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

The multinomial distribution has probability mass functionIt is zero outside this count simplex. The multinomial coefficient counts the individual category sequences giving the same aggregate counts; the probability vector satisfies ${\sum }_k{p}_{ik}=1$.

Set ${\eta }_{ik}={\beta }_k^T{\mathbf{x}}_i$, so ${\eta }_{i1}=0$. The multinomial logistic regression implies $p_{ik}=e^{{\eta }_{ik}}{p}_{i1}$. Normalizing givesMultiply the multinomial likelihoods for conditionally independent groups. The multinomial coefficients are constant in $\beta$, leavingThe reference constraint ${\beta }_1=0$ identifies the coefficients: a common shift of every category coefficient otherwise leaves all probabilities unchanged. The PDF places the full sum inside the numerator exponent; the TeX breaks that expression.

Assume conditional independence of the Poisson distributions, and write $u_i={\mu }_{i1}$, $S_i={\sum }_k{e}^{{\eta }_{ik}}$. The correct Poisson mass function has $e^{-\mu }$; the positive sign in the PDF's reminder is erroneous. The group likelihood isMultiply by the shape–rate Gamma distribution prior $b^a{u}_i^{a-1}{e}^{-b{u}_i}/\Gamma (a)$ and integrate. The gamma integral givesIndependent baseline priors give the product over groups. For fixed $a,b$, the leading factors do not depend on $\beta$, proving the requested Gamma-integrated baseline Poisson likelihood.

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.

The first coefficient is the posterior mean log odds of choosing invertebrates rather than fish for smaller alligators at Lake Hancock. It corresponds tofor those invertebrate-to-fish odds. This is not the absolute probability of choosing the second category, because the other categories also enter the normalization.

The size coefficient adds $-1.524$ to the invertebrate-to-fish log odds on changing to the larger class, holding lake fixed. Its common-across-lakes odds ratio isabout a 78 percent reduction in the relative odds. Exponentiating a posterior mean log odds is a geometric summary, not the arithmetic posterior mean of the odds.

There are $(K-1)p=4\cdot 5=20$ free category coefficients plus $I=8$ group intercepts, giving 28 free parameters in the fitted Poisson model. The effective parameter count in DIC, $26.8$, is close to 28 and slightly smaller, consistent with some regularization or incomplete information. Comparing it with 20 would omit the nuisance intercepts. A direct conditional multinomial fit has 20 coefficients but a different observational likelihood, since it conditions on the totals.