No description available.

Time series models are used to characterise uncertainty in many real-world dynamical phenomena. A time series model typically contains a static variable, called parameter, which parametrizes the joint law of the random variables involved in the definition of the model. When a time series model is to be fitted to some sequentially observed data, it is essential to decide on the value of the parameter that describes the data best, a procedure generally called parameter estimation. This thesis comprises novel contributions to the methodology on parameter estimation in time series models. Our primary interest is online estimation, although batch estimation is also considered. The developed methods are based on batch and online versions of expectation-maximisation (EM) and gradient ascent, two widely popular algorithms for maximum likelihood estimation (MLE). In the last two decades, the range of statistical models where parameter estimation can be performed has been significantly extended with the development of Monte Carlo methods. We provide contribution to the field in a similar manner, namely by combining EM and gradient ascent algorithms with sequential Monte Carlo (SMC) techniques. The time series models we investigate are widely used in statistical and engineering applications. The original work of this thesis is organised in Chapters 4 to 7. Chapter 4 contains an online EM algorithm using SMC for MLE in changepoint models, which are widely used to model heterogeneity in sequential data. In Chapter 5, we present batch and online EM algorithms using SMC for MLE in linear Gaussian multiple target tracking models. Chapter 6 contains a novel methodology for implementing MLE in a hidden Markov model having intractable probability densities for its observations. Finally, in Chapter 7 we formulate the nonnegative matrix factorisation problem as MLE in a specific hidden Markov model and propose online EM algorithms using SMC to perform MLE.

Doctor of Philosophy / Department of Statistics / Paul Nelson / Panels of judges are often used to estimate consumer preferences for m items such as food products. Judges can either evaluate each item on several ordinal scales and indirectly produce an overall ranking, or directly report a ranking of the items. A complete ranking orders all the items from best to worst. A partial ranking, as we use the term, only reports rankings of the best q out of m items. Direct ranking, the subject of this report, does not require the widespread but questionable practice of treating ordinal measurement as though they were on ratio or interval scales. Here, we develop and study segmentation models in which the panel may consist of relatively homogeneous subgroups, the segments. Judges within a subgroup will tend to agree among themselves and differ from judges in the other subgroups. We develop and study the statistical analysis of mixture models where it is not known to which segment a judge belongs or in some cases how many segments there are. Viewing segment membership indicator variables as latent data, an E-M algorithm was used to find the maximum likelihood estimators of the parameters specifying a mixture of Mallow’s (1957) distance models for complete and partial rankings. A simulation study was conducted to evaluate the behavior of the E-M algorithm in terms of such issues as the fraction of data sets for which the algorithm fails to converge and the sensitivity of initial values to the convergence rate and the performance of the maximum likelihood estimators in terms of bias and mean square error, where applicable. A Bayesian approach was developed and credible set estimators was constructed. Simulation was used to evaluate the performance of these credible sets as confidence sets. A method for predicting segment membership from covariates measured on a judge was derived using a logistic model applied to a mixture of Mallows probability distance models. The effects of covariates on segment membership were assessed. Likelihood sets for parameters specifying mixtures of Mallows distance models were constructed and explored.

This thesis is concerned with methods for Bayesian inference and their applications in astrophysics. We principally discuss two related themes: advances in nested sampling (Chapters 3 to 5), and Bayesian sparse reconstruction of signals from noisy data (Chapters 6 and 7). Nested sampling is a popular method for Bayesian computation which is widely used in astrophysics. Following the introduction and background material in Chapters 1 and 2, Chapter 3 analyses the sampling errors in nested sampling parameter estimation and presents a method for estimating them numerically for a single nested sampling calculation. Chapter 4 introduces diagnostic tests for detecting when software has not performed the nested sampling algorithm accurately, for example due to missing a mode in a multimodal posterior. The uncertainty estimates and diagnostics in Chapters 3 and 4 are implemented in the $\texttt{nestcheck}$ software package, and both chapters describe an astronomical application of the techniques introduced. Chapter 5 describes dynamic nested sampling: a generalisation of the nested sampling algorithm which can produce large improvements in computational efficiency compared to standard nested sampling. We have implemented dynamic nested sampling in the $\texttt{dyPolyChord}$ and $\texttt{perfectns}$ software packages. Chapter 6 presents a principled Bayesian framework for signal reconstruction, in which the signal is modelled by basis functions whose number (and form, if required) is determined by the data themselves. This approach is based on a Bayesian interpretation of conventional sparse reconstruction and regularisation techniques, in which sparsity is imposed through priors via Bayesian model selection. We demonstrate our method for noisy 1- and 2-dimensional signals, including examples of processing astronomical images. The numerical implementation uses dynamic nested sampling, and uncertainties are calculated using the methods introduced in Chapters 3 and 4. Chapter 7 applies our Bayesian sparse reconstruction framework to artificial neural networks, where it allows the optimum network architecture to be determined by treating the number of nodes and hidden layers as parameters. We conclude by suggesting possible areas of future research in Chapter 8.