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

机译：通过多项式展开的logit模型的封闭式贝叶斯推断

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摘要

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.
机译：市场营销和选择文献中的文章证明了将人际异质性纳入行为模型的必要性（例如，本文研究的用于多种二元结果的logit模型）。但是，随着异质性总体分布而扩展的对数似然性不会产生封闭形式的推论，因此依赖于数值积分技术（例如MCMC方法）。在这里，我们为logit模型提供了另一种封闭形式的贝叶斯推断，它是通过多项式展开来近似logit可能性，然后从现在可以共轭和可积分的灵活族中得出异质性分布。对于响应系数是独立的问题，选择Gamma分布会导致快速收敛的闭式展开。如果系数之间存在相关性，则仍可以通过从多变量Gamma分布中假设异质性分布来获得快速收敛的闭合形式展开。然后，解决方案来自多元伽马分布的矩生成函数，或者通常来自假定的多元异质性分布。因此，可以直接获得闭合形式的贝叶斯推论，导数（用于弹性计算），总体分布参数估计值（用于汇总）和起始值（用于复杂算法）。两项仿真研究证明了我们方法的有效性。

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