probability of sampling a new v that falls between v$_{i}$ and v$_{i}$$_{+}$$_{1}$ is now so we can sample an interval bounded by v$_{i}$ and v$_{i}$$_{+}$$_{1}$ from a multinomial density with weights proportional to v $_{i}$. Once we have the i that corresponds to that interval, we can sample the con- tinuous v by sampling e from a standard exponential density, truncated on the right at v$_{i}$$_{+}$$_{1}$ − v$_{i}$ , and setting v = v$_{i}$ +$_{e}$ . Thus, we can sample v by first sampling i with weight v $_{i}$, then sampling a standard uniform random variable h , and finally setting To sample R independent draws from the target posterior, we need R “threshold” draws of v . Then, for each v , we repeatedly sample from g ($_{q}$) until − log ( F ($_{q}$ | y )) < v . Once we have a q that meets this criterion, we save it as a valid sample from $_{p}$($_{q}$ | y ) . The complete algorithm is summarized as Algorithm 1. The only restriction on g ($_{q}$) is that the inequality 0 ≤ F ($_{q}$ | y ) ≤ 1 must hold, at least for any q with a non-negligible posterior density. Because v > 0, we must have u < 1. Thus, any q for which F ($_{q}$ | y ) > 1 would always be accepted, no matter how small $_{p}$($_{q}$ | Y ) might be. By construction, F ($_{q}$ $^{∗}$| y ) = 1, meaning that no candidate q will have a higher acceptance probability than the q with the highest posterior density. This is an intuitively appealing property. In principle, it is up to the researcher to choose g ($_{q}$) , and some choices may be more efficient than others. We have found that a multivariate normal (MVN) proposal distribution, with mean at q $^{∗}$, works well for the kinds of continuous posterior densities that marketing researchers typically encounter. The MVN density, with a covariance matrix equal to the negative inverse Hessian of the log posterior at q $^{∗}$, is an asymptotic approximation (specifically, a second-order Taylor series) to the posterior density itself (Carlin and Louis 2000, sec. 5.2). By multiplying that covariance matrix by a scaling constant s , we can derive a proposal distribution that has the general shape 7 2.3 The proposal distribution v = v$_{i}$ − log [ 1 −$_{h}$ ( 1 − exp ( v$_{i}$ − v$_{i}$$_{+}$$_{1}$ )] . (15)