  In this paper, we revisit McFadden (1978)'s correction factor for sampling of
alternatives in multinomial logit (MNL) and mixed multinomial logit (MMNL)
models. McFadden (1978) proved that consistent parameter estimates are obtained
when estimating MNL models using a sampled subset of alternatives, including
the chosen alternative, in combination with a correction factor. We decompose
this correction factor into i) a correction for overestimating the MNL choice
probability due to using a smaller subset of alternatives, and ii) a correction
for which a subset of alternatives is contrasted through utility differences
and thereby the extent to which we learn about the parameters of interest in
MNL. Keane and Wasi (2016) proved that the overall expected positive
information divergence - comprising the above two elements - is minimised
between the true and sampled likelihood when applying a sampling protocol
satisfying uniform conditioning. We generalise their result to the case of
positive conditioning and show that whilst McFadden (1978)'s correction factor
may not minimise the overall expected information divergence, it does minimise
the expected information loss with respect to the parameters of interest. We
apply this result in the context of Bayesian analysis and show that McFadden
(1978)'s correction factor minimises the expected information loss regarding
the parameters of interest across the entire posterior density irrespective of
sample size. In other words, McFadden (1978)'s correction factor has desirable
small and large sample properties. We also show that our results for Bayesian
MNL models transfer to MMNL and that only McFadden (1978) correction factor is
sufficient to minimise the expected information loss in the parameters of
interest. Monte Carlo simulations illustrate the successful application of
sampling of alternatives in Bayesian MMNL models.
