Methodology for inference on the Markov modulated Poisson process and theory for optimal scaling of the random walk Metropolis

Two distinct strands of research are developed: new methodology for inference on the Markov modulated Poisson process (MMPP), and new theory on optimal scaling for the random walk Metropolis (RWM). A novel technique is presented for simulating from the exact distribution of a continuous time Markov chain over an interval given the start and end states and the infinitesimal generator. This is used to create a Gibbs sampler which samples from the exact distribution of the hidden Markov chain in an MMPP. The Gibbs sampler is compared with several Metropolis-Hastings algorithms on a variety of simulated datasets. It is found that the Gibbs sampler is more efficient than all but one of the Metropolis-Hastings algorithms, sometimes by an order of magnitude. One alternative algorithm, with reparameterisation motivated by a Taylor expansion of the MMPP log-likelihood, outperforms the Gibbs sampler when the different Poisson process intensities are similar. The Gibbs sampler is applied to modelling the occurrence of a rare DNA motif. IV Two Lemmas are derived that apply to stationary Metropolis-Hastings Markov chains and simplify the analytical forms for expected acceptance rate and expected square jump distance (ESJD), a measure of efficiency. These are applied to the RWM for elliptically symmetric unimodal targets, and the existence, subject to conditions, of at least one finite optimal scaling is proved in finite dimension d. A one-to-one relationship between acceptance rate and scale parameter is also established. Asymptotic forms for ESJD and expected acceptance rate as d → ∞ are then derived and conditions under which the limiting optimal acceptance rate is 0.234 are obtained. It is also shown that in a more general setting the limiting optimal acceptance rate is ≤ 0.234. Limiting efficiency results are also obtained for partial-blocking and for the exploration of elliptical targets with elliptical proposals of the same shape.