## 5 Discussion 

Variational methods allow estimation of hierarchical discrete choice models in a small fraction of the time required for MCMC. They open Bayesian inference to a wide class of problems for which resource constraints make MCMC intractable, such as the MML model with many heterogeneous units. For now, variational methods appear to be the only viable option in these cases. 

Of course, one can use variational methods to estimates many more types of models than the MML model examined here. Within the MML family, it would be straightforward to add a utility scaling parameter, or allow the heterogeneous coefficients themselves to depend on observed covariates. The value of the variational approach is greatest when sub- sampling of data is ill-advised. For example, consider a linear model with heterogeneous coefficients on exogenous and endogenous covariates, where the available instrumental variables only weakly explain the endogenous part. To draw inferences about the covari- ances of the heterogeneous parameters, we may need a large amount of data to achieve reasonable power in hypothesis testing. MCMC is untenable here, but variational methods have promise. When factorized variational distributions are inadequate, alternatives such as mixtures of normals or Dirichlet processes (Blei and Jordan 2006) can be applied. 

We emphasize that we do not advocate abandoning MCMC in favor of variational meth- ods. On the contrary, we suggest using MCMC when possible. MCMC offers consistency guarantees with no analog in variational methods, and it has withstood decades of scrutiny. But we advise against subsampling the data (i.e., throwing out information), or discarding key modeling elements, simply to make the problem fit within the time and resource con- straints of MCMC. The possibility of applying variational methods to previously intractable problems makes them an important addition to the statistician’s toolkit. 

