Simulation Based Optimal Design

Transcription

1 BAYESIAN STATISTICS 6, pp J. M. Bernard, J. O. Berger, A. P. Dawid and A. F. M. Smith (Eds.) Oxfrd University Press, 1998 Simulatin Based Optimal Design PETER MULLER Duke University SUMMARY We review simulatin based methds in ptimal design. Expected utility maximizatin, i.e., ptimal design, is cncerned with maximizing an integral expressin representing expected utility with respect t sme design parameter. Except in special cases neither the maximizatin nr the integratin can be slved analytically and apprximatins and/r simulatin based methds are needed. On ne hand the integratin prblem is easier t slve than the integratin appearing in psterir inference prblems. This is because the expectatin is with respect t the jint distributin f parameters and data, which typically allws efficient randm variate generatin. On the ther hand, the prblem is difficult because the integratin is embedded in the maximizatin and has t pssibly be evaluated many times fr different design parameters. We discuss fur related strategies: prir simulatin; smthing f Mnte Carl simulatins; Markv chain Mnte Carl (MCMC) simulatin in an augmented prbability mdel; a simulated annealing type apprach. Keywrds: DECISION PROBLEMS; EXPECTED UTILITY MAXIMIZATION; STOCHASTIC OPTIMIZATION.

2 1. INTRODUCTION Optimal design prblems are cncerned with maximizing expected utility in a statistical experiment. The maximizatin is ver sme design parameter. The experiment is defined by a mdel, i.e., a distributin f a vectr f bservables cnditinal n sme unknwn parameter vectr. The mdel may depend n the design parameter, hence the subscript. The mdel is cmpleted by a prir distributin fr the parameter vectr. Utility is a functin!"", fr example negative squared errr lss! # $% $ & " '(, r smething mre prblem specific like! # ttal number f successfully treated patients, etc. Since the design parameter has t be chsen befre bserving the experiment, we need t maximize the expectatin f!) with respect t ". We can frmally state the design prblem as * # +,-. /0 1 " where 1 # 2!"" " 789:7;8< 3 45 =9: 6 >? 1 is the expected utility fr actin. Unless mdel and likelihd are chsen t allw analytic evaluatin f the expected utility integral, the ptimal design prblem requires numerical slutin strategies. Chalner and Verdinelli (1995) and Verdinelli (1992) prvide an extensive review f analytical and apprximate slutins t Bayesian ptimal design prblems, fcusing n the traditinal experimental design questin f chsing cvariates in a regressin prblem, including nn-linear regressin and linear regressin with interest n a nn-linear functin f the parameters. In this paper we explre different, simulatin based strategies. Sectin 2 reviews the basic cncept f Mnte Carl simulatin fr the evaluatin f 1, including a prpsal f brrwing strength acrss simulatins under different designs by smthing thrugh simulated pairs f design and bserved utilities. Sectin 3 appraches the prblem with very different strategies, using a mdel augmentatin which defines an artificial prbability mdel n the triple f design, data and parameters. Simulatin in the augmented mdel is shwn t be equivalent t slving the ptimal design prblem (1). Critical shrtcmings f the prpsed apprach are prblems arising with flat and high dimensinal expected utility surfaces. Sectin 4 prpses an idea reminiscent f simulated annealing which replaces the expected utility surface by a mre peaked surface withut changing the slutin f the ptimal design prblem. 2. PRIOR SIMULATION Except in special prblems, the stchastic ptimizatin prblem (1) des nt allw a clsed frm slutin. Often the utility functin!) is chsen t allw analytic evaluatin even if a mre prblem specific utility/lss functin were available. A typical example is the use f prepsterir variance n sme parameters f interest instead f ttal treatment success in medical decisin prblems. Mre realistic utility functins can be used in simulatin based slutins t the ptimal design prblem. Mst simulatin based methds fr ptimal design are based n the bservatin that the integral in 1 is easily evaluated by Mnte Carl simulatin. In mst prblems " # is available fr efficient randm variate generatin, allwing an apprximatin f 1 by 1 # A? C DEF B!" D"D "

3 where %D"D, #?">>>" A ' is a Mnte Carl sample generated by D, D D. Since the prir prbability mdel typically des nt depend n the chsen design, we d nt use a subscript fr. But n additinal difficulties wuld arise if the prir were t depend n in a given applicatin. Mnte Carl apprximatins f this type are rutinely used in Bayesian ptimal design prblems. In many examples part f the expected utility integral can be slved analytically, leaving Mnte Carl simulatin nly fr the remaining nn-analytic integratin. Sme recent examples in the literature are the fllwing studies. Sun, Tsutakawa and Lu (1996) cnsider ptimal design fr quantal respnses, specifying 1 as expected psterir variance f sme functin f a binary respnse curve. Amngst ther strategies t evaluate 1, they use a cmbinatin f Mnte Carl integratin with respect t and numerical quadrature fr integratin with respect t. Carlin, Kadane and Gelfand (1998) discuss ptimal design in a sequential experiment. The setup allws up t bservatins (data mnitring pints). The design defines a stpping rule which gives at each time a decisin f whether t stp the experiment r cntinue sampling t the next step. Given that the trial is stpped at a certain stage, the terminal decisin prblem f chsing ne f the tw available actins can be slved analytically. The design is parameterized in terms f a vectr f upper and lwer bunds which decide cntinuatin f the experiment at each step by specifying cntinuatin if the psterir mean n the parameter f interest falls between the bunds, and terminatin therwise. Mnte Carl simulatin frm " is used t evaluate expected utility fr a given design. In the cntext f the sequential design this strategy is referred t as frward simulatin. A mre traditinal methd t evaluate sequential designs uses backward inductin (chapter 12, DeGrt 1970). Vlachs and Gelfand (1996) use Mnte Carl simulatin t evaluate the cntinuatin risk in a backward inductin fr a sequential design prblem. If at the time f the decisin sme data is already available then the decisin needs t cnditin n. The prir is replaced by. In the evaluatin f (2), generating pssibly requires Markv chain Mnte Carl (MCMC) simulatin. Fr example, Wakefield (1994) cnsiders ptimal design strategies in a ppulatin pharmackinetic study. The prblem is t find the ptimal dsage regimen fr a new patient. The decisin is made cnditinal n already bserved data fr earlier patients wh were administered the same treatment. Wakefield prpses Mnte Carl evaluatin f expected utilities using a Mnte Carl average as in (2), replacing the prir by the relevant psterir. The required psterir sample # %D" #?">>>" A ' is generated by MCMC simulatin. Cmpared t parameter estimatin, the slutin f the ptimal design prblem cmes at n additinal cmputatinal cst, since the MCMC psterir sample is already generated. In a similar decisin prblem, Palmer and Muller (1998) cnsider the questin f chsing an ptimal schedule f bld stem cell cllectins. The decisin is a vectr f indicatrs fr each f 10 pssible days with D #? if a stem cell cllectin is scheduled fr day, and D # therwise. The design space is the set f all pssible F vectrs f such indicatrs. We evaluate 1 by a Mnte Carl apprximatin like in (2). The decisin is made fr a new patient, cnditinal n data frm the previusly bserved patients. Thus replaces as the relevant distributin n. As in Wakefield (1994), we prceed by generating via MCMC a sample %D " #?">>>" A ' and evaluating 1 by (2). In many prblems the expected utility surface 1 can be assumed cntinuus. The use f (2) in the cntext f the ptimal design maximizatin fails t explit such cntinuity. Separate

4 large scale Mnte Carl simulatin fr each new value f includes inefficient duplicatin f effrt in the fllwing sense. In the curse f the simulatin, assume we have already evaluated 1 and cnsider nw anther design which is clse t (in sme apprpriate metric). Repeating the full Mnte Carl simulatin fr 1 entirely neglects the already available evaluatin f 1. In Muller and Parmigiani (1996) we prpse a numerical ptimal Bayesian design scheme which des explit such cntinuity if present. First select sme designs D (pssibly n a grid). Then simulate experiments D"D ", ne fr each chsen design. Fr each simulated experiment D"D"D we evaluate the bserved utility!d #!D"D"D. In a scatterplt f D and!d the integratin in (1) can be replaced by a simple scatterplt smthing 1, and the ptimal design can be read ff as the mde f the smth curve. Figure 1 shws tw typical examples. Under specific assumptins n the design space, the utility functin and the chsen methd f scatter plt smthing we shw that the ptimal design based n 1 is a cnsistent estimatr fr *. SAMPLE SIZE UTILITY INITIAL DOSE D PERC CHANGE -0.4 Figure 1. Simulated utilities fr sme designs. The left panel shws simulated utilities fr a design prblem with a ne-dimensinal design parameter ( sample size n the hrizntal axis). The right panel crrespnds t a bivariate design parameter ( % change, initial dse). In bth panels, the pints shw the bserved utilities fr the simulated experiments. The smth curve/surface is the fitted functin. Smthing replaces the integratin in (1). The mde f gives the ptimal design. In the right panel, the estimated mde is indicated with a white triangle. The left figure shws simulated utilities in Example 1 f Muller and Parmigiani (1996). The example in the right panel is taken frm Clyde, Muller and Parmigiani (1995b). Erkanli, Syer and Angld (1998) use the scheme in a prblem cncerning the estimatin f the prevalence f a rare disrder in a tw phase design. In a first phase an inexpensive screening test is administered. A prprtin F f the patients wh screened psitively are subjected t a mre expensive diagnstic test. Fr the patients wh tested negative n the screening test, a prprtin ( is chsen fr the diagnstic test. The design prblem is the chice f ptimal prprtins # F" ( subject t an expected budget cnstraint. The design criterin is minimum prepsterir expected variance fr. We will return t this prblem as

5 Example 3 belw. 3. AUGMENTED PROBABILITY SIMULATION In Bielza, Muller and Ris Insua (1996) and Clyde, Muller and Parmigiani (1995a) we prpse t slve the ptimal design prblem (1) by recasting the prblem as a prblem f simulatin frm an augmented prbability mdel "". A prbability mdel ) is defined in such a way that the marginal distributin in is prprtinal t the expected utility 1. Specifically, assume is bunded,!"" is nn-negative and bunded and define an artificial distributin ""!"" n "". Under, the marginal distributin in is prprtinal t!"" " # 1, i.e., is prprtinal t the expected utility 1, as desired. The definitin f "" is reminiscent f data augmentatin as used in psterir inference (Tanner and Wng, 1987). But in cntrast t data augmentatin the marginal distributin f the riginal randm variables " is changed in the augmented mdel. Als, interest in the augmented mdel fcuses n the induced distributin f the artificially added latent variable, nt n the riginal parameters. Simulatin frm ) can be used t slve the ptimal design prblem. "" We use super-indices t indicate design parameters and crrespnding utilities after iteratins f the Markv chain. 1. Start with a design. Simulate " frm " #. Evaluate! #! "". 2. Generate a candidate frm a prbing distributin -. Details f the prbing distributin will be discussed belw. 3. Simulate ", as in 1, under design. Evaluate! #!"". 4. Cmpute 5. Set + #?" "" - "" - " "! #?"! -! - F "! F # "! with prbability + "! with prbability? $ +! 6. Repeat steps 2 thrugh 5 until the chain is judged t have practically cnverged. The algrithm defines an MCMC scheme t simulate frm "", using a Metrplis- Hastings chain with an independence prpsal t update ". There still remains the specificatin f the prbing distributin -. Chsing a symmetric prbing distributin, that is ne fr which -, leads t a simple expressin fr the acceptance prbability, # - + #?"!"!. Fr later reference we nte an alternative prbing distributin leading t an algrithm which is similar t a Gibbs sampler fr psterir simulatin. Dente with the dimensin f the design space, and let # F">>>"7. Replace steps 2 thrugh 5 by a single steps 2 :

6 2. Fr #?">>>", generate a new value F F">>>" F F" F">>>" 7. T (apprximately) generate frm D" # we use the fllwing randm walk Metrplis use a prbing distributin - which changes nly the -th crdinate, i.e., D # D, #. Simulating sufficiently many iteratins f steps 2 thrugh 5 with - as prbing distributin will generate an apprximate draw frm F">>>" F F" F">>>" 7. Algrithm 1, as well as the abve described variatin, are special cases f MCMC simulatin as described in Tierney (1994). If the design space 7 is an interval in and!"" A A fr sme bund, then it fllws frm results in Tierney (1994) that the abve described schemes define Markv chains in "" with statinary distributin. The utput frm the MCMC simulatin can be used in several ways t derive the ptimal design. First, the sample f simulated can be used t derive an estimate f. The mde crrespnds t the ptimal design *. This is illustrated in the fllwing Example 1. chain. Prceed like in steps 2 thrugh 5 f Algrithm 1, but instead f - Example 1. In Bielza et al. (1996) we illustrated the scheme in an example taken frm Cvaliu and Oliver (1995). An electric cmpany has t decide whether t cmmissin a cnventinal r an advanced reactr. The cmpany has the ptin f rdering a test f the advanced reactr. The prblem invlves three decisin ndes # F" ( " representing decisins whether r nt t cmmissin the test (F ), and which reactr type t decide fr under the pssible utcmes f the test, including a dummy utcme crrespnding t n test ( ( " ). There are tw randm variables # F" (, the utcme f the test (F ) and the ccurrence f an accident ( (. The utility functin includes a cst fr cmmissining the test and fr rdering the tw reactr types, and a payff as a functin f pssible accidents and reactr types. Decisin space and sample space are finite discrete with few pssible realizatins nly, allwing many alternative slutin methds including straightfrward analytic evaluatin f expected utility under all pssible decisins. Fr illustratin we slve the prblem using simulatin n the augmented prbability mdel " (there are n unbservable parameters in this prblem). Figure 2a shws the histgram f simulated values fr the decisin nde #? Figure 2. Example 1. The left panel shws estimated by a histgram f simulated values. The right panel shws the same estimate based n given in (3). Simulatin frm is discussed in Sectin 4. In many cases the apprach used in Example 1 will be impracticable because the expected utility functin, and thus, are t flat arund the mde, requiring prhibitive simulatin

7 0 sample sizes t estimate the mde. Instead we cnsider a scatter plt f all simulated pairs "! and prceed as prpsed in the previus sectin. A smth fit thrugh the simulated pints prvides an estimate f 1, and the ptimal design can be readily read ff. In this case, the MCMC simulatin serves t stir sampling twards areas f high expected utility and replaces sampling n a grid. Nte that the pairs "! used fr the smthing have t include simulated designs, including prpsals which were rejected as states f the Markv chain. Cmpared t simulating an equal number f experiments fr designs n a regular grid, as prpsed in Sectin 2, the cmputatinal effrt invlved in Algrithm 1 is the same. The evaluatin f the acceptance prbabilities in step 5 requires n additinal cmputatin beynd the evaluatin f the bserved utilities!). Example 2 illustrates this strategy D1K (*1000) D1C (*1000) D2K (*1000) 1F"* ( " "* 1*F" ( "* " D2C (*1000) Figure 3. Example 2. The estimated expected utility surface. The plts shw as a functin f and, respectively, keeping the ther tw crdinates fixed at the estimated ptimal value. The slid diamnd marks the ptimal design. Example 2. In Ris Insua et al. (1997), we discuss a prblem cncerned with the peratin represent the amunt f water released ( f tw reservirs. Fur parameters # F " ( F " " thrugh turbines (subscript?) and spill gates (subscript ) n tw reservirs ( and ), respectively. A prbability mdel defines a distributin n inflws int the reservirs. A utility functin frmally cmbines a variety f gals related t energy deficit, final strage and water spillage. Using Algrithm 1 we simulated experiments fr 100,000 design chices. Figure 3 shws a smth fit thrugh the simulated pairs "!. The value in each cell f the grid is the average ver all simulated utilities with designs falling within the respective grid cell. Using the MCMC simulatin incurs n additinal cmputatinal cst beynd evaluating the utilities!"" fr each cnsidered design. The MCMC simulatin keeps sampling fcused in areas f high expected utility, and avids unnecessary sampling in areas f lw expected utility. Figure 4 shws a histgram f the number f times that each f 10,000 cells in an equally spaced???? grid ver the design space were visited. Evenly spreading simulatins acrss the design space, as prpsed in Sectin 2, we wuld have in all cells an

8 equal number f simulatins. Using the MCMC scheme, designs clse t * get heavily versampled, as desirable fr a mre reliable recnstructin f the expected utility surface clse t the mde N Figure 4. Example 2. A histgram f the number f designs which were cnsidered in each f the 10,000 cells f the grid used in Figure 3. If the designs were evenly spread ver the grid all cells wuld include an equal number f simulatins. The strategy used in Example 2 is feasible nly fr a lw dimensinal decisin space, say up t arund fur dimensins. The next sectin describes a mre generally applicable variatin f the algrithm. 4. TIGHTENING THE EXPECTED UTILITY SURFACE Simulatin frm the augmented mdel ) des nt entirely slve the riginal stchastic ptimizatin prblem. It nly transfrms it t a prblem f estimating the mde f a distributin based n a simulated sample. In place f (1) we face the prblem f deducing the mde f frm the simulatin utput. In the last tw examples this was pssible by cnsidering the histgram f simulated s, r a surface thrugh all simulated pairs "!. Hwever, simple inspectin f a histgram r surface can nt prvide a general slutin fr at least tw reasns. First, in high dimensinal design spaces density estimatin becmes impracticable, using histgrams r any ther methd. Similarly, fitting a surface thrugh the simulated "! pairs becmes difficult fr high dimensinal. And secndly, expected utility surfaces in mst applicatins are very flat ver a wide range f designs. This wuld require unreasnably large simulatin sample sizes t estimate the mde f. Bth prblems can be addressed by replacing the target functin with a pwer transfrmatin. This is the generic idea f simulated annealing (van Laarhven and Aarts, 1987). In the cntext f simulated annealing the reciprcal #?" is referred t as annealing temperature. Cnsidering the limit replaces the riginal target functin with a pint mass at the mde. Fr sufficiently large, simulatins frm a prbability density prprtinal t the -th pwer f the riginal target functin cluster tightly arund the mde. Direct applicatin f this scheme t the ptimal design prblem (1) is hindered by the fact that we d nt get t evaluate itself, but nly the augmented mdel "". Taking a pwer f "" r f!"" wuld, f curse, nt achieve the desired transfrmatin f the marginal. Hwever,

9 simulated annealing mtivates the fllwing related scheme. Cnsider "F"F">>>" " EF!" " " " i.e., fr each generate experiments % " " #?">>>" ' and cnsider the prduct f the bserved utilities. The implied marginal in is prprtinal t the -th pwer f the expected utility, 1. Nte that substituting the average instead f the prduct f the bserved utilities in (3) wuld imply 1 as marginal distributin n, nt the desired pwer 1. The mdificatin t the earlier MCMC algrithm t generate frm is minimal. We replace steps 1, 3 and 4 by: 1. Simulate " ", #?">>>". Fr each simulated experiment evaluate! #! " ". Define! #!. 3. Simulate " ", #?">>>". Fr each simulated experiment evaluate! #!" ". Define! #!. 4. Cmpute + #?"!"!. The mdified algrithm can be used t replace 1 by a mre peaked transfrmatin 1. This can be dne withut any ntin f an annealing schedule, i.e., using ne fixed value nly. Fr illustratin, cnsider Example 1. Because f the relatively small differences in expected utility, a disprprtinately large simulatin sample size is needed t detect the ptimal design in this simple prblem. Using 1 the differences becme exacerbated. Figure 2b shws 1, using #?. The fllwing Example 3 illustrates the same strategy in a mre cmplex decisin prblem D D D2 1 % 1 '( D1 Figure 5. Example 3. The estimated expected utility surface and the distributin frm which designs were simulated. Sampling frm fcuses simulatins arund the mde.

10 ? Example 3. We implement the mdified algrithm fr the tw phase design prblem described in Erkanli, Syer and Angld (1998). The prblem was described earlier in Sectin 2. Figure 5a shws estimated expected utilities as a functin f # F" (. We use the MCMC scheme f Algrithm 2 t select the designs at which experiments were simulated. This achieves a simulatin which cncentrates effrts arund the mde, and uses nly few simulatins fr areas f the parameter space away frm the ptimal design. This is illustrated in Figure 5b. The strategy used in Example 3 relied upn the lw dimensinality f the decisin vectr, and wuld nt be feasible in higher dimensins. Fr higher dimensinal applicatins Algrithm 2 can be embedded in an annealing scheme. Starting with #? use an annealing schedule t increase ver iteratins, i.e., #. In Sectin 3 we described a Gibbs sampler like variatin f Algrithm 1. We can define an analgus variatin f Algrithm 2. If the decisin vectr is a discrete -dimensinal vectr # F">>>"7 with %?">>>" ', then direct applicatin f Therem B frm Geman and Geman (1984) establishes cnvergence t the ptimal design. Let 1* # 1, 1* #. The result requires 1 and # 1* $ 1* that the annealing schedule be bunded by " ). Fr practical use we d nt prpse a frmal implementatin f an annealing scheme. Rather, we suggest t increase nly until the simulated designs are sufficiently tightly clustered such that the (trimmed) sample mean f the simulated designs is a reasnable apprximatin f the ptimal design. Figure 6 illustrates this scheme in Example 2. The circles indicate the designs generated in the simulated annealing scheme. The sample mean f the simula- D1K D2K D1C F" ( " D2C Figure 6. Example 2. Simulatin utput in a simulated annealing scheme. Nte hw the simulated pints cluster arund the mde f expected utility surface shwn in Figure 3. tin utput prvides a gd apprximatin f the ptimal design withut the intermediate step f recnstructing. Still, prblems where cntinuity f 1 is an unreasnable assumptin defy this strategy. Designs with high expected utility can nt necessarily be expected t cluster tgether. In such prblems we are nly left with the ptin f frmal simulated annealing.

11 ?? Example 4. In Sans and Muller (1999) we address a prblem f chsing rainfall mnitring statins. Out f an existing netwrk f 80 statins a subset f arund 40 statins needs t be selected. We define a utility functin t frmalize the aim f making inference abut rainfall ver the area n ne hand, and minimizing cst n the ther hand. The utility functin includes a payff each time the predictive residuals at any f the current 80 lcatins is in abslute value belw a certain threshld. The predictive residual at a statin which is actually included in the subset is defined as zer. The design parameter is a vectr # F">>>" f indicatrs with D #? if statin is included in the chsen subset, and D # therwise. The high dimensinality f the decisin vectr cmplicates the design prblem. In Sans and Muller (1999) we used Algrithm 1, tgether with an idea prpsed in Bielza et al. (1996). Using a hierarchical clustering tree f the simulated designs we find areas where simulatins cluster. Alternatively, Figure 7 illustrates the use f the simulated annealing algrithm described abve. ITERATION U(d) Figure 7. Example 4. Estimated fr simulated designs,. Evaluating is nt part f the algrithm and was separately cmputed fr this plt, using a Mnte Carl apprximatin (2) with. The numerical standard errrs are apprximately. The white line is a less fit thrugh the scatterplt shwing the steady increase in expected utility. 5. DISCUSSION We have prpsed several schemes t implement simulatin based ptimal Bayesian design. Nne f the prpsed algrithms is sufficiently general and rbust t allw rutine applicatin withut adaptatin t the specific prblem. Rather, the prpsed algrithms shuld be understd as examples fr pssible strategies. The discussin was in the cntext f slving ptimal design prblems specified in the frm f a prbability mdel, a decisin space, and a utility functin, but culd f curse be used t slve decisin prblems defined in any equivalent alternative way. Fr example, influence diagrams are used fr a graphical representatin f decisin prblems. See, fr example, Shachter (1986) r Sheny (1995) fr a definitin and review f slutin strategies. The algrithms defined in this paper can be used as simulatin based slutin strategies fr influence diagrams,

12 ? including arbitrary, pssibly cntinuus randm variables and decisin ndes. In fact, Jenzarli (1995) prpsed the use f Gibbs sampling estimates (2) t evaluate 1 in the cntext f influence diagrams (Gibbs sampling rather than independent sampling is required since the decisin is pssibly made cnditinal n sme data knwn at the time f decisin making). An interesting variatin in the simulated annealing scheme discussed in Sectin 4 is the use f prbing distributins which incrprate sme apprximate knwledge f the ptimal design. Laud, Berliner and Gel (1992) and Fei and Berliner (1991) discuss such strategies in a mre general cntext. The ptimal design prblem (1) is frmally similar t the prblem f maximum likelihd estimatin with nrmalizatin described in Geyer (1994). Given a family % 9 ' f nn-negative integrable functins, find t maximize the nrmalized likelihd 9 " 9. Geyer (1994) discusses a slutin based n replacing the integral # 9 by a Mnte Carl estimate based n Mnte Carl simulatins. The paper gives cnditins under which the resulting apprximatin hypcnverges t, and maximum likelihd estimates derived frm ) cnverge t the maximum likelihd estimate derived frm ). Hypcnvergence is a type f cnvergence f a sequence f functins defined and explained in Geyer (1994). ACKNOWLEDGMENT Research was supprted by NIH/NCI under grant CA75981 and by NSF under grant NSF/DMS REFERENCES Bielza, C., Muller, P., and Ris Insua, D. (1996). Mnte Carl Methds fr Decisin Analysis with Applicatins t Influence Diagrams, Tech. Rep., Duke University.. Carlin, B., Kadane, J., and Gelfand, A. (1998). Appraches fr ptimal sequential decisin analysis in clinical trials., 54, Chalner, K. and Verdinelli, I. (1995). Bayesian experimental design: a review, Statist. Sci. 10, Clyde, M., Muller, P., Parmigiani, G. (1995a). Explring the expected utility surfaces by Markv changes, Tech. Rep., Duke University. Clyde, M., Muller, P., and Parmigiani, G. (1995b). Inference and Design Strategies fr a Hierarchical Lgistic Regressin Mdel. In Bayesian Bistatistics, (Berry D.A. and Stangl, D. eds.), New Yrk: Marcel Dekker, Cvaliu, Z., and Oliver, R. (1995). Representatin and slutin f decisin prblems using sequential decisin diagrams. Manag. Sci. 41, 12, DeGrt, M. (1970). Optimal Statistical Decisins. New Yrk: McGraw Hill. Erkanli, A., Syer, R., and Angld, A. (1998). Optimal Bayesian tw-phase designs fr prevalence estimatin. J. Statist. Planning and Inference 66, Fei, L. and Berliner, L.M. (1991). Asympttic prperties f stchastic prbing fr glbal ptimizatin: The finite case. Tech. Rep. 474, Ohi State University. Geman, S. and Geman A. (1984). Stchastic relaxatin, Gibbs distributins and the Bayesian restratin f images. IEEE Trans. Patt. Anal. Mach. Intelligence 6(6), Geyer, C.J. (1994). On the cnvergence f Mnte Carl maximum likelihd calculatins, J. Ry. Statist. Sc. B 56, Jenzarli, A. (1995). Slving Influence Diagrams using Gibbs sampling. Tech. Rep., Cllege f Business, University f Tampa. Laarhven, van P.J.M. and Aarts, E.H.L. (1987). Simulated Annealing: Thery and Applicatins. Drdrecht: D. Reidel.

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