Allenby and Lenk 1994; Allenby and Rossi 1999). MCMC provides approximate draws from the joint posterior distribution on GLYPH<18>$_{1}$ ; : : : ; GLYPH<18>$_{H}$ and GLYPH<30> . The draws enable the estimation of (1) and related integrals. However, the more agents there are in the data set, the more MCMC output we need to collect and store – even if we are only interested in GLYPH<30> , we still need repeated draws for all of GLYPH<18>$_{1}$ ; : : : ; GLYPH<18>$_{H}$ . Variational methods (Jordan et al. 1999; Wainwright and Jordan 2003) offer a deter- ministic alternative for approximate inference. With variational inference, we maximize a data-dependent lower bound on the marginal likelihood (1), over a set of auxiliary pa- rameters distinct from the model parameters. In the fully Bayesian specification, the end result is an approximate joint posterior distribution for GLYPH<18>$_{1}$ ; : : : ; GLYPH<18>$_{H}$ and GLYPH<30> . For empirical Bayes, variational techniques lead to a point estimate O GLYPH<30> as well as an approximate posterior distribution for the GLYPH<18>$_{h}$ ’s. The main advantage of variational methods versus MCMC is computational efficiency. Variational inference algorithms typically converge to their final approximation in far less time than it takes to generate an adequate number of MCMC draws. This advantage comes at the cost of a biased approximation, in contrast to the consistency guarantees that accom- pany MCMC. We give evidence in Section 4 that, for our random utility discrete choice model, variational bias is negligible, and the computational speedup is very large. Varia- tional convergence is also easy to assess, in contrast to MCMC. Furthermore, the size of the variational representation is fixed, while the size of the MCMC approximation increases with the number of draws. Variational techniques can therefore be applied to much larger data sets. For example, with 10,000 decision-makers and 20,000 total MCMC draws, using a 25-dimensional GLYPH<18>$_{h}$ and corresponding GLYPH<30> , the MCMC representation of the posterior exceeds 2 GB—if we discard half the draws for burn-in of the chain. In fact, many data sets today contain observations on millions of agents, mod- els can contain far more than 25 preference parameters, and MCMC chains may require 3