1 leapm.tex, 4/21/1999 STRAIGHT LINE FITTING-- A BAYESIAN SOLUTION
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Abstract:
Abstract. Fitting the "best " straight line to a scatter plot of data D j f(x 1; y 1) : : : (x n; y n)g in which both variables x i; y i are subject to unknown error is undoubtedly the most common problem of inference faced by scientists, engineers, medical researchers, and economists. The problem is to estimate the parameters ff; fi in the straight line equation y = ff + fix, and assess the accuracy of the estimates. Whenever we try to discover or estimate a relationship between two factors we are almost sure to be in this situation. But from the viewpoint of orthodox statistics the problem turned out to be a horrendous can of worms; generations of efforts led only to a long line of false starts, and no satisfactory solution. We give the Bayesian solution to the problem, which turns out to be eminently satisfactory and straightforward, although a little tricky in the derivation. However, not much of the final result is really new. Arnold Zellner (1971) gave a very similar solution long ago, but it went unnoticed by those who had the most need to know about it. We give a pedagogical introduction to the problem and add a few final touches, dealing with choice of priors and parameterizations. In any event, whether or not the following solution has anything new in it, the currently great and universal importance of the problem would warrant bringing the result to the attention of the
