A sequential Monte Carlo algorithm with transformations for Bayesian model exploration: applications in population genetics

Given a statistical model that attempts to explain the data, calculating the Bayes’ posterior distribution of the models parameters is desirable. The marginal likelihood of the model is also of interest, which is used for model comparison. However, for most applications, only estimates of these two measurements can be obtained with a class of methods that give consistent estimates being Monte Carlo algorithms. This thesis attempts to improve both the process in inferring a high-dimensional posterior distribution and the corresponding model marginal likelihood, on the condition that we can define an ordered set of statistical models in which deterministic transformations between each adjacent model can be applied. We propose an adaption of the sequential Monte Carlo algorithm, which we term the “transformation Sequential Monte Carlo” algorithm. The key feature of this algorithm is by defining a series of target distributions, that make use of said mentioned model transformations, we aim to infer high dimensional models by using easier to estimate posteriors from lower dimensional models with a model transformation applied. Our proposed algorithm has advantages over many established MC methods. One notable advantage is that we can tailor the algorithm if we wish to update a posterior distribution by including additional observations, but these observations also correspond to a new parameter set that needs to be inferred. Alternatively it is useful where the parameter space can become too large to explore using basic MC methods, for example if there exists an exponential or factorial relationship with observation size and the number of discrete values, but using a lower dimensional model and incorporating it into the model exploration assists with convergence. We test these strengths of tSMC under three applications, which include two population genetics applications being ancestral reconstruction under the coalescent and the other being the Structure algorithm