# Computational Methods for the Bayesian Analysis of Dynamic Models

Stavropoulos, Photis (1998) Computational Methods for the Bayesian Analysis of Dynamic Models. PhD thesis, University of Glasgow.

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## Abstract

The topic of dynamic models has been extensively studied in statistics, both from classical and Bayesian perspectives. Some of the unknown components of a dynamic model evolve in time according to a probability model and give rise to observed data according to a second model. If one adopts a Bayesian point of view, as we do in this thesis, these models combined with any available prior information give rise to a sequence of posterior distributions for all the unknowns. They encompass all our knowledge about them but in practice the cases where they are analytically available are the exception to the rule. A wealth of methods for approximating them have been developed, originating mainly from the engineering community. They may however lead to unsatisfactory results and the only remaining resort will be to represent the posterior distributions by samples. As in all areas of Bayesian statistics Markov Chain Monte Carlo methods are the most widespread tool for this purpose. The aim of this thesis is to present and study a group of techniques for sampling from intractable distributions; namely, the importance resampling methods. They are fast and easy to implement, and some of them have nice theoretical properties. Moreover, by their design they are well suited for application in dynamic model contexts. The thesis is divided into two parts. The first one comprises Chapters 1 and 2. Chapter 1 offers an introduction to dynamic models and presents in detail the original resampling method, the weighted bootstrap. We examine how it can be applied in dynamic model problems and prove some of its theoretical properties. In Chapter 2 we present more recently suggested techniques that try to improve the characteristics of weighted bootstrap. We prove that one of them, the smooth bootstrap, has the same properties as weighted bootstrap. We also compare all the methods in several simulation experiments. The second part consists of the three remaining chapters. Their unifying element is that they all deal with versions of the same problem: how to follow a moving particle in space when all the data we receive are noisy measurements of our squared distance from it. This is a geometrical description of problems that arise in industry. For example, the "moving particle" may be the changing optimal conditions for production of a commodity and the "squared distance" from the particle may be the cost we pay for not producing at these conditions. The problem can be expressed as a dynamic model with intractable posteriors and we want to see how resampling will perform in such a difficult situation. In Chapter 3 we suggest several solutions. They can all be adapted to any version of the problem. Resampling encounters some difficulties for which we manage to find an ad hoc solution. In Chapter 4 we deal with two more complicated versions of the problem. One of them causes resampling to break down. This leads us to Chapter 5, where we present and examine more recently proposed resampling algorithms. The conclusion is that there is still the need for further improvement of the resampling techniques.

Item Type: Thesis (PhD) Doctoral Adviser: Michael Titterington Statistics 1998 Enlighten Team glathesis:1998-76454 Copyright of this thesis is held by the author. 19 Nov 2019 14:19 19 Nov 2019 14:19 http://theses.gla.ac.uk/id/eprint/76454

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