Computational Approach to Poisson Regression Model

We recognize the equation in step 2 as the weighted least squares estimate on the adjusted observations. Iterating through these two steps until convergence finds the iteratively reweighted least squares estimates. 

This is the usual frequentist approach to this model. The maximum likelihood vector is given by 

In the computational Bayesian approach, we want to draw a sample from the actual posterior, not its approximation. As we noted before, we know its shape. Our approach will be to use the Metropolis-Hastings algorithm with an independent candidate density. We want a candidate density that is as close as possible to the posterior so many candidates will be accepted. We want the candidate density to have heavier tails than the posterior, so we move around the parameter space quickly. That will let us have shorter burn-in and use less thinning. We use the maximum likelihood vector 0ml matched curvature covariance matrix Vj fz, as the 

We approximate the likelihood function by a multivariate normal 0ML where 0ml the MLE and is the matched curvature covariance matrix that is output by the iteratively reweighted least squares. We use a multivariate nor-ma/[bo, Vo] prior for or we can use a flat prior if we have no prior information. The approximate posterior will be 

Since both the prior and (approximate) likelihood are multivariate normal, the approximate posterior will be multivariate normal also. The updated constants will be 

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COMPUTATIONAL APPROACH TO POISSON REGRESSION MODEL 207 2. Then update the parameter vector to step n by... 

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