The least-squares method reconsidered

Our proposed prior (8.68) makes use of the assumption of prior independence, 

Assuming the Gauss-Markov conditions hold and that the errors are normally distributed, the hkelihood function is 

as we have assmned in (8.68), the prior is uniform in the region of appreciable nonzero likelihood, p(0) c, then the most probable value of 0, for any value of cr, is that which minimizes S(0), (8.58). Therefore, the least-squares method is justified statistically, as long as the Gauss Markov conditions hold and the errors are normally distributed. 

It is shown in the supplemental material in the accompanying website that, in the ffe-quentist view, least squares is an unbiased estimator of the true value (i.e., if we repeat the set of experiments many times, the average estimate is the true value) if merely the zero-mean Gauss-Markov condition (8.50) is satisfied. 

Numerical treatment of nonlinear least-squares problems 

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