ClosedForm Bayesian Inferences for the Logit Model via Polynomial Expansions
Abstract
Articles in Marketing and choice literatures have demonstrated the need for incorporating personlevel 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 closedform inferences, and therefore numerical integration techniques are relied upon (e.g., MCMC methods). We present here an alternative, closedform 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 closedform expansions; if there are correlations among the coefficients one can still obtain rapidly convergent closedform 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. Closedform 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 December 2005
 arXiv:
 arXiv:math/0512444
 Bibcode:
 2005math.....12444M
 Keywords:

 Mathematics  Statistics Theory;
 Mathematics  Probability
 EPrint:
 30 pages, 2 figures, corrected some typos. Appears in Quantitative Marketing and Economics vol 4 (2006), no. 2, 173206