posal draws. 

Table 3 presents the true log marginal likelihood (MVT), along with estimates using our method, the importance sampling method in Lenk (2009), and the harmonic mean estimator (Newton et al. 1994). We also included the mean acceptance probabilities, and the standard deviations of the various estimates across the simulated datasets. We can see that our estimates for the log marginal likelihood are remarkably close to the MVT densities, and are robust when we use different scale factors. Accuracy appears to be better for larger datasets than smaller ones, while improving the approximation of p ( u | y ) by increasing the number of proposal draws offers negligible improve- ment. Note that the performance of our method is comparable to that of Lenk, but is much better than the harmonic mean estimator. Our method is similar to Lenk’s in that it computes the prob- ability that a proposal draw falls within the support of the posterior density. However, the inputs to the estimator of the marginal likelihood are intrinsically generated as the algorithm progresses. In contrast, the Lenk estimator requires an additional importance sampling run after the MCMC draws are collected. 

## 6 Discussion of practical considerations and limitations 

To those who have spent long work hours dealing with MCMC convergence and efficiency issues, the utility of an alternative algorithm is appealing. Ours allows for sampling from a posterior density in parallel, without having to worry about whether an MCMC estimation chain has con- verged. If heterogeneous units (like households) are conditionally independent, then the sparsity of the Hessian of the log posterior lets us construct a sampling algorithm whose complexity grows only linearly in the number of units. This method makes Bayesian inference more attractive to practitioners who might otherwise be put off by the inefficiencies of MCMC. 

This is not to say that our method is guaranteed to generate perfect samples from the target posterior distribution. One area of potential concern is that the empirical distribution ̂ q$_{v}$ ( v ) is only a discrete approximation to q$_{v}$ ( v ) . That discretization could introduce some error into the estimate of the posterior density. This error can be reduced by increasing M (the number of proposal draws that we use to compute ̂ q$_{v}$ ( v ) ), at the expense of costlier computation of ̂ q$_{v}$ ( v ) and, possibly, lower acceptance rates. In our experience, and consistent with Figure 3, we have not found this to be a 

