  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-form expansions; if there are correlations among the
coefficients one can still obtain rapidly convergent closed-form 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 multivariate
heterogeneity distribution assumed.
  Closed-form Bayesian inferences, derivatives (useful for elasticity
calculations), population distribution parameter estimates (useful for
summarization) and starting values (useful for complicated algorithms) are
hence directly available. Two simulation studies demonstrate the efficacy of
our approach.
