Closed-form Bayesian inferences for the logit model via polynomial expansions

Articles in Marketing and choice literatures have demonstrated the need for incorporating person-level heterogeneity into behavioral models (e.g., logit models for multiple binary outcomes as studied here). However, the logit likelihood extended with a Population distribution of heterogeneity doesn't yield closed-form inferences, and therefore numerical integration techniques are relied upon (e.g., MCMC methods). We present here an alternative, closed-form Bayesian inferences for the logit model, which we obtain by approximating the logit likelihood via a polynomial expansion, and then positing a distribution of heterogeneity from a flexible family that is now conjugate and integrable. For problems where the response coefficients are independent, choosing the Gamma distribution leads to rapidly convergent closed-forin expansions; if there are correlations among the coefficients one can still obtain rapidly convergent closed-forin expansions by positing a distribution of heterogeneity from a Multivariate Gamma distribution. The solution then comes from the moment generating function of the Multivariate Gamma distribution or in general from the inultivariate heterogeneity distribution assumed. Closed-form Bayesian inferences, derivatives (useful for elasticity calculations), POPL[lation distribution parameter estimates (useful for summarization) and starting values (useful for complicated algorithins)