Applications of Statistics

Lars Syll has commented on a book by David Salsburg, criticising workaday applications of statistics. Lars has this quote:

Kolmogorov established the mathematical meaning of probability: Probability is a measure of sets in an abstract space of events.

This is not quite right.

  • Kolmogorov established a possible meaning, not ‘the’ meaning. (Actually Wittgenstein anticipated him.)
  • Even taking this theory, it is not clear why the space should be ‘measurable‘. More generally one has ‘upper’ and ‘lower’ measures, which need not be equal. One can extend the more familiar notions of probability, entropy, information and statistics to such measures. Such extended notions seem more credible.
  • In practice one often has some ‘given data’ which is at least slightly distant from the ‘real’ ‘events’ of interest. The data space is typically rather a rather tame ‘space’, so that a careful use of statistics is appropriate. But one still has the problem of ‘lifting’ the results to the ‘real events’.

These remarks seem to cover the criques of Syll and Salsburg, but are more nuanced. Statistical results, like any mathematics, need to be interpreted with care. But, depending on which of the above remarks apply, the results may be more or less easy to interpret: not all naive statistics are equally dubious!

Dave Marsay