Direct Simulation Methods for Multiple Changepoint Problems.

The multiple changepoint model has been considered in a wide range of statistical modelling, as it increases the flexibility to simple statistical applications. The main purpose of the thesis enables the Bayesian inference from such models by using the idea of particle filters. Compared to the existed methodology such as RJMCMC of Green (1995), the attraction of our particle filter is its simplicity and efficiency. We propose an on-line algorithm for exact filtering for a class of multiple changepoint problems. This class of models satisfy an important conditional independence property. This algorithm enables simulation from the true joint posterior distribution of the number and position of the changepoints for a class of changepoint models. The computational cost of this exact algorithm is quadratic in the number of observations. We further show how resampling ideas from particle filters can be used to reduce the computational cost to linear in the number of observations, at the expense of introducing small errors; and propose two new, optimum resampling algorithms for this problem. In practice, large computational savings can be obtained whilst introducing negligible error. We demonstrate how the resulting particle filter is practicable for segmentation of human GC content. We then generalise our method to models where the conditional independence property does not hold. In particular we consider models with dependence of the parameters across neighbouring segments. Examples of such models are those with unknown hyper-parameters, and piecewise polynomial regression models which assume continuity of the regression function. The particle filter we propose is based on a simple approximation to the filtering recursion. We show that the error introduced by the approximation can be small. We demonstrate our method on the problem of Bayesian curve fitting. The novelty of our model is that we fit a piecewise polynomial function and allow for both discontinuity and continuity at changepoints. This method is compared to existing Bayesian curve fitting method, and applied to the analysis of well-log data.