Tracy Harms wrote: > My impression is that one of the things John Randall > was pointing out is that the mathematics of statistics > is devised for dealing with uncertainty and other > limitations in what is, or can be, exactly known. This > contrasts with most of mathematics, where perfect > knowledge is presumed, axiomatically. He pointed > toward techniques from numerical analysis for those > cases where the subject matter falls, at least in > effect, within that axiomatic framework. > > The implication of this is that we should not strive > for consistent terminology to deal with all things > that can be thought of in terms of populations. > (Consider applications of set theory.) We should > instead let statistics have its own terms, and > recognize that those terms apply where "normal" math > cannot reach, i.e. into hazardous estimations of > empirically encountered patterns. >
Mathematical statistics is perfectly precise: it is its interpretation that may be off. This has less to do with the uncertainty in probability than with the pitfalls of any kind of modelling. In statistics, all of the uncertainty is relegated to the probability function or probability density function. This is the modelling step. After that, everything is completely precise and, viewed from a far enough distance, is the elaboration of the theory of measurable functions. To take a very simple example: suppose I toss 10 fair coins. Then I can get a probability function for the number of heads and perform any calculation I like with no uncertainty. The applicability of this depends on how accurate my assumptions are, that is, are the coins fair? Numerical analysis is subject to the same constraints. For example, I can precisely fit a polynomial to the data. Interpreting if this makes sense depends on extra-mathematical concerns such as whether I have enough points, or whether a polynomial will give a reasonable approximation to the original function. I would argue that mathematical statistics is firmly within the canon of "normal" math: it is the modelling that may produce imprecise or inapplicable results. Best wishes, John ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
