Optimisation of the population Monte Carlo algorithm : application to cosmology.

Abstract

In this thesis we study the Population Monte Carlo (PMC) algorithm and utilise simulations to improve the efficiency of the algorithm by optimising the algorithm parameters. We then ap ply these optimisation results to a cosmological parameter estimation problem, specifically that of determining the initial conditions for structure formation. We accomplish this by using cos mic microwave background (CMB) data to constrain models with an admixture of adiabatic and isocurvature modes. We review the standard cosmological model and current cosmological probes used for cosmol ogy and discuss the CMB anisotropy spectrum, which forms the basis for our cosmological parameter estimation study. We briefly outline linear perturbation theory and initial conditions that form the basis of the inflationary models considered in this thesis. We describe the adiabatic and isocurvature perturbations and investigate their effect on the CMB anisotropy spectrum. We outline the Bayesian parameter estimation methodology adopted in our study and review Monte Carlo sampling, specifically the Markov Chain Monte Carlo (MCMC) and PMC algo rithms explaining why these methods are used in Bayesian parameter estimation. We discuss recent developments to the PMC and MCMC algorithms and discuss various applications of these algorithms in cosmology.

We focus on optimising the performance of the PMC algorithm with respect to its algorithm pa rameters that are specified initially. However, we first define a measure of efficiency, related to the computational cost of the sampling algorithm and then use simulations to maximise this mea sure with respect to the algorithm parameters. These algorithm parameters include the sample size drawn at each iteration, the number of importance functions used, and the parameters that characterise the importance functions. Before this though, we will first investigate the optimi sation of the PMC algorithm for a multivariate Gaussian target distribution, and present results for choosing the optimal algorithm parameters that maximise efficiency. We will also explore the performance of PMC on more complex distributions such as the banana shaped, bimodal and hypercube distribution, and discuss the advantages and shortfalls for these distributions. We incorporate the results from the previous optimisation study by applying the PMC algorithm to a cosmological parameter estimation problem. We constrain models with an admixture of adiabatic and isocurvature perturbations using the nine-year data release from the Wilkinson Mi crowave Anisotropy Probe (WMAP) experiment. We discuss challenges faced in sampling such complex distributions, the modifications to the PMC sampler needed to achieve convergence, and the efficiencies achieved in sampling these distributions. We present results on the marginal and joint parameter distributions for all possible admixtures of adiabatic and isocurvature modes. We then perform a principal component analysis to determine the degeneracies that arise from the introduction of isocurvature modes. In comparison to similar studies undertaken with the WMAP one-year and three-year datasets, we find that the allowed isocurvature fraction is more tightly constrained than in previous studies.