Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 34 / 3 / d / Solution

The Poisson trick introduces ${\lambda }_i=\log {\mu }_{i1}$. With an exactly flat log-rate prior, the baseline prior measure is $d{u}_i/u_i$. For $n_i>0$,Thus flat log-rate marginalization gives the multinomial likelihood:Alternatively, independent Poisson counts conditional on their total are exactly multinomial distributions with probabilities $e^{{\eta }_{ik}}/S_i$. Locally uniform coefficient priors make the coefficient posterior proportional to this likelihood within their flat region.

The printed BUGS prior is a proper normal distribution on ${\lambda }_i$, with variance $s^2=10^5$, since BUGS uses precision as its second argument. It is broad but not exactly flat. Substitution $v=S_i{u}_i$ shows that, apart from a coefficient-independent constant, the integrated likelihood is the multinomial kernel times the Gaussian log-rate correction to the Poisson trickIt depends on the coefficients through $S_i$. Dominated convergence gives $R_i\to 1$ as $s^2\to \infty$, so the finite-variance code yields an approximation to the multinomial posterior, not exact equality. Large log rates can make prior sensitivity relevant.

The cell-wise factors form a standard Poisson regression, convenient for BUGS. They avoid an explicitly constrained count vector and permit scalar log-concave updates for intercepts and coefficients; an exactly flat-log-rate implementation also gives gamma conditional baseline rates. This can be computationally efficient, although the actual log-normal baseline prior is not gamma-conjugate and introduces $I$ nuisance intercepts. Efficiency depends on the update scheme.