A Bayesian perspective on sampling of alternatives

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.