TABLE OF CONTENTS Here we briefly summarize the main results of this chapter:
We saw how the Bayesian treatment leads to the familiar method of least-squares when we are interested in the question of the most probable set of model parameters (see Equation (10.34)), assuming an IID normal distribution for our knowledge of the measurement errors and a flat prior for each parameter.
We then relaxed the IID requirement for our knowledge of the measurement errors by introducing E, the covariance matrix for the errors. Equation (10.48) gives the revised solution for the most probable set of parameters. Weighted linear least-squares can be seen as a special case of this equation.
A full description of our knowledge of the model parameters is given by the joint posterior distribution for the parameters. For a linear model, and a flat prior for each parameter, this distribution is particularly simple, namely a multivariate Gaussian. Equation (10.75) or Table 10.2 defines the boundary, ? 
, of a (joint) credible region for one or more of the parameters that contains a specified probability. Also, it turns out that for Gaussian posteriors, in any number of dimensions, the marginal PDF is also equal to the projected distribution (projected PDF).
A useful summary of the parameter errors is given by the model parameter covariance matrix, V = ? 2 ? ?1. If we are employing the covariance matrix, E, for our knowledge of the measurement errors, then simply replace ? ?1 by ?
