Constrained space MCMC methods for nested sampling Bayesian computations

Examensarbete för masterexamen
Olander, Jacob
Natural phenomena can in general be described using several different scientific models, which creates a need for systematic model selection. Bayesian model comparison assigns relative probabilities to a set of possible models using the model evidence (marginal likelihood), obtained by an integral that in general needs to be evaluated numerically. Nested sampling is a conceptual framework that efficiently estimates the model evidence and, additionally, provides samples from the model parameter posterior distribution used in Bayesian parameter estimation. A vital step of nested sampling is the likelihoodconstrained sampling of the model parameter prior distribution, a task that has proven particularly difficult and that is subject to ongoing research. In this thesis we implement, evaluate and compare three methods for constrained sampling in conjunction with a nested sampling framework. The methods are variants of Markov chain Monte Carlo algorithms: Metropolis, Galilean Monte Carlo and the affine-invariant stretch move, respectively. The latter is applied in the context of nested sampling for the first time in this work. The performances of the methods are assessed by their application to a reference problem that has a known analytical solution. The problem is inspired by effective field theories in subatomic physics where the model parameters take the form of coefficients that are of natural size. We conclude that the efficiency and computational accuracy of nested sampling is strongly dependent on the choice of sampling method and the settings of its associated hyperparameters. In certain cases, especially for high-dimensional parameter spaces, the implementations of this work are seen to achieve better computational accuracy than MultiNest, a state-of-the-art nested sampling implementation extensively used in astronomy and cosmology. Generally for nested sampling, we observe that it is possible to obtain an inaccurate result without receiving any clear warning signs indicating that this is the case. However, we demonstrate that the validity of the computational results can be assessed by monitoring the sampling process.
Bayesian inference, parameter estimation, model comparison, evidence, nested sampling, MCMC, Metropolis, Galilean Monte Carlo, affine-invariant sampling
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