APPLIED MATHEMATICS & STATISTICS (AMS)
APPLIED MATHEMATICS & STATISTICS (AMS)
Statistics is the study of uncertainty -- how to measure it, and what to do about it. The "how to measure it" part of this sentence relates to probability, the branch of mathematics devoted to uncertainty quantification, and the "what to do about it" part connects with decision theory, the study of how to make choices in the face of uncertainty.
Over the last 350 years two main probabilistic approaches have been developed: frequentist (which equates probabilities with relative frequencies and only applies to inherently repeatable phenomena) and Bayesian (which defines probabilities subjectively through betting odds and can, in principle, be applied to all phenomena, repeatable or not).
The Bayesian approach, which quantifies uncertainty through the use of probability distributions for all unknown quantities, is more general, but for centuries its progress was held back by a formidable technical challenge: the accurate numerical evaluation of high-dimensional integrals defined by these probability distributions.
Around 1990 the statistics profession learned about a class of methods -- invented by physicists in the early 1950s -- for solving this problem by simulating from the relevant probability distributions, and noticed that computers had finally become fast enough for this approach to become practical. The resulting sets of Markov chain Monte Carlo (MCMC) techniques, and refinements upon them over the past decade, have created a revolution in Bayesian methods and applications.
The current members of the Statistics Group within AMS -- David Draper, Raquel Prado, and
Bruno Sansó -- all have active research agendas in Bayesian
methods.
Our work at present mainly concentrates in five areas, as follows.
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