Acceptance-rejection-sampling

Draw a random value of 9 from the candidate density go( ). 

Calculate the weight for that value as the ratio of the target to the scaled up candidate density 

An accepted 0 is a random draw from the exact posterior distribution g 0 y), even though we only knew the unsealed posterior distribution f y O)g 0). We can draw a random sample from the posterior by repeating the steps. 

The next step is to draw a uniform 0,1) random variable. Suppose we draw u =. 435198. Since u w(2.6), we do not accept the value 2.60. Suppose the second candidate value drawn is 9 = -.94. The value of the unsealed target is 

Suppose the next uniform(0,1) value drawn is u =. 577230. We have u (—.94), JO —.94 is accepted as a draw from the target. 

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Steps for Drawing a Random Sample from the Posterior Using Acceptance-Rejection-Sampling... 

Table 2.1 Table for performing acceptance-rejection-sampling... 

Acceptance-rejection-sampling is a method for drawing a random sample from the exact posterior (the target), even though all we know is the unsealed posterior. 

Acception-rejection-sampling and sampling-importance-resampling will work for multiple parameters, but they become inefficient as the number of parameters increases. 

The main use of acceptance-rejection-sampling and adaptive-rejection-sampling will be to sample single parameters as part of a larger Markov-chain Monte Carlo process. 

We want to generate a random sample from the posterior given in Exercise 3.1. First we draw a random sample of size 100000 from a Laplace 0,l) candidate density. Use acceptance-rejection-sampling to reshape it to be a random sample from the posterior. 

Markov chain Monte Carlo samples are not independent random samples. This is unlike the case for samples drawn directly from the posterior by acceptance-rejection sampling. This means that it is more difficult to do inferences from the Markov chain Monte carlo sample. In this section we discus the differing points of view on this problem. 

In other cases, we only know the proportional form of the full conditional density for that node. We would use one of the direct methods from Chapter 2 such as acceptance-rejection-sampling or adaptive-iejection-sampling to sample from that node. 

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