Closed-form Bayesian inferences for the logit model via polynomial expansions |
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Authors: | Steven J Miller Eric T Bradlow Kevin Dayaratna |
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Institution: | (1) Brown University, Providence;(2) Wharton Small Business Development Center, Wharton School of the University of Pennsylvania, USA;(3) Marketing Department, University of Maryland, USA |
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Abstract: | 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.
JEL Classification C6 · C8 · M3 |
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Keywords: | Closed-Form Bayesian Inferences Logit model Generalized Multivariate Gamma Distribution |
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