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Consider a set of categorical variables where at least one of them is binary.The log-linear model that describes the counts in the resulting contingencytable implies a specific logistic regression model, with the binary variable asthe outcome. Within the Bayesian framework, the $g$-prior and mixtures of$g$-priors are commonly assigned to the parameters of a generalized linearmodel. We prove that assigning a $g$-prior (or a mixture of $g$-priors) to theparameters of a certain log-linear model designates a $g$-prior (or a mixtureof $g$-priors) on the parameters of the corresponding logistic regression. Byderiving an asymptotic result, and with numerical illustrations, we demonstratethat when a $g$-prior is adopted, this correspondence extends to the posteriordistribution of the model parameters. Thus, it is valid to translate inferencesfrom fitting a log-linear model to inferences within the logistic regressionframework, with regard to the presence of main effects and interaction terms.

On the correspondence from Bayesian log-linear modelling to logistic regression modelling with $g$-priors