problem, but some applications for which this may be an issue might exist. Like many other methods that collect random samples from posterior distributions, its effi- ciency depends in part on a prudent selection of the proposal density g ($_{q}$) . For the examples in this paper, we use a MVN density that is centered at the posterior mode with a covariance matrix that is proportional to the inverse of the Hessian at the mode. One might then wonder if there is an optimal way to determine just how “scaled out” the proposal covariance needs to be. At this time, we think that manual search is the best alternative. If we start with a small M (say, 100 draws), and find that F ($_{q}$ | y ) > 1 for any of the M proposals, we have learned that the proposal density is not valid, with little computational or real-time cost. We can then re-scale the proposal until F ($_{q}$ | y ) < 1, and then gradually increase M until we get a good approximation to p ( u ) . This is no different, in principle, than the tuning step in a Metropolis-Hasting algorithm. However, our method has the advantage that we can make these adjustments before the posterior sampling phase begins. In contrast, with adaptive MCMC methods an improperly tuned sampler might not be apparent until the chain has run for a substantial period of time. Also, even if an acceptance rate appears to be low, we can still collect draws in parallel, so the “clock time” remains much less than the time we spend trying to optimize selection of the proposal. There are many popular models, such as multinomial probit, for which the likelihood of the observed data is not available in closed form. When direct numerical approximations to these like- lihoods (e.g., Monte Carlo integration) are not tractable, MCMC with data augmentation is a possi- ble alternative. Recent advances in parallelization using graphical processing units (GPUs) might make numerical estimation of integrals more practical than it was even 10 years ago (Suchard et al. 2010). If so, then our method could be a viable, efficient alternative to data augmentation in these kinds of models. Multiple imputation of missing data could suffer from the same kinds of problems, since a latent parameter, introduced for the data augmentation step, is only weakly identified on its own. If the number of missing data points is small, one could treat the represen- tation of the missing data points as if they were parameters. But the implications of this require additional research. Another opportunity for further research involves the case of multimodal posteriors. Our method does require finding the global posterior mode, and all the models discussed in this paper have unimodal posterior distributions. When the posterior is multimodal, one might instead use 31