These notes were written to accompany a sequence of workshop lectures at the Indian Institute of Science, Bangalore in January 2013. My lectures had two purposes. The first was to introduce some of the main ideas of exchangeability theory at a level suitable for graduate students (assuming familiarity with measure-theoretic probability). The second was to show the key rôle exchangeability has p...

Sets of desirable gambles constitute a quite general type of uncertainty model with an interesting geometrical interpretation. We study inĄnite exchangeability assessments for them, and give a counterpart of de FinettiŠs inĄnite representation theorem. We show how the inĄnite representation in terms of frequency vectors is tied up with multivariate Bernstein (basis) polynomials. We also lay bar...

Although there have been a number of available tests bivariate exchangeability, i.e. symmetry for distributions, the literature is void whether multivariate distribution with more than two dimensions exchangeable or not. In this paper, permutation exchangeability distributions are proposed, which based on non-parametric combination methodology, combining tests. Numerical experiments real as wel...

We consider inference for high-dimensional separately and jointly exchangeable arrays where the dimensions may be much larger than sample sizes. For both arrays, we first derive central limit theorems over rectangles subsequently develop novel multiplier bootstraps with theoretical guarantees. These results rely on new technical tools such as Hoeffding-type decomposition maximal inequalities de...

In cohort and case-control studies, confounding that arises as a result of differences in the distribution of determinants of the outcome between exposure groups leading to non-exchangeability are addressed by restriction, matching or with statistical models. In case-only studies, this issue is addressed by comparing each individual with his/herself. Although case-only designs use self-matching...