Candidate density Metropolis-Hastings

The Metropolis-Hastings algorithm uses an auxiliary probability density function, (P P), from which it is easy to obtain sample values. Assuming that the chain is in a state P, a new candidate value, P, is generated from the auxiliary distribution (P P), given the current state of the chain P. 

In Chapter 7 we develop a method for finding a Markov chain that has good mixing properties. We will use the Metropolis-Hastings algorithm with heavy-tailed independent candidate density. We then discuss the problem of statistical inference on the sample from the Markov chain. We want to base our inferences on an approximately random sample from the posterior. This requires that we determine... 

In Section 6.1 we show how the Metropolis-Hastings algorithm can be used to find a Markov chain that has the posterior as its long-run distribution in the case of a single parameter. There are two kinds of candidate densities we can use, random-walk candidate densities, or independent candidate densities. We see how the chain... 

Figure 6.1 Six consecutive draws from a Metropolis-Hastings chain with a random-walk candidate density. Note the candidate density is centered around the current value.

Figure 6.2 Trace plot and histogram of 1000 Metropolis-Hastings values using the random-walk candidate density with standard deviation 1.

Figure 6.6 Histograms for 5000 and 20000 draws from the Metropolis-Hastings chain using independent candidate gencaating density with mean 0 and standard deviation 3.

Figure 6.8 Trace plots of 6i and 02 for 1000 steps of the Metropolis-Hastings chain with the random-walk candidate density.

Figure 6.14 Histograms of 0 and 02 for 5000 and 20000 draws of the Metropolis-Hastings chain with the independent candidate density.

In the blockwise Metropolis-Hastings algorithm the candidate density for the block of parameters Oj given all the other parameters 0-j and the data y must dominate the true conditional density in the tails. That is... 

The Gibbs sampling algorithm is a special case of blockwise Metropolis-Hastings where the candidate density for each block of parameters is its correct conditional distribution given all other parameters not in its block and the observed data. The acceptance probability is always 1, so every candidate is accepted. 

The appearance of the traceplot of a Metropolis-Hastings chain depends on the type of candidate density used. 

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