Retrospective sampling in MCMC with an application to COM‐Poisson regression

Abstract

The normalisation constant in the distribution of a discrete random variable may not be available in closed form; in such cases the calculation of the likelihood can be computationally expensive. Approximations of the likelihood or approximate Bayesian computation (ABC) methods can be used; but the resulting MCMC algorithm may not sample from the target of interest. In certain situations one can efficiently compute lower and upper bounds on the likelihood. As a result, the target density and the acceptance probability of the MetropolisHastings algorithm can be bounded. We propose an efficient and exact MCMC algorithm based on the idea of retrospective sampling. This procedure can be applied to a number of discrete distributions, one of which is the COM-Poisson distribution. In practice the bounds on the acceptance probability do not need to be particularly tight in order to accept or reject a move. We demonstrate this method using data on the emergency hospital admissions in Scotland in 2010, where the main interest lies in the estimation of the variability of admissions, since it is considered as a proxy for health inequalities.

Efficient Bayesian inference for COM-Poisson regression models

“…In this paper, we presented a computationally more efficient MCMC algorithm for COM-Poisson regression compared to the alternative in Chanialidis et al (2014). We showed how rejection sampling, combined with the exchange algorithm, can be used to overcome the problem of an intractable likelihood in the COM-Poisson distribution.…”

“…-Use of the asymptotic approximation by Minka et al (2003). -Estimate upper and lower bounds for the value of the normalisation constant and use these in an MCMC algorithm (Chanialidis et al 2014).…”

“…is known in closed form. The instrumental distribution in (4) is based on the same principle as the upper bounds used in the retrospective sampling algorithm proposed by Chanialidis et al (2014). In contrast to the arbitrarily precise upper bound required for the retrospective algorithm, the bounds set out above are based on a trade-off between achieving a high acceptance rate while at the same time keeping the instrumental distribution simple so that sampling from it is computationally efficient.…”

“…-Ifπ n < U <π n , refine the bounds, i.e increase n and return to step 2. We will now compare the algorithm presented in Chanialidis et al (2014) with the MCMC algorithm presented in this paper, using the publications data discussed in Sect. 4.2.…”

“…Finally, in Sect. 5 we compare the proposed MCMC algorithm with the one in Chanialidis et al (2014) and show that the newly proposed MCMC samples from the correct posterior distribution and is computationally more efficient.…”