TABLE OF CONTENTS In the last chapter, we learned that the posterior distribution for the parameters in a linear model with Gaussian errors and flat priors is itself a multivariate Gaussian. The topology for this distribution in the multi-dimensional parameter space is very simple. In contrast, even for flat priors, the topology of the posterior for a nonlinear model can be very complex with many hills and valleys.
Examples of nonlinear models:
f i = A 1 cos ?t i + A 2 sin ?t i
where A 1, A 2 are linear parameters,
and ? is a nonlinear parameter.
where A 1, A 2, A 3 are linear parameters,
and C 1, C 2, 
, 
are nonlinear parameters.
In this chapter, we will let ? represent the set of all parameters both linear and nonlinear and 
the most probable set of the parameters. Again, the problem is to find the most probable set of parameters together with an estimate of their errors. (Of course, if the posterior has several maxima of comparable magnitude then it doesn t make sense to talk about a single best set of parameters.) The Bayesian solution to the problem is very simple in principle but can be very difficult in practice. The calculations require integrals over the parameter space which can be difficult to evaluate.
The brute force approach is as follows: for a one-parameter model, the most robust...
