Chapter 9 Subjective Probability and Bayesian Methodology

Abstract

Subjective probability and Bayesian methods provide a unified approach to handle not only randomness from stochastic sample-paths, but also uncertainty about input parameters and response metamodels. The chapter surveys some basic concepts, principles and techniques useful for a subjective Bayesian approach to uncertainty analysis, data collection plans to reduce input uncertainty, response surface modeling, and expected value-of-information approaches to experimental designs for selection procedures. Some differences from the classical technique are identified.

Introduction

If simulation is defined to be the analysis of stochastic processes through the generation of sample paths of the process, then Bayesian and subjective probability methods apply in several ways for the modeling, design and analysis of simulation experiments. By Bayesian methods, we refer here to parameter inference through repeated observations of data with Bayes' rule. Examples in simulation are input parameter inference using field data or the inference of metamodel parameters from simulation replications. The Bayesian approach entails postulating a ‘prior probability’ model that describes a modeler's initial uncertainty about parameters, a likelihood function that describes the distribution of data, given that a parameter holds a specific value, and Bayes' rule, which provides a coherent method of updating beliefs about uncertainty when data becomes available. By subjective probability, we refer to probability assessments for all unknown quantities, including parameters that can be inferred with Bayes' rule, as well as unknown quantities for which parameters cannot be inferred from repeated sampling of data (e.g., one-shot deals like the total potential market size for a particular new product from a simulated manufacturing facility). By frequentist, we mean methods based on sampling statistics from repeated observations, such as maximum likelihood (MLE) methods to fit input parameters, or ranking and selection procedures that provide worst-case probability of correct selection guarantees based on repeated applications of the procedure. The chapter describes applications of Bayesian and subjective probability methods in simulation, and identifies some ways that the Bayesian approach differs from the frequentist approach that underlies much of simulation theory.

In the simulation community, Glynn (1986) first suggested Bayesian applications of uncertainty analysis for statistical input parameter uncertainty. In that paper, the traditional role of estimating $\alpha =h(\mathrm{E}[Y])$ is extended to account for statistical input parameter uncertainty, so $\alpha (\theta )=h(\mathrm{E}[Y|\theta ])$ depends upon unknown parameters with distribution $p(\theta )$ that can be updated with data from the modeled system. Three questions he poses are: (i) how to estimate the distribution of $\alpha (\Theta )$ induced by the random variable Θ, (ii) how to estimate the mean $\mathrm{E}[\alpha (\Theta )]$, and (iii) estimation of credible sets, e.g., finding a, b so the probability $\mathrm{Pr}(\alpha (\Theta )\in [a,b])$ equals a pre-specified value, like 0.95. Chick (1997) provided a review of the few works to that date that applied Bayesian ideas to simulation, then suggested a broader range of application areas than uncertainty analysis, including ranking and selection, response surface modeling, and experimental design.

The basic goal is to understand how uncertainty and decision variables affect system performance, so that better decisions can be made. The premise in this chapter is that representing all uncertainty with probability can aid decision-makers that face uncertainty. Stochastic uncertainty, the randomness in simulation models that occurs even if all parameters are known, is already widely modeled with probability. The subjective Bayesian approach also models input parameter and response surface uncertainty with probability distributions, a practice that has been less common in stochastic process simulation.

Probabilistic models for uncertainty are increasingly employed for at least three reasons. One, doing so allows the modeler to quantify how parameter uncertainty influences the performance of a simulated system. Parameters of models of real systems are rarely known with certainty. The Bayesian approach for uncertainty analysis overcomes some limitations of the classical approach for parameter and model selection (Chick, 2001, Barton and Schruben, 2001, Draper, 1995). Two, simulation experiments can be designed to run more efficiently (Chick and Inoue, 2001a, Santner et al., 2003). And three, Bayesian and subjective probability methods are not new but are increasingly implemented due to the development of improved computing power and Markov chain Monte Carlo (MCMC) methods (Gilks et al., 1996).

This chapter describes the subjective Bayesian formulation for simulation. Section 1 presents the basics of subjective probability and Bayesian statistics in the context of quantifying uncertainty about one statistical input parameter. Section 2 summarizes the main ideas and techniques for addressing three main challenges in implementing Bayesian inference: maximization, integration, and sampling variates from posterior distributions. Section 3 addresses input distribution selection when multiple candidate distributions exist. Section 4 presents a joint formulation for input and output modeling, and reviews applications for data collection to reduce input uncertainty in a way that reduces output uncertainty, and for response surface modeling and simulation experiments to reduce response surface uncertainty. Section 5 describes applications of Bayesian expected value of information methods for efficiently selecting the best of a finite set of simulated alternatives.

Simulation research with Bayesian methods has grown rapidly since the mid to late 1990s. A partial reference list is Chen and Schmeiser (1995), Chen (1996), Scott (1996), Nelson et al. (1997), Chen et al. (1999), Cheng (1999), Lee and Glynn (1999), Andradóttir and Bier (2000), Chick and Inoue, 2001a, Chick and Inoue, 2001b, Chick (2001), Cheng and Currie (2003), Steckley and Henderson (2003), Chick et al. (2003), Zouaoui and Wilson, 2003, Zouaoui and Wilson, 2004, Ng and Chick (2004), as well as applications to insurance, finance, waterway safety, civil engineering and other areas described in the Winter Simulation Conference Proceedings. Work on deterministic simulation with potentially important implications for stochastic simulation includes O'Hagan et al. (1999), Kennedy and O'Hagan (2001), Craig et al. (2001), Santner et al. (2003). Excellent references for subjective probability and Bayesian statistics in general, not just in simulation, include Lindley (1972), Berger (1985), Bernardo and Smith (1994), with special mention for de Finetti (1990), Savage (1972) and de Groot (1970).