By now maybe you're feeling pretty good. The least-squares algorithms which I outlined in the preceding two lectures are very powerful and, to the extent practical under the current circumstances, I have tried to build them up from mathematical first principles. Even where I have waved my arms around and tried to make common-sense arguments, mathematically rigorous proofs can be found in any number of real statistics texts. It seems as though these least-squares methodologies represent the one true path to the best answers in almost any data-reduction problem you could imagine. That just isn't the case. Least-squares may be the part-time statistician's dream, because the algorithm is straightforward and easily applied, and it carries the endorsement of numerous texts on "Statistics for Scientists." Thus, the method of least squares apparently allows the astronomer to claim before the astronomical community and (more important?) before the referee, that the very best that can be done with the data has been done. Not so! These algorithms, heavily documented and widely used though they be, are not up to handing a couple of simple problems that we encounter every day when reducing real data. In fact, I believe I can say with some possibility of being right, that modern astronomers use the method of least squares improperly more often than not.
There are two situations in particular that I would like to discuss now. First, I have mentioned once or twice that the development of these least-squares solutions rests upon the assumption that only one of the variables can contain error, and that this variable can be isolated on the left side of the equals sign. If this condition is not met - and it very often isn't - the equations we have used so far are invalid. Second, real data do not in general have a truly Gaussian error distribution. In fact, in almost every case that you will encounter in your careers, the true error distribution is not even known. Again, blind application of the algorithms we have developed so far will not guarantee the "best" answer; they may not even guarantee an answer that is "close enough for government work"; I will try to convince you that under some real-life circumstances they will not even guarantee a unique answer.
Let's get to it.