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proposal draws less than v , we have q$_{v}$ ( v$_{i}$ ) = i / M . Therefore, 

<img src='content_image/325545.jpg'>

Putting all of this together, we can estimate the marginal likelihood as 

<img src='content_image/325544.jpg'>

As a demonstration of the accuracy of this estimator, we use the same linear regression exam- ple that Lenk (2009) uses. 

<img src='content_image/325542.jpg'>

<img src='content_image/325543.jpg'>

For this model, L ( y ) is a multivariate-t density (MVT), which we can compute analytically. This allows us to compare the estimates of L ( y ) with the “truth.” To do this, we conducted a simulation study for simulated datasets of different numbers of observations n ∈ { 200, 2000 } and numbers of covariates k ∈ { 5, 25, 100 } . For each n , k pair, we simulated 25 datasets. For each dataset, each vector x$_{i}$ includes an intercept and k iid samples from a standard normal density. Thus, there are k + 2 parameters, corresponding to the elements of b , plus s . The true intercept term is 5, and the remaining true b parameters are linearly spaced from − 5 to 5. In all cases, there are T = 25 observations per unit. Hyperpriors are set as r = 2, a = 1, b 0 as a zero vector and V$_{0}$ = 0.2 · I$_{k}$ . 

For each dataset, we collected 250 samples from the posterior density, with different numbers of proposal draws ( M = 1, 000 or 10, 000), and different scale factors ( s = 0.5, 0.6, 0.7, or 0.8) on the Hessian ( − sH is the precision matrix of the MVN proposal density, and lower scale factors generate more diffuse proposals). We excluded the s = 0.8, n = 200 case because the proposal density was not sufficiently diffuse to ensure that F ($_{q}$ | y ) was between 0 and 1 across the M pro- 

