Numerical integration in Bayesian analysis

Peter Muller, Purdue University

Abstract

This thesis contains contributions to extend approaches to numerical posterior integration in the context of Bayesian analysis in dynamic models. Accept-reject and importance sampling are combined in a mixed algorithm, which applies importance weighting for Monte Carlo sample points with high weight and accept-reject selection of sample points with low weights. Asymptotic variance and expected computation cost for the resulting integral estimate are derived and minimized to determine an optimal cutoff value to choose between the two algorithms. A generic algorithm to generate posterior Monte Carlo samples is proposed. Posterior samples can be used to simultaneously estimate virtually any posterior measure of interest. No approximation or envelope function for the posterior density is needed. A Monte Carlo integration algorithm for dynamic models is suggested. Let y$\sb{t}$ and $\theta\sb{t}$ denote observation vector and unknown parameter vector at time $t$. A general dynamic time series model is specified by a sampling equation y$\sb{t}=f\sb{t}(\theta\sb{t})+ \nu\sb{t}$ and an evolution equation $\theta\sb{t}=g\sb{t}(\theta\sb{t-1})+{\bf \omega}\sb{t}$ which specifies how the parameter vector evolves between time periods. Here $f\sb{t}$ and $g\sb{t}$ are known, non-linear functions and ${\bf \nu}\sb{t}$ and $\omega\sb{t}$ are random variables with arbitrary specified distributions. We simulate the model with a Monte Carlo sample from the initial prior distribution on $\theta\sb0$, making at each time step $t$ a Monte Carlo sample from the posterior on $\theta\sb{t}$ available, which can then be used to estimate posterior integrals, marginal densities etc. The main problem in simulating this model is to transform a given Monte Carlo sample from the posterior distribution at time $t$ $-$ 1 into a sample from the posterior distribution at time $t$. This is solved without decreasing the size of the Monte Carlo sample by using a variation of the generic posterior generation algorithm. We examine changes in foreign exchange rates using a Bayesian version of a vector ARCH (autoregressive conditional heteroscedasticity) model with time-varying parameters. One part of the model is a vector autoregression of the considered exchange rates. The other part models the covariance matrix of this vector autoregression. All parameters are modeled to evolve over time by allowing at each time step of the model some "evolution noise". (Abstract shortened with permission of author.)

Degree

Ph.D.

Advisors

Berger, Purdue University.

Subject Area

Statistics

COinS