| Summary: | This thesis discusses methods for Bayesian parameter estimation, particularly in the case of state space models (SSMs). We begin by reviewing established methods for filtering in SSMs, and by examining the graphical model structure of a parameterized SSM. Then we discuss established methods for estimating the parameters of such an SSM, making use of its graphical structure. Next we employ monotone triangular transport maps as a method of estimating conditional probability densities and performing conditional sampling, and relate these tasks to the original filtering problem. We provide some practical results and experiments for employing these maps for inference, particularly examining the map parameterization for this function approximation problem. Using these ingredients, we introduce and discuss an algorithm that uses transport to perform online inference of the static parameters of an SSM, and relate this algorithm to prior methods. Finally, we tie the problems of function approximation and static parameter inference together with numerical examples of transport for sequential inference.
Most of the results in this thesis are powered by two software packages that were developed at length over the course of the thesis work: EnsembleFiltering.jl, written in Julia for performing automatically-differentiable ensemble-based filtering on the CPU and GPU; and MParT, written in C++ for evaluating and training monotone triangular transport maps.
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