COM-Poisson regression is an increasingly popular model for count data. Its main advantage is that it permits to model separately the mean and the variance of the counts, thus allowing the same covariate to affect in different ways the average level and the variability of the response variable. A key limiting factor to the use of the COM-Poisson distribution is the calculation of the normalisation constant: its accurate evaluation can be time-consuming and is not always feasible. We circumvent this problem, in the context of estimating a Bayesian COM-Poisson regression, by resorting to the exchange algorithm, an MCMC method applicable to situations where the sampling model (likelihood) can only be computed up to a normalisation constant. The algorithm requires to draw from the sampling model, which in the case of the COM-Poisson distribution can be done efficiently using rejection sampling. We illustrate the method and the benefits of using a Bayesian COM-Poisson regression model, through a simulation and two real-world data sets with different levels of dispersion. B Charalampos Chanialidis

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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.

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