, L ) plae, where L s the loss (or expected loss, the case of a radomzed acto) curred whe that acto s take ad s the state of ature. The fgure below shows pots correspodg to actos a*, a ad a, for whch losses are such that a* domates both a ad a. I the case of a the losses are greater tha for a* for both states of ature; the case of a oly the loss for s strctly greater tha wth a*. Icdetally the acto a* would strctly domate every acto represeted by pots the shaded quadrat (cludg the boudares), wth the excepto of the pot a* tself. (I ths kd of represetato, a acto that domates acto a but does ot strctly domate t would be represeted by the same pot as acto a, sce the losses would be the same.) Fgure -5: L a a* a L If a acto a* domates a acto a there s o eed to leave a the competto for a best acto. If the domace s ot strct, the the losses are the same for a as for a*, ad a ca be dspesed wth; f t s strct, the oe may actually do worse by usg a tha by usg a*. Before edg ths secto, we ca re-terate ths cocept terms of rsk fuctos ad decso rules. I Chapter 5 of Frech ad Isua (000), t s stated that rsk fuctos duce a atural orderg amog decso rule: a decso rule whch performs uformly better terms of rsk tha aother for each value of seems better overall. Let δ ( ) ad δ ( ) be two decso rules. The δ ( ) domates δ ( ) f (, ) (, ),, R δ R δ wth strct equalty. Smlarly, t ca be stated that the δ ( ) ad δ ( ) are equvalet f R δ, ) = R( δ, ),. (

23 Part I: Decso Theory Cocepts ad Methods Defto: A decso rule (or acto) s admssble f there exsts o R-better decso rule where R s referrg to the rsk fucto. A decso rule s admssble s admssble f there exsts a R-better decso rule. The above defto was provded by Berger (985). A acto that s ot admssble s sad to be admssble, ad ca be dspesed wth because there s a acto that does at least as well uder the crcumstaces. O the other had, a acto that s close to the lower left boudary of a set of mxed actos ( the L, L represetato) may ot be so bad as the ame admssble would mply (Ldgre, 97). That s, there are degrees of admssblty whch the termology gores. I statstcal problems, for example, there may be solutos that are slghtly admssble but are preferred for some reaso to those that domate-because of computablty, for stace. Secto.7 Bayesa ad Classcal Approaches Bayes versus Mmax Prcple Ths secto explores the coectos betwee Bayesa ad classcal approaches through ther respectve forms of decso prcples, amely the Bayes ad Mmax Prcple. The latter s essetally based o the rsk fucto R( δ, ) whch duces a partal orderg amog the decso rules 3, leadg to the cocept of admssblty. Sce ths s oly a partal orderg, the Bayesa approach troduces a pror dstrbuto whch may be used to weght the rsk fucto ad orders accordg to Bayes rsk (Frech ad Isua, 000). Much of ths dscusso ceters aroud the cocept of admssblty. From the precedg three sub-sectos, some possble geeraltes ca be made () Bayes solutos are usually admssble; () A mmax acto s a Bayes acto; ad (3) Admssble actos are Bayes, for some pror dstrbuto (Ldgre, 97). These are o-formal statemets; however they are dscussed detal wth proofs 4.8 (Bayes admssblty) ad 5.5 (Mmax admssblty ad Comparso wth Bayes) of Berger (985). (See also Chapter 6 of Frech ad Isua (000). Aga a geometrc represetato of ust two states of ature s structve as the set of all possble (radomzed) actos s represeted the L L plae as a covex set. That a mmax soluto s Bayes for some pror dstrbuto s evdet, for the case of two states of ature, from the geometrcal represetato the L L plae (Ldgre, 97). The fgures below llustrate four cases that may arse. It s mportat to recall here that rsk fucto s the expected loss (as defed by the Frequetst or classcal approach). The rsk fucto s further explored Secto 3. 3 It s mportat to ote that decsos ad actos are sometmes terchageable, especally the o-data decso problems. Thus, although the oto of decso rules has bee brought ths secto t s dscussed more detal the ext secto.

24 Part I: Decso Theory Cocepts ad Methods Fgure.6: (a) The mmax acto occurs o the boudary of the set of actos, ad s a mxture of two pure actos; ths acto would be Bayes for the pror dstrbuto (w, - w) such that w/( w) s equal to the slope of the le through the two pure actos volved. (b) The mmax acto, whch s a pure acto, would be Bayes for ay pror dstrbuto such that w/( w) s a slope betwee the slopes of the le segmets whch meet that pure acto. (c) The pror dstrbuto whch produces the mmax acto as a Bayes acto correspods to the taget le at L = L ; ths kd of set of radomzed actos would oly occur f there are ftely may pure actos at the outset. (d) The mmax acto would be ay that yelds a (L, L ) o the bottom edge of the acto set; these are Bayes for the pror dstrbuto, whch assgs probablty to. 3

25 Part I: Decso Theory Cocepts ad Methods Example -3: (Ldgre, 97) Revstg the problem troduced Secto.5 produces the followg Bayes losses wth gve pror probabltes ( w, w). The loss table has bee repeated for ease. a a a B( a ) = w + 3( w) = 3 w B( a B( a 3 ) = 5w + ( w) = + 4w ) = 3w + 5( w) = 5 w. The graphs of these are show fgure.7 below. The lowest pot of tersecto ad the hghest mmum s at w =. That s, of all the pror dstrbutos that Nature mght 5 choose, the hghest mmum Bayes loss (loss curred by usg the Bayes acto) s 3 acheved for the pror dstrbuto (, ) Fgure.7: 6 0 /5 w The mmax radomzed acto s easly foud (by the graphcal procedure descrbed 4 Secto.4) to be (,,0) wth losses L = L = 3/5. The pot o the le through (, 5 5 3) ad (5, ) wth equal co-ordates: 4 Ths s sad to be a least favorable pror dstrbuto (Ldgre, 97). 4

26 Part I: Decso Theory Cocepts ad Methods L 3 L =, 3 or L = 3 5. A pror dstrbuto that would yeld ths radomzed acto as Bayes acto s defed by a w such that w =, w 3 where -/3 s the slope of the le through (5, ) ad (, 3). Notce that ths w, whch s w = / 5, s precsely the w of the least favorable dstrbuto 5 determed above. A graphcal represetato of ths has ot bee show here but ca be acheved much the same maer descrbed Secto.4 ad.5. The Bayes Perspectve Mathematcally the Bayes approach s ust postulatg a weghtg fucto that provdes a orderg amog the actos; practcally, the decso maker s corporatg to that weghtg fucto persoal prefereces about what that ukow state of ature s lkely to be. The cocept of subectve probablty s dscussed Part C o utlty theory. I the role of ratoal decso makg, the cluso of such persoal prefereces deems reasoable. However, o-bayesas do crtcze ths approach due to ts vew of subectvty. I defece of ths approach, t s ot terrbly sestve; therefore, small accuraces specfcato s ot treacherous. Furthermore, the Bayes solutos are usually admssble. The Mmax Approach The mmax approach, o the other had, s ot early so easy to defed. Berger (985) goes o to say that whe cosdered from a Bayesa vewpot, t s clear that the mmax approach ca be ureasoable. It s pessmstc vew - makg the assumpto that the worst wll happe. Although t s frequetly admssble (ad Bayes for some pror dstrbuto), the dstrbuto s least favorable. A more sgfcat obecto s that the mmax prcple, by cosderg sup R (, δ ), may volate the ratoalty prcples (Berger, 985). Sometmes a mmax soluto ca be computed by determg amog the Bayes solutos oe for whch the losses uder the varous states of ature are equal. That s, f * * p* = ( p,..., pk ) s a radomzed acto that s Bayes wth respect to some pror g( ) ad s such that the (expected) loss fucto 5 The oto of a least favorable dstrbuto gvg rse to the mmax acto as a Bayes acto s geeral, but ot trval to establsh. Ths dea s dscussed more depthly sources o game theory whch s ot cluded ths report. 5

27 Part I: Decso Theory Cocepts ad Methods L(, p *) = E[ l(, a)] = l(, a ) p s costat, the p* s mmax. The developmet follows (Ldgre, 97): The assumpto that p* s Bayes for g( ) meas that for ay radomzed acto p, B( p*) = m B( p) B( p). P But sce L(, p*) s costat, ts mea value wth respect to (wrt) the weghtg g( ) s ust that costat value: B ( p*) = E[ L(, p*)] = L(, p*). O the other had, sce B(p) s the expected value of L (, p), whch caot exceed the maxmum value of L(, p) B( p) max L(, p) ad t follows for ay p that L(, p*) max L(, p). Smlarly, the costat L(, p*) caot exceed the smallest of these maxma L(, p*) m max L(, p), where the rght sde of ths equalty s the mmax loss. P Sce L(, p*) s defed to be costat, t s equal to ts maxmum value, whch tur s greater tha or equal to the smallest such maxmum: L(, p*) = max L(, p*) m max L(, p). Ad the because L(, p*) s ether less tha or greater tha the mmax loss, t must be equal to t-ad ths meas that p* s a mmax soluto, as was asserted. Ths ca also be vsually easg wth ust two states of ature. Ldgre (985) has descrbed ths pheomeo more depth both algebracally ad graphcally. A smlar proof ca be foud Berger (985) ad s dscussed bref Frech ad Isua (000). Before closg ths secto, I would lke to make oe ote. Two of three key decso rules have bee dscussed here. The thrd, the Ivarace Prcple, has ot bee cluded a theoretcal dscusso but for completeess s metoed. The Ivarace Prcple, as the ame would have t mply, bascally states that f two problems have detcal formal structures (sample sze, parameter space, destes ad loss fucto), the the same decso rule should be used each problem. A etre chapter Berger s 985 book Statstcal Decso Theory ad Bayesa Aalyss s devoted to ths prcple. P * 6

28 Part I: Decso Theory Cocepts ad Methods Secto 3: Statstcal Decso Processes Statstcal decso problems clude data - meag the results of makg oe or more observatos o some radom quatty that s thought to be tmately related to the state of ature some decso problem (Ldgre, 97). The avalablty of such data provdes some llumato the selecto of a acto such that the state of ature s ot completely ukow. It wll be show that the use of data wll defe procedures whch result a expected loss that s lower tha what would be curred f the data were ot avalable. However, eve the avalablty of data wll ot avod completely the kd of stuato ecoutered the o-data case, whch there s o clear-cut crtero for ratg the varous caddate procedures as a bass for choosg oe of them as best. Secto 3. Data ad the State of Nature To obta data for use makg decsos a approprate expermet of chace should be performed oe whch the state of ature determes the geerato of the data, ad so the probablty dstrbuto for the data depeds o that state of ature. The data of a gve problem may cosst of a sgle umber (value of a radom varable), or a sequece of umbers - usually resultg from performg the same expermet repeatedly, or sometmes a result or results that are ot umercal. I geeral, a radom varable X wll be employed to refer to the data, ad the problems cosdered here X wll deote ether a sgle radom varable, or a sequece of radom varables: (X,,X ). I ay case X wll have certa possble values ad a probablty for each, accordg to the state of ature. Thus, for each value of x of the radom quatty X there s a probablty f ( x; ) P ( X = x) = assged to that value x by the state of ature. (The otato P (E) wll mea the probablty of the evet E whe the state of ature s.) 6 Secto 3. Decso Fuctos Secto troduced the cocept of a decso rule wth the o-data cotext. It s relevace develops ths secto as a procedure for usg data as a ad to decso makg volvg a rule, or set of structos, that assgs oe of the avalable actos to each possble value of the data X. Thus, whe the pertet expermet s performed ad a value of X obtaed, say X = x, a acto has bee assged by the rule to that value, ad 6 It s mportat to realze that order for the data to be of value makg a decso, the depedece of the probablty dstrbuto for X o the state of ature must be kow. That s, f ( x; ) s assumed to be gve or kow. 7

29 Part I: Decso Theory Cocepts ad Methods that acto s take. A decso rule s a fucto a = δ(x), ad s called a decso fucto, or statstcal decso fucto. Berger (985) gves the followg defto. Defto: A (oradomzed) decso rule δ (x) s a fucto from X to A. (It s always assumed that the fuctos are measureable.) If X = x s the observed value of the sample formato, the δ (x) s the acto that wll be take. (Recall for a o-data problem, a decso rule s smply a acto.) Two decso rules, δ ( x ) ad δ ( x), are cosdered equvalet f P ( δ ( X ) = δ ( X )) = for all. Questo: How may dstct rules are there? If there are ust k avalable actos (a, a,, a k ), ad f the data X ca have oe of ust m possble values (x, x,, x m ), the there are precsely k m dstct decso fuctos that ca be specfed. Of the k m possble decso fuctos, some are sesble, some are foolsh; some gore the data, ad some wll use t wrogly (Ldgre, 97). Example 3-: (Ldgre, 97) Cosder the followg table wth 8 decso rules to betwee two actos ad where the observed value X={0,,}. The frst decso rule gores the data as t takes acto a regardless; the last decso rule performs much the same way. The secod decso rule suggests takg acto a s X = 0 or. x δ δ δ δ δ δ δ δ 0 a a a a a a a a a a a a a a a a a a a a a a a a Decso rules ca be radomzed much the way actos are radomzed a extraeous radom devce to choose amog the avalable rules. A radomzed decso fucto s the a probablty dstrbuto over the set of pure decso fuctos, assgg probablty p to decso fucto δ (x). A radomzed decso rule ca be defed alteratvely by attachg a outcome x of the data X (from a expermet), whch selects oe of the actos (a, a,, a k ) accordg to some probablty dstrbuto (g, g,, g k ) (Ldgre, 97). Ths s rule s equvalet to a radomzato of pure decso fuctos, whe the umbers of actos ad possble values of X are fte. The followg defto provded by Berger (985) closes ths part. Defto: A radomzed decso rule δ ( x, ) s, for each x, a probablty dstrbuto o A, wth the terpretato that f x s observed, δ ( x, a) s the probablty that ac acto A wll be chose. (Aga a radomzed decso rule o-data problems s smply referred to as a 8

30 Part I: Decso Theory Cocepts ad Methods radomzed acto.) Noradomzed rules ca be cosdered a specal case of radomzed rules, that they correspod to the radomzed rules whch, for each x, so that a specfc acto chose has probablty oe. Let δ deote the equvalet radomzed rule (at ths tme) for the oradomzed rule δ (x) gve by δ ( x, a) = I Secto 3.3 The Rsk Fucto A fδ ( x) A, ( δ ( x)) = 0 fδ ( x) A. Secto frst troduced the cocept of rsk fucto as that uderstood by the classcal approach. Ths secto defes ad explores ths cocept more explctly. Followg from the prevous sub-secto, whe a gve decso fucto δ(x) s used, the loss curred depeds ot oly o the state of ature that govers t, but also o the value of X that s observed. Sce X s radom, the loss curred ca be restated as, l(, δ ( x)), ad s a radom varable. The frequetst decso-theoretc evaluates, for each, the expected value of loss f such a decso rule δ(x) was used repeatedly wth varyg X the decso problem. Defto: The rsk fucto of a decso rule δ(x) s defed by R(, δ ) = E[ l(, δ ( x))] whch depeds o the state of ature ad o the decso rule δ. (Notably, for a o-data problem, R(, δ ) = L(, a) ). Assumg a dstrbuto for X defed by P ( X = x) = f ( x; ), the rsk fucto would be calculated as R (, δ ) = l(, δ ( x )) f ( x ; ). Or smply stated, the rsk fucto s the weghted average of the varous values of the radom loss. Notce that the depedece of rsk o the state arses because of two facts: the loss for a gve acto depeds o, ad the probablty weghts used computg the expected loss depeds o. Thus, whe the decso s based o observed results of a expermet, a decso rule (pure or radomzed) ca be selected from those avalable, kowg the rsk fucto but gorat of the true state of ature. The problem of selectg a decso rule kowg the rsk fucto s mathematcally exactly the same as that of selectg a acto the 9