Acceptance Probabilities

In the context of Metropolis Hastings, in general, an acceptance probability needs to be calculated for the proposed state. Let the target distribution, the posterior, be denoted and the prior distribution denoted . Let the parameter or parameters of interest be denoted , with chain samples of denoted x. A proposed new state will be denoted , and the current state denoted . The proposal density is just . The data, as before, may be denoted , which consist of lengths for I cracks at times .

It is known that the acceptance probability is:

Using Bayes, the form of the posterior is obtainable from the likelihood and prior as:

and the constant of proportionality is

which is clearly independent of x. Since this is the case, the ratio of the posteriors is the ratio of the products of the priors and likelihoods.

• Growth Model for with Multiplicative Variance
• Growth Model for Constant Variance