A Bayesian perspective on sampling of alternatives

In this paper, we apply a Bayesian perspective to sampling of alternatives for multinomial logit (MNL) and mixed multinomial logit (MMNL) models. We find three theoretical results -- i) McFadden's correction factor under the uniform sampling protocol can be transferred to the Bayesian context in MNL; ii) the uniform sampling protocol minimises the loss in information on the parameters of interest (i.e. the kernel of the posterior density) and thereby has desirable small sample properties in MNL; and iii) our theoretical results extend to Bayesian MMNL models using data augmentation. Notably, sampling of alternatives in Bayesian MMNL models does not require the inclusion of the additional correction factor, as identified by Guevara and Ben-Akiva (2013a) in classical settings. Accordingly, due to desirable small and large sample properties, uniform sampling is the recommended sampling protocol in MNL and MMNL, irrespective of the estimation framework selected.