Minimum Variance Estimation—How the Theory Fits

Other than intellectual curiosity, we can only think of two reasons why a scientist or engineer might study mathematics. The first is to obtain procedures or algorithms to solve a problem or class of problems. The second is to clearly and precisely conceptualize an idea; to capture its essence. At first glance, the former may appear to be the more important of the two. After all, aren’t solutions to problems what we are really after? Yes, indeed! But the two reasons above are not really dichotomous. Certainly, the history of probability and statistics shows that viable solutions to many problems were not forthcoming until people like Borel and Kolmororov laid the proper foundations so that others could technically formulate their problems and rigorously check them. For example, it is hard to imagine formulating and proving the Kalman theorem with the probability theory of 1850. It would be misleading, however, to suggest that the only reason for precisely formulating ideas is to obtain solutions to problems. There are other, very practical, reasons for doing so. Let us briefly explore this.