Strathmann, Heiko; (2018) Kernel methods for Monte Carlo. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

This thesis investigates the use of reproducing kernel Hilbert spaces (RKHS) in the context of Monte Carlo algorithms. The work proceeds in three main themes. Adaptive Monte Carlo proposals: We introduce and study two adaptive Markov chain Monte Carlo (MCMC) algorithms to sample from target distributions with non-linear support and intractable gradients. Our algorithms, generalisations of random walk Metropolis and Hamiltonian Monte Carlo, adaptively learn local covariance and gradient structure respectively, by modelling past samples in an RKHS. We further show how to embed these methods into the sequential Monte Carlo framework. Efficient and principled score estimation: We propose methods for fitting an RKHS exponential family model that work by fitting the gradient of the log density, the score, thus avoiding the need to compute a normalization constant. While the problem is of general interest, here we focus on its embedding into the adaptive MCMC context from above. We improve the computational efficiency of an earlier solution with two novel fast approximation schemes without guarantees, and a low-rank, Nyström-like solution. The latter retains the consistency and convergence rates of the exact solution, at lower computational cost. Goodness-of-fit testing: We propose a non-parametric statistical test for goodness-of-fit. The measure is a divergence constructed via Stein's method using functions from an RKHS. We derive a statistical test, both for i.i.d. and non-i.i.d. samples, and apply the test to quantifying convergence of approximate MCMC methods, statistical model criticism, and evaluating accuracy in non-parametric score estimation.

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