Poisson Log-Linear Models• Suppose data consist of counts, y1, . . . , yn.• In addition, we have a vector of predictors xi= (xi1, . . . , xip)0• Poisson log-linear model:yi∼ Poisson(ηi), ηi= exp(x0iβ).• Bayesian inferences can proceed by specifying a prior for β (e.g.,multivariate normal), and basing inferences on summaries of theposterior obtained from a Gibbs sampler.Posterior Computation for Poisson Log-Linear Mode ls• Unlike in the normal linear regression or probit model cases, thefull conditional posterior distribution of βjdoes not (in general)have a form that is easy to sample from directly.• However, for typical choices of prior (such as normal), the fullconditional distribution is log-concave and adaptive rejectionsampling can be used to implement Gibbs sampling.• For example, this can be done easily in WinBUGS. Alternative,one can use a random walk Metropolis-Hastings algorithm orsome other black-box scheme.Poisson-Gamma Log-Linear Models• Often, count data may be over-dispersed re lative to the Poissondistribution, meaning that the variance is greater than the mean.• To account for this over-dispersion, the following Poisson-Gammamodel is commonly used:yi∼ Poisson(ηi), ηi= ξiexp(x0iβ),where ξi∼ G(φ−1, φ−1) is a gamma distributed frailty multiplier• The marginal expectation and variance integrating out ξiare asfollows:E(yi| xi) = exp(x0iβ) and V(yi| xi) = exp(x0iβ)+φ exp(x0iβ)2,so it is c lear that φ measures the degree of overdispersionPosterior Computation in Poisson-Gamma Log-LinearModelsAlthough the conditional posterior distributions of βjand φ typicallydo not follow a simple form, ξihas a conjugate gamma full conditionaldistribution simplifying efficient computation.Posterior distribution is proportional tonYi=1{ξiexp(x0iβ)}yiexp { − ξiexp(x0iβ)}(φ−1)φ−1Γ(φ−1)ξφ−1−1iexp(−ξiφ−1)×π(β, φ)From this expression, the conditional p osterior distribution of ξiis∝(ξi)φ−1+Piyi−1exp [ − ξi{φ−1+Xiexp(x0iβ)}],and we haveπ(ξi| β, φ, y, x) = Gξi; φ−1+nXi=1yi, φ−1+nXi=1exp(x0iβ).Thus, in implementing Gibbs sampling we can sample directly fromthe conditional of ξi.Log-Linear Models with Categorical Predictors• When one or more of the elements of xiare binary indicatorvariables, conditionally-conjugate priors can be defined.• This simplifies efficient computation and facilitates the devel-opment of methods for variable selection and order restrictedinference in log-linear models.• Suppose for simplicity that all elements of xiare 0/1, and letγj= exp(βj), for j = 1, . . . , p.• We propose using the following prior for γ = (γ1, . . . , γp)0:π(γ) =pYj=1G(γh; ah, bh),where ah, bhare pre-specified hyperparametersThe resulting full conditional posterior distribution is proportionaltoπ(φ)nYi=1{ξipYj=1γxijj}yiexp { − ξipYj=1γxijj}pYj=1γaj−1jexp(−γjbj).From this expression, it can be easily shown that the full conditionalof γjisGaj+nXi=1xijyij, bj+nXi:xij=1ξipYh6=jγxihh.Hence, we can sample directly from the conditional distribution forthe exponentiated intercept and for the exponentiated regression co-efficients for binary predictorsBayesian Order Restricted Inference for Log-LinearModels• Now suppose that we are interested in assessing the relationshipbetween an ordered categorical predictor wi∈ {1, . . . , k}, anda count outcome yi• Initially focusing on the case where wiis the only predictor, wecan letηi= ξiγ1k−1Yj=1γxijj,where ηiis the Poisson mean, and xij= 1(wi> j) for j =1, . . . , k − 1.• Suppose we choose gamma priors for the γj’s - does the incor-porated the ordered structure of