content_image/325403.jpg
<img src='content_image/325403.jpg'>

of the target posterior near its mode. That proposal distribution will be valid as long as s is large enough so that F ($_{q}$ | y ) is between 0 and 1 for any plausible value of q , and that the mode of g ($_{q}$) is at q $^{∗}$. 

We illustrate the idea of scaling the proposal density in Figure 1. The solid line (the same in all three panels) is a “target” posterior density. The dotted lines plot potential normal proposal densities, multiplied by the corresponding c$_{2}$ / c$_{1}$ ratio. The covariance of the proposal density in the left panel is the negative inverse Hessian of the log posterior density. Samples from the left tail of the posterior distribution will have F ($_{q}$ | y ) > 1. In the middle and right panels, the covariance is the same as in the left panel, but multiplied by 1.4 and 1.8, respectively. As the covariance increases, more of the posterior support will have F ($_{q}$ | y ) ≤ 1. 

It is possible that g ($_{q}$) could under-sample values of q in the tails of the posterior. However, if M is sufficiently large, and F ($_{q}$ | y ) ≤ 1 for all M proposals, then it is unlikely that we would see F ($_{q}$ | y ) > 1 in the rejection sampling phase of the algorithm. If that were to happen, we can stop, rescale, and try again. Any values of q that we might miss would have such low posterior density that there would be little meaningful effect on inferences we might make from the output. 

We believe that the potential cost from under-sampling the tails is dwarfed by our method’s relative computational advantage, we recognize that there may be some applications for which sampling extreme values of q may be important. In that case, this may not be the best estimation method for those applications. Otherwise, there is nothing special about using the MVN for g ($_{q}$) . It is straightforward to implement with manual adaptive selection of s . This is similar, in spirit, to the concept of tuning a Metropolis-Hastings algorithm. Heavier-tailed proposals, such as the multivariate-t distribution, can fail because of high kurtosis at the mode. 

