Dynamical Conditional Independence Models and Markov Chain Monte Carlo Methods

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1. INTRODUCTION

We develop sampling-based methods for models of observations that arise sequentially. Our interest is in applications where analysis of incoming data is required in real time, such as in clinical monitoring. We suppose that the model expands by progressively incorporating new data and new parameters. For example, in clinical monitoring, new patient-specific parameters are introduced with each new patient. Without loss of generality, we imagine that observations [F.sub.1], [F.sub.2], . . ., [F.sub.t], . . . arrive at integer times t = 1, 2, . . ., t, . . .. At each time t, the new data [F.sub.t] are accompanied by a (possibly empty) set of new model parameters or missing data [[Phi].sub.t]. Thus the model for [F.sub.1], . . ., [F.sub.t] comprises unknowns [[Phi].sub.1], . . ., [[Phi].sub.t]. In such a dynamic model (DM), data or parameters incorporated at one expansion stage may at later stages become uninteresting in themselves, such as when a patient dies or is discharged. (Herein, we use the word "parameter" to mean any model unknown, including missing data.)

Sampling-based methods of Bayesian inference and prediction include importance sampling and Markov chain Monte Carlo (MCMC). Suppose that at time t we have a sample [H.sub.t] of values of ([[Phi].sub.1], . . ., [[Phi].sub.t]) from the posterior distribution [Pi]([[Phi].sub.1], . . ., [[Phi].sub.t][where][F.sub.1], . . ., [F.sub.t]). Arrival of a new data item [F.sub.t+1] shifts interest to the new posterior [Pi]([[Phi].sub.1], . . ., [[Phi].sub.t+1][where][F.sub.1], . . ., [F.sub.at+1]), prompting us to generate a new sample [H.sub.t+1] of values of ([[Phi].sub.1], . . ., [[Phi].sub.t+1]) from [Pi]([[Phi].sub.1], . . ., [[Phi].sub.t+1][where][F.sub.1], . . ., [F.sub.t+1]). When computing the new sample [H.sub.t+1], it seems sensible to try to use information contained in the available sample [H.sub.t]. Under conventional MCMC sampling, this is not possible; with each new data item, the available sample of parameter values must be discarded, and a new sample must be created by restarting the MCMC from scratch on the entire model. This waste of information causes responses to new data to become slow. In particular, it hampers application of the method in real-time contexts.

The aforementioned difficulty can be avoided by adopting sampling methods other than MCMC. Kong, Liu, and Wong (1993; henceforth KLW) proposed a method for sequential updating of posterior distributions based on importance sampling. They retained the original parameter sample [H.sub.0] throughout and took incoming information into account by dynamically adapting the importance weights associated with elements of [H.sub.0]. However, their method is not directly applicable to DMs with an expanding parameter space. Smith and Gelfand (1992) proposed a sampling - importance resampling (SIR) sequential updating scheme. Gamerman and Migon (1993) discussed sequential analysis of data within a dynamic hierarchical model that is a special case of our DMs. They obtained closed forms for the posterior and predictive distributions of interest. In doing this, they assumed knowledge of variance matrices (up to a scalar factor), linearity of the structural equations, and error normality. West (1991, 1993) considered sequential analysis of a special case of our DMs, through a sampling-based method that uses kernel density reconstruction techniques coupled with importance resampling.

We propose two methods that are in some respects developments of KLW's work. The first adapts an importance sampling approach to expanding parameter spaces, and the second combines importance sampling and MCMC sampling. Both methods exploit conditional independence between groups of model parameters, allowing sampled values of parameters that are no longer of interest to be discarded.

In Section 2.1 we assume a general conditional independence structure for a DM, which we describe using a graph (as in Whittaker 1990). …