Monte Carlo methods for intractable and doubly intractable density estimation

This thesis is concerned with Monte Carlo methods for intractable and doubly intractable density estimation. The primary focus is on the likelihood free method of Approximate Bayesian inference, where the presence of an intractable likelihood term necessitates the need for various approximation procedures. We propose a novel Sequential Monte Carlo based algorithm and demonstrate the significant efficiency (computational and statistical) improvements compared to the widely used SMC-ABC, in numerical experiments for a simple Gaussian model and a more realistic random network model. Further, we investigate a recently proposed algorithm, called SAMC-ABC, an adaptive MCMC algorithm where we also demonstrate some advantages over ABC-MCMC; primarily in the reduction of variance of the estimated means although at a cost of increased bias for which we propose a potential correction. In addition, we provide theoretical guarantees of ergodicity and convergence of another newly proposed algorithm termed Adaptive Noisy Exchange, that is aimed at problems of intractable normalising constants where regular MCMC cannot be employed. Finally, we propose potential improvements and future research directions for all of the considered algorithms

Rare event ABC-SMC2^{2}

Approximate Bayesian computation (ABC) is a well-established family of Monte Carlo methods for performing approximate Bayesian inference in the case where an ``implicit'' model is used for the data: when the data model can be simulated, but the likelihood cannot easily be pointwise evaluated. A fundamental property of standard ABC approaches is that the number of Monte Carlo points required to achieve a given accuracy scales exponentially with the dimension of the data. Prangle et al. (2018) proposes a Markov chain Monte Carlo (MCMC) method that uses a rare event sequential Monte Carlo (SMC) approach to estimating the ABC likelihood that avoids this exponential scaling, and thus allows ABC to be used on higher dimensional data. This paper builds on the work of Prangle et al. (2018) by using the rare event SMC approach within an SMC algorithm, instead of within an MCMC algorithm. The new method has a similar structure to SMC$^{2}$ (Chopin et al., 2013), and requires less tuning than the MCMC approach. We demonstrate the new approach, compared to existing ABC-SMC methods, on a toy example and on a duplication-divergence random graph model used for modelling protein interaction networks