A Forward Monte Carlo Method For Solving Influence Diagrams Using Local Computation

The main goal of this paper is to describe a new Monte Carlo method for solving influence diagrams using local computation. We propose a forward Monte Carlo sampling technique that draws independent and identically distributed observations. Methods that have been proposed in this spirit sample from the entire distribution. However, when the number of variables is large, the state space of all variables is exponentially large, and the sample size required for good estimates may be too large to be practical. In this paper, we develop a forward Monte Carlo method, which generates observations from only a small set of chance variables for each decision node in the influence diagram. We use methods developed for exact solution of influence diagrams to limit the number of chance variables sampled at any time. Because influence diagrams model each chance variable with a conditional probability distribution, the forward Monte Carlo solution method lends itself very well to influence-diagram representations.

[1]  James E. Smith.  Moment Methods for Decision Analysis , 1993 .

[2]  David L. Poole,et al.  A NEW METHOD FOR INFLUENCE DIAGRAM EVALUATION , 1993, Comput. Intell..

[3]  C. Robert Kenley.  INFLUENCE DIAGRAM MODELS WITH CONTINUOUS VARIABLES , 1986 .

[4]  Charles J. Geyer,et al.  Practical Markov Chain Monte Carlo , 1992 .

[5]  C. Bielza,et al.  Decision Analysis by Augmented Probability Simulation , 1999 .

[6]  William A. Miller,et al.  A comparison of approaches and implementations for automating decision analysis , 1990 .

[7]  Frank Jensen,et al.  From Influence Diagrams to junction Trees , 1994, UAI.

[8]  W. K. Hastings,et al.  Monte Carlo Sampling Methods Using Markov Chains and Their Applications , 1970 .

[9]  Ronald A. Howard,et al.  Readings on the Principles and Applications of Decision Analysis , 1989 .

[10]  H. Brachinger,et al.  Decision analysis , 1997 .

[11]  C. N. Morris,et al.  The calculation of posterior distributions by data augmentation , 1987 .

[12]  R. M. Oliver,et al.  Representation and solution of decision problems using sequential decision diagrams , 1995 .

[13]  James E. Smith,et al.  Structuring Conditional Relationships in Influence Diagrams , 1993, Oper. Res..

[14]  Howard Carter,et al.  Risk Analysis and Its Applications , 1984 .

[15]  Max Henrion,et al.  Propagating uncertainty in bayesian networks by probabilistic logic sampling , 1986, UAI.

[16]  Thomas D. Nielsen,et al.  Well-Defined Decision Scenarios , 1999 .

[17]  R. Howard.  Proximal Decision Analysis , 1971 .

[18]  Ross D. Shachter,et al.  Dynamic programming and influence diagrams , 1990, IEEE Trans. Syst. Man Cybern..

[19]  Scott M. Olmsted.  On representing and solving decision problems , 1983 .

[20]  Ross D. Shachter.  An ordered examination of influence diagrams , 1990, Networks.

[21]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[22]  C. Goutis.  A graphical method for solving a decision analysis problem , 1995, IEEE Trans. Syst. Man Cybern..

[23]  P. Glasserman,et al.  Pricing American-style securities using simulation , 1997 .

[24]  Judea Pearl,et al.  Evidential Reasoning Using Stochastic Simulation of Causal Models , 1987, Artif. Intell..

[25]  Nevin Lianwen Zhang,et al.  Probabilistic Inference in Influence Diagrams , 1998, Comput. Intell..

[26]  Ross D. Shachter.  Bayes-Ball: The Rational Pastime (for Determining Irrelevance and Requisite Information in Belief Networks and Influence Diagrams) , 1998, UAI.

[27]  Jim Q. Smith.  Influence diagrams for Bayesian decision analysis , 1989 .

[28]  J. Neumann,et al.  Theory of games and economic behavior , 1945, 100 Years of Math Milestones.

[29]  Nevin Lianwen Zhang,et al.  Solving Asymmetric Decision Problems with Influence Diagrams , 1994, UAI.

[30]  Ross D. Shachter,et al.  Simulation Approaches to General Probabilistic Inference on Belief Networks , 2013, UAI.

[31]  H. Johnson,et al.  The American Put Option Valued Analytically , 1984 .

[32]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[33]  Prakash P. Shenoy,et al.  Game Trees For Decision Analysis , 1998 .

[34]  Robert Wilson.  Computing Equilibria of Two-Person Games from the Extensive Form , 1972 .

[35]  Nevin Lianwen Zhang,et al.  A computational theory of decision networks , 1993, Int. J. Approx. Reason..

[36]  Prakash P. Shenoy,et al.  Representing and Solving Asymmetric Decision Problems Using Valuation Networks , 1995, AISTATS.

[37]  Pierre Claver Ndilikilikesha.  A study of influence diagrams and their generalizations , 1992 .

[38]  Adrian F. M. Smith,et al.  Sampling-Based Approaches to Calculating Marginal Densities , 1990 .

[39]  D. Rubin,et al.  Inference from Iterative Simulation Using Multiple Sequences , 1992 .

[40]  Ross D. Shachter.  Evaluating Influence Diagrams , 1986, Oper. Res..

[41]  Ross D. Shachter,et al.  Decision Making Using Probabilistic Inference Methods , 1992, UAI.

[42]  Stuart German,et al.  Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images , 1988 .

[43]  Adrian F. M. Smith,et al.  Bayesian computation via the gibbs sampler and related markov chain monte carlo methods (with discus , 1993 .

[44]  Prakash P. Shenoy,et al.  Valuation-Based Systems for Bayesian Decision Analysis , 1992, Oper. Res..

[45]  Robert G. Sargent,et al.  Statistical analysis of simulation output data , 1976, SIML.

[46]  Prakash P. Shenoy,et al.  A New Pruning Method for Solving Decision Trees and Game Trees , 1995, UAI.

[47]  L. Tierney.  Markov Chains for Exploring Posterior Distributions , 1994 .

[48]  W. David Kelton.  Analysis of simulation of output data , 1985, WSC '85.

[49]  C. Robert Kenley,et al.  Gaussian influence diagrams , 1989 .

[50]  D. Keefer.  Certainty Equivalents for Three-Point Discrete-Distribution Approximations , 1994 .

[51]  A. C. Miller,et al.  Discrete Approximations of Probability Distributions , 1983 .

[52]  Thomas D. Nielsen,et al.  Welldefined Decision Scenarios , 1999, UAI.

[53]  Joseph A Tatman,et al.  Decision Processes In Influence Diagrams: Formulation and Analysis , 1985 .

[54]  Prakash P. Shenoy,et al.  Valuation network representation and solution of asymmetric decision problems , 2000, Eur. J. Oper. Res..

[55]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[56]  Pierre Ndilikilikesha,et al.  Potential influence diagrams , 1994, Int. J. Approx. Reason..