# deFinetti’s Theorem Part II

## Some further references on deFinetti’s Theorem

This paper shows how the problem of exchangeability can be interpreted geometrically and proves some “approximate” versions of the theorem for finite sequences.

- G. R. Wood, Binomial Mixtures and Finite Exchangeability.

This paper considers the problem of extending a finite exchangeable sequence to an infinite one; and also the problem of determining how likely it is that a randomly chosen distribution on $0,\ldots, n$ will be a mixture of binomial distributions. The questions turn out to be closely related. The paper gives a formula computing the fraction of binomial mixtures among all distributions (which turns out to go to zero VERY quickly with increasing $n$) and then relates that to the probability that an $n$ exchangeable sequence is infinitely extendible (which also goes to zero very quickly with $n$).

The results amount to some (non-trivial!) calculations of volumes of regions in simplices.

Apparently as of the time of this paper there was a conjecture due to Crisma giving a formula for the probability that an exchangeable sequence of $n$ random variables can be extended to one of length $r$ that was unsolved, what is its current status?

- P. Diaconis and D. Freedman, Finite Exchangeable Sequences

This paper computes the distance between the distribution of $k$ exchangeable random variables taking values in a (finite) set $S$ and the closest mixture of IID random variables. To explain (one of) their theorems, represent a probability distribution on $S$ as a point in the $|S|$-dimensional simplex. Suppose $\mu$ is a probability distribution on this simplex. Given such a probability distribution $\mu$, let $P(k,\mu)$ be the distribution on $S$ from $\mu$, and then making $k$ iid choices from this $\mu$.

**Theorem:**
Let $S$ be a finite set with $|S|$ elements. Let $P$ be an exchangeable probability on $S^{n}$. Then there is a probability $\mu$
on the $|S|$-simplex so that $| P_{k}-P(k,\mu ) | \le 2|S| k/n$ for all $k\le n$. Here $P_{k}$ is the marginal probability of $P$ on sequences of length $k\le n$.

In other words, if $P$ is a distribution on sequences of length $k$ that can be extended to an exchangeable sequence of length $n$, then $P$ is within distance $k/n$ of a “mixture.”

The distance here is “variation distance” $|P-Q|=2\sup_A|P(A)-Q(A)|$ over Borel sets $A$.