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In this paper, we revisit McFadden (1978)'s correction factor for sampling ofalternatives in multinomial logit (MNL) and mixed multinomial logit (MMNL)models. McFadden (1978) proved that consistent parameter estimates are obtainedwhen estimating MNL models using a sampled subset of alternatives, includingthe chosen alternative, in combination with a correction factor. We decomposethis correction factor into i) a correction for overestimating the MNL choiceprobability due to using a smaller subset of alternatives, and ii) a correctionfor which a subset of alternatives is contrasted through utility differencesand thereby the extent to which we learn about the parameters of interest inMNL. Keane and Wasi (2016) proved that the overall expected positiveinformation divergence - comprising the above two elements - is minimisedbetween the true and sampled likelihood when applying a sampling protocolsatisfying uniform conditioning. We generalise their result to the case ofpositive conditioning and show that whilst McFadden (1978)'s correction factormay not minimise the overall expected information divergence, it does minimisethe expected information loss with respect to the parameters of interest. Weapply this result in the context of Bayesian analysis and show that McFadden(1978)'s correction factor minimises the expected information loss regardingthe parameters of interest across the entire posterior density irrespective ofsample size. In other words, McFadden (1978)'s correction factor has desirablesmall and large sample properties. We also show that our results for BayesianMNL models transfer to MMNL and that only McFadden (1978) correction factor issufficient to minimise the expected information loss in the parameters ofinterest. Monte Carlo simulations illustrate the successful application ofsampling of alternatives in Bayesian MMNL models.

Revisiting McFadden's correction factor for sampling of alternatives in multinomial logit and mixed multinomial logit models