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

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. Copyright Springer Science + Business Media, LLC 2006Closed-Form Bayesian Inferences, Logit model, Generalized Multivariate Gamma Distribution,

Count Models Based on Weibull Interarrival Times

“…This is accomplished by deriving our model using a polynomial expansion (which can be expressed in closed form). See Bradlow, Hardie, and Fader (2000), Everson and Bradlow (2002), and Miller, Bradlow, and Dayaratna (2006) for similar polynomial expansion solutions for the negative binomial, beta-binomial, and binary logit models, respectively. Fifth, and related to the previous point, once the model is expressed as a closed-form sum of polynomial terms, we can easily introduce a conjugate mixing distribution (the gamma distribution) to capture the underlying dispersion in incidence rates across individuals.…”

Perspectives on Bayesian Methods and Big Data

“…In Bradlow et al [2], these approximation methods are applied to count data in the form of a negative-binomial distribution. This was subsequently extended by Everson and Bradlow [5] to the Beta-Binomial model and then more generally to choice models [15]. This research is all related to recent research on Variational Bayesian Methods (http://www.variational-bayes.org/; [10]) that provide more general approximation methods.…”

Closed Form Bayesian Inferences for Binary Logistic Regression with Applications to American Voter Turnout

“…In subsequent research, McShane et al (2008) used similar techniques to improve on Weibull count model estimation [35]. Finally, used polynomial expansions to examine the same problem examined here, namely for binary logistic regression [36]. Miller et al's approach, however, suffered from a serious limitation by requiring that the prior distribution be single-sided.…”