Abstract

Suppose we represent observational data as a sequence of random variables. If the distribution of this sequence does not depend on the order of the variables, the sequence is conditionally independent given a random probability measure. This result is de Finetti’s theorem, which forms the formal basis of Bayesian statistics and also plays a pivotal role in applied probability. Various generalizations exist, both to other types of invariances than exchangeability, and to other random structures than sequences. A beautiful theorem of Lauritzen provides a common framework for these results and directly links them to statistics, by formalizing invariance by means of a sufficient statistic.

I will discuss the relation between symmetry and sufficiency, sketch the general theorem, and discuss how it can be used to determine the fundamental independence structure of Bayesian models when the data is not an exchangeable sequence, but rather some other exchangeable random structure—the examples I will consider in the talk are random partitions and random graphs. The theorem can also be used to derive parametric exponential family models as a special case.

With this talk, I will stand in for a cancelled RCC on short notice, and I will use a talk I recently gave in the Statistical Laboratory; this talk has evolved from an RCC titled “Exchangeability” which I gave last spring. The talks differ quite a bit and there is plenty of new material, but if you attended my talk last year and thought it was boring, this one is bound to be worse.