Raftery (1986) A Note on Bayes Factors for Log-linear Contingency Table Models with Vague Prior Information

Summary

A two-page technical note making two points about the Spiegelhalter-Smith (1982) approximate Bayes factor $B_{01}$ for comparing a log-linear contingency-table model $M_0$ against the saturated model $M_1$. First, the SS approximation is indeterminate when any cell frequency is zero (their prior $\propto\prod\phi_i^{-1}$); switching to the standard Jeffreys prior $\propto\prod\phi_i^{-1/2}$ cures this — the same formula holds with $x_i$ replaced by $x_i+\tfrac12$, and the prior is then proper so no arbitrary constant is needed. Second, $-2\log B_{01}$ is asymptotically equivalent to $\chi^2 - s\log n$, i.e. Schwarz's (1978) BIC — an early statement of the Bayes-factor/BIC connection later systematized by Kass-Raftery (1995).

Key Claims

• Zero-cell indeterminacy. The Spiegelhalter-Smith Bayes factor SS(32) breaks down if any $x_i=0$, because their prior density $\propto\prod_i(1/\phi_i)$ is improper in a way that leaves the answer undefined at empty cells.
• Jeffreys-prior fix. Using the Jeffreys prior $\propto\prod_i(1/\phi_i)^{1/2}$ instead, the Bayes factor is still given by SS(32) but with every $x_i$ replaced by $x_i+\tfrac12$ (and the normalizing constant SS(33) adjusted). Because this prior is proper, the arbitrary multiplicative constant that the SS "imaginary observations" device was invented to handle need not arise at all — in principle one could apply Bayes' theorem directly (though exact contingency-table Bayes factors remain hard in practice).
• BIC as an approximate Bayes factor. Conditional on $M_0$, $-2\log B_{01}\;\overset{p}{\simeq}\;\chi^2 - s\log n = \text{BIC}$, where $\chi^2$ is the likelihood-ratio goodness-of-fit statistic, $s$ the number of constraints, and $n=\sum x_i$. This is Schwarz's (1978) criterion; replacing $\log n$ by $2$ gives AIC.
• Consistency. BIC $\to-\infty$ in probability under $M_0$ and $\to+\infty$ under $M_1$, so the BIC-based selection is consistent in the contingency-table setting.

Concepts Introduced or Extended

• Bayes Factor

Entities Mentioned

• Adrian E. Raftery

Quotes

"The approximate Bayes factor, $B_{01}$, for log-linear contingency table models proposed by Spiegelhalter and Smith (1982) is indeterminate if any of the cell frequencies is zero. It is noted that use of a standard Jeffreys prior overcomes this difficulty."

"$-2\log B_{01}\;\simeq\;\chi^2 - s\log n = \text{BIC}$ ... equivalent to the Schwarz (1978) model selection criterion."

My Take

A minor note in length but a meaningful waypoint in Raftery's program: the observation that $-2\log B_{01}\approx$ BIC is exactly the bridge — approximate a hard-to-compute Bayes factor by an easy penalized-likelihood criterion — that he would generalize a decade later into the widely-cited Kass-Raftery (1995) treatment and the BIC-based BMA machinery. The zero-cell / Jeffreys fix is the practically useful bit for anyone doing Bayesian log-linear analysis (empty cells are the norm in sparse tables), and it is the same improper-prior-calibration concern that recurs in Raftery-Akman (1986) via the Spiegelhalter-Smith imaginary-observation device — here dissolved simply by using a proper Jeffreys prior. For the wiki it grounds the "BIC is an asymptotic Bayes factor" claim on the Bayes Factor page with a concrete, early derivation.