Abstract

Once upon a time most statistical inference was carried out by regarding the data being analyzed as realizations of random variables whose joint distribution was determined up to some unknown parameters, usually but not always finite-dimensional. The challenge lay in dealing with the unknowns when making appropriate inferences. This view enabled us to draw on a fine body mathematical theory, which was comforting in that there seemed to be a solid foundation for what we were doing, going back to Kolmogorov’s 1933 axiomatization of probability. (Of course this is a gross oversimplification, and ignores major philosophical issues.) In his famous 1962 paper “The future of data analysis” Tukey questioned this orthodoxy, and promoted data analysis, a subject related to statistics, but one far less governed by mathematical theory, and which did not appear to have any foundations. Tukey’s view is flourishing today, yet mathematical statistics lives, and may itself be flourishing. We now have many processes – assays, devices, technologies – which can generate large amounts of data very quickly, data for which a realistic joint distribution is unimaginable, no matter how we might parameterize. By this I simply mean that we could never pass the statistician’s Turing test – to simulate data indistinguishable from the real thing – with such data. What do we do? Well, we combine statistics with data analysis (as perhaps we always have), doing things that seem appropriate, alongside with things that would be correct, given certain assumptions that are patently false. I’ll be illustrating these ideas with examples from biology, more precisely, high-throughput biology. It often seems to work, sometimes rather well, and one day we may understand why.