Metropolis-Hastings Algorithm for Multiple Parameters

Let q 6, 6) be the candidate density when the chain is at 6 and let g 0 y) be the posterior density. The reversibility condition will be 

Multiple Parameters with a Random-Walk Candidate Density 

For a random-walk candidate distribution the candidate is drawn from a symmetric distribution centered at the current value. Suppose we have p parameters represented by the parameter vector 6. Since we are using a random-walk candidate density it given by 

This means that a eandidate 0 that has a higher value of the target density than target density of the current value 0 will always be accepted. The chain will always move uphill. On the other hand, a candidate with a lower target value will only be accepted with a probability equal to the proportion of target density value to current density value. There is a certain probability that the chain will move downhill. This allows a chain with a random-walk candidate density to move around the whole parameter space. 

Example 9 Suppose there are two parameters, 0 and 62- It is useful to try a target density that we know and could approach analytically, so we know what a random sample from the target should look like. We will use a bivariate normal fi, V) distribution with mean vector and covariance matrix equal to 

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