Bayesian density regression for count data

Despite the increasing popularity of quantile regression models for continuous responses, models for count data have so far received lit-tle attention. The main quantile regression technique for count data involves adding uniform random noise or “jittering”, thus overcoming the problem that the conditional quantile function is not a continuous function of the parameters of interest. Although jittering allows esti-mating the conditional quantiles, it has the drawback that, for small values of the response variable Y, the added noise can have a large influence on the estimated quantiles. In addition, quantile regression can lead to “crossing ” quantiles. We propose a Bayesian Dirichlet pro-cess (DP)-based approach to quantile regression for count data. The approach is based on an adaptive DP mixture (DPM) of COM-Poisson regression models and determines the quantiles by estimating the den-sity of the data, thus eliminating all the aforementioned problems. Taking advantage of the exchange algorithm, the proposed MCMC al-gorithm can be applied to distributions on which the likelihood can only be computed up to a normalising constant. 1 Quantile regression Quantile regression was introduced as a nonparametric method for mod-elling a variable of interest as a function of covariates [6]. By estimating the conditional quantiles rather than the mean, it gives a more complete de-scription of the conditional distribution of the response variable than least squares regression, and is especially relevant in certain types of applications. 1 a