I study predictive evaluation for credit rating migrations where each time period t and origin grade m produces a multinomial row of counts N\_{j m,t} over destinations j=1,\\dots,q with cohort size n\_{m,t}=\\sum\_{j} N\_{j m,t}. For a Bayesian transition model I compute the pointwise log posterior predictive density for that row as $$ \mathrm{lppd}{t,m} =\log\left(\frac{1}{S}\sum{s=1}^{S}\frac{n!}{\prod_{j=1}^{q}N_j!}\prod_{j=1}^{q}\bigl(p^{(s)}*{j}\bigr)^{N_j}\right), $$ where p^{(s)} are draws from the posterior for the row probabilities and I write n=n*{m,t} and N_j=N\_{j m,t} for brevity.
Using Stirling expansions with uniform remainder bounds for the combinatorial factor \\log n!-\\sum\_{j}\\log N_j! together with method of types bounds for multinomial probabilities, I arrive at the per cohort normalization $$ \frac{\mathrm{lppd}{t,m}}{n} =\sum{j=1}^{q} f_j\log \bar p_j ;-; \sum_{j=1}^{q} f_j\log f_j ;+; \mathcal{O}!\left(\tfrac{\log n}{n}\right) + r_n, $$ where f_j=N_j/n are empirical proportions and \\bar p_j=\\tfrac{1}{S}\\sum\_{s=1}^{S} p^{(s)}*j. Under posterior concentration at p^\\star with strictly positive coordinates on the support of f I get $$ \frac{\mathrm{lppd}*{t,m}}{n} \to \sum_{j} f_j \log p^\star_j ;-; \sum_{j} f_j \log f_j, $$ and replacing f by the true probabilities q yields the expected limit -D\_{\\mathrm{KL}}(q\\Vert p^\\star). Without assuming concentration, a Laplace or Varadhan argument suggests that \\tfrac{1}{n} times the log of the posterior mixture converges to the supremum of \\sum\_{j} f_j \\log p_j over the posterior support.
Robbins gives uniform Stirling bounds that imply the entropy term plus an order (\\log n)/n correction. Cover and Thomas and also Csiszár give method of types bounds for multinomial probabilities which translate into the same per cohort order when divided by n. Dembo and Zeitouni state the Laplace Varadhan principle which yields the supremum limit for log integrals of the form \\tfrac{1}{n}\\log \\int \\exp{n,g(p)},\\Pi(\\mathrm{d}p). Clarke and Barron prove that in regular parametric models the average log posterior predictive converges to the expected log likelihood at the true parameter with the familiar dimension over two times \\log n over n correction.
Is there a single reference that states a theorem for multinomial tables or discrete exponential families that directly yields the normalization above for \\mathrm{lppd}, including the explicit order (\\log n)/n remainder, with assumptions on boundary behavior made clear. Alternatively, is there a standard citation that combines the method of types rate with the Stirling remainder to give this per cohort expansion in one place. In the absence of a row wise statement, is it acceptable to cite Clarke and Barron for the general predictive \\mathrm{lppd} limit together with Robbins for Stirling and Cover and Thomas for the multinomial rate, and present the combination as a short sketch.
Thanks!
PS. I can’t seem to get it all in mathmode, sorry for that :-)