Oral Prelim: Joonha Park, Computational sampling methods in high dimension and applications to modelling of infectious diseases
Stochastic properties of a random variable such as mean or variance are, except in the cases where it follows a theoretically analyzable distribution, generally investigated by Monte Carlo methods where samples are drawn from its distribution or an approximation thereof. The efficacy of such methods hinges on how well the distribution can be represented by sample draws and how much computational costs are needed to obtain the samples. Common sampling techniques, such as the Markov Chain Monte Carlo (MCMC), Importance sampling, or Sequential Monte Carlo (SMC) in the case of inference on dynamic systems, share the difficulty that the performance deteriorates farily rapidly in high dimension. In this prospectus, we propose two novel methods that aim to at least partly overcome such difficulty. The first method concerns sampling from high dimensional discrete space. It is in principle an application of Hamiltonian Monte Cralo, which is a kind of Markov Chain Monte Carlo method developed for continuous space which is known to perform relatively well in high dimension, to discrete spaces. The second is about running Sequential Monte Carlo in high dimension under the assumption of weak interaction between dimensions. The deterioration of SMC in high dimension is particularly well known, and for this reason inference for nonlinear dynamic systems of high dimension has been regarded as computationally infeasible problems. The new method will be applied to make a joint inference on measles epidemics in multiple cities using the UK data from 1944 to 1965.