N 500 1,000 2,000 5,000 10,000 15,000 20,000 30,000 40,000 50,000 scale factor 1.22 1.16 1.10 1.08 1.04 1.03 1.03 1.03 1.03 1.02 acc. rate ( × 10 − $^{5}$) 2.1 1.5 2.6 0.7 2.9 2.6 3.6 3.9 2.7 3.3 Table 2: Acceptance rates for binary choice simulation. For each condition, k = 3, so there are six population-level parameters and 3 N heterogeneous parameters. proposed methods to approximate the marginal likelihood, L ( y ) from MCMC output (Gelfand and Dey 1994; Newton and Raftery 1994; Chib 1995; Raftery et al. 2007), but none has achieved universal acceptance as being consistent, stable and easy to compute. In fact, Lenk (2009) demon- strated that methods that depend solely on samples from the posterior density could suffer from a “pseudo-bias,” and he proposes an importance-sampling method to correct for it. This pseudo- bias arises because the convex hull of MCMC samples defines only a subset of the posterior sup- port, while L ( y ) is defined as an integral of the data likelihood over the prior distribution. Lenk demonstrates that his method dominates other popular methods, although with substantial com- putational effort. Thus, the estimation of the marginal likelihood remains a difficult problem in MCMC-based Bayesian statistics. We estimate the marginal likelihood using quantities that we already collect during the course of the estimation procedure. Recall that q ( u ) is the probability that, given a threshold value u , a proposal from g ($_{q}$) is accepted as a sample from $_{p}$($_{q}$ | y ) . Therefore, after substituting in Equation 8, we can express the expected marginal acceptance probability for any one posterior sample as Applying a change of variables so v = − log u , and then rearranging terms, The values for c$_{1}$ and c$_{2}$ are immediately available from the algorithm. A reasonable estima- tor of g is ˆ g , the inverse of the mean of the observed average number of proposals per accepted sample. What remains is estimating the integral in Equation 19, for which we use the same pro- posal draws that we already collected for estimating ̂ q$_{v}$ ( v ) . The empirical CDF of these draws is discrete, so we can partition the support of q$_{v}$ ( v ) at v$_{1}$ . . . v$_{M}$ . Also, since ̂ q$_{v}$ ( v ) is the proportion of 27 g = ∫ 1 0 q ( u ) p ( u | y ) du = c$_{1}$ c$_{2}$ L ( y ) ∫ 1 0 q $^{2}$( u ) du (18) L ( y ) = − c$_{1}$ c$_{2}$$_{g}$ ∫ ¥ 0 q $^{2}$( v ) exp ( − v ) dv . (19)