# Bayesian Reliability Analysis

How do I derive the posterior density of a Weibull Distribution?

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#### Solution Preview

The question is very general. Therefore, I've got to start from Bayesian estimation and an example follows. Please refer to the attachment.

Bayesian estimation

Now let's focus on the problem of estimation of a 1 dimensional parameter. Mean Squared Error corresponds to using

The risk function of a procedure (estimator) is

Now consider using a prior with density . The Bayes risk of is

How should we choose to minimize ? The solution lies in recognizing that is really a joint density

For this joint density the conditional density of X given is just the model . From now on I write the model as to emphasize this fact. We can now compute a ...

#### Solution Summary

The solution answers the question(s) below. Posterior density of Weibull distribution