The confidence intervals derived above are based upon a quadratic expansion for S( ?) about ? _M that is only approximate for a nonlinear model. The exact single-response posterior without this approximation is

While analytical manipulation of this formula is difficult, MCMC simulation is a powerful tool to obtain posterior expectations of the form

Many statistical questions can be posed in this form. For example, let us say that we wish to compute the probability that some hypothesis H _? is true. Let ? be the region in ( ?, ?) space in which the hypothesis H _? is true, and outside of ?, the hypothesis is false. Let I _?( ?, ?) be the indicator function

For example, if we wish to test the hypothesis that ? _lo ? ?j ? ? _hi, we use the indicator function

The probability that the hypothesis is true then takes the form of a posterior expectation of the indicator function:

When the dimension of ( ?, ?) space is small, it may be possible to compute (8.152) by quadrature, but MCMC simulation is generally more efficient.

Such calculations also arise when we wish to use knowledge gained from analysis of the data to choose an optimal course of action, and form the basis of statistical decision theory. We want to consider which of a set { a _j}...