In this paper, we apply a Bayesian perspective to the sampling of alternatives for multinomial logit (MNL) and mixed multinomial logit (MMNL) models. A sampling of alternatives reduces the computational challenge of evaluating the denominator of the logit choice probability for large choice sets by only using a smaller subset of sampled alternatives including the chosen alternative. To correct for the resulting overestimation of the choice probability, a correction factor has to be applied. McFadden (1978) proposes a correction factor to the utility of each alternative which is based on the probability of sampling the smaller subset of alternatives and that alternative being chosen. McFadden's correction factor ensures consistency of parameter estimates under a wide range of sampling protocols. A special sampling protocol discussed by McFadden is uniform conditioning, which assigns the same sampling probability and therefore the same correction factor to each alternative in the sampled choice set. Since a constant is added to each alternative the correction factor cancels out, but consistent estimates are still obtained. Bayesian estimation is focused on describing the full posterior distributions of the parameters of interest instead of the consistency of their point estimates. We theoretically show that uniform conditioning is sufficient to minimise the loss of information from a sampling of alternatives on the parameters of interest over the full posterior distribution in Bayesian MNL models. Minimum loss of information is, however, not guaranteed for other sampling protocols. This result extends to Bayesian MMNL models estimated using the principle of data augmentation. The application of uniform conditioning, a more restrictive sampling protocol, is thus sufficient in a Bayesian estimation context to achieve finite sample properties of MNL and MMNL parameter estimates.
翻译:在本文中,我们从巴伊西亚角度出发,对多种代谢逻辑(MNL)和混合的多种统一逻辑(MMNL)模型的替代物取样。对替代品的抽样,通过只使用一小部分抽样替代物(包括所选替代物)来减少对大选择组的逻辑选择概率进行计算评估的计算挑战。为了纠正由此造成的过高估计选择概率,必须使用一个校正系数。McFadden(1978年)对每种替代物的实用性提出了一个校正系数,该系数基于对较少量替代物进行取样的概率和选择的替代物进行选择。McFadden(MMM)的校正系数校正系数确保了在一系列广泛抽样协议下对参数的估算的一致性。 McFadden(MFadden)讨论的特别抽样协议对大型选择组群选择组的分类选择的概率概率进行了统一的调整。由于每种替代物的校正系数被抵消,但仍得到一致的估计。Bayesian(MM)的估算值的完整海边框值分布,因此,在最起码的估算值的数值中可以得出对最低的数值的数值的数值的精确值的数值。