Selecting a prior for single-response data

We note again that convergence of Newton s method does not require the use of this exact Hessian, but convergence to a minimum does require the approximate Hessian to be positive-definite at each iteration. If we define the linearized design matrix with the elements 

If we approximate the Hessian by retaining only the first contribution, we have an approximation that is always at least positive-semidefinite. 

The gradient components also are expressed simply in terms of X g, 

We now return to the question of proposing a prior, based first upon tiie assumption of prior 

Using the likelihood function that follows from die Gauss-Markov conditions and the assumption of normally-distributed errors, we have 

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