proposal draws less than v , we have q$_{v}$ ( v$_{i}$ ) = i / M . Therefore, Putting all of this together, we can estimate the marginal likelihood as As a demonstration of the accuracy of this estimator, we use the same linear regression exam- ple that Lenk (2009) uses. 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- 28 y$_{it}$ ∼ N ( x ′ i b , s $^{2}$) , i = 1 . . . n , t = 1 . . . T (24) $_{b}$|$_{s}$ ∼ N ($_{b}$$_{0}$ , s $^{2}$V$_{0}$ ) s 2 ∼ IG ( r , $_{a}$) (25) L ( y ) ≈ c$_{1}$ M $^{2}$c$_{2}$ ˆ g M GLYPH<229> i = 1 ( 2 i − 1 ) exp ( − v$_{i}$ ) (23) ∫ ¥ 0 q $^{2}$( v ) exp ( − v ) dv ≈ M GLYPH<229> i = 1$^{∫}$ v$_{i}$$_{+}$$_{1}$ v$_{i}$ ( i M$^{$_{)}$2}$ exp ( − v$_{i}$ ) dv (20) = 1 M 2 M GLYPH<229> i = 1 i $^{2}$[ exp ( − v$_{i}$ ) − exp ( − v$_{i}$$_{+}$$_{1}$ )] (21) = 1 M 2 M GLYPH<229> i = 1 ( 2 i − 1 ) exp ( − v$_{i}$ ) (22)