Problems in Bayesian statistics relating to discontinuous phenomena, catastrophe theory and forecasting

Abstract

The aim of this thesis is to generalise Bayesian Forecasting processes to models where normality assumptions are, not appropriate. In particular I develop models that can change their minds and I utilise Catastrophe Theory in their description. Under squared-error loss types of criteria the estimates will be smoothed out, so for model description and prediction I need to use bounded loss functions. Unfortunately the induced types of estimators have not been investigated very fully and so two chapters of the thesis represent an attempt to develop theory up to a necessary level to be used on Times Series models of the above kind. An introduction to Catastrophe Theory is then given. Catastrophe Theory is basically a classification of C∞-potential functions and since the expected loss function is in fact itself a potential function, I can use the classification on them. Chapters 6 and 7 relate the topologies of the posterior distribution and loss function to the topologies of the posterior expected loss hence a Bayes classification of posterior distributions is possible. In Chapter 8, I relate these results to the forecasting of non-stationary time series obtaining models which are very much akin to the simple weighted moving average processes under which lies this firm mathematical foundation. From this I can generate pleasing models which adjust in a "Catastrophic" way to changes in the underlying process generating the data