# Intuitive understanding of the Aldous-Hoover representation theorem for row-column exchangeable arrays

I would like to ask a couple of questions about the Aldous-Hoover theorem for the representation of probability distributions over (2D) arrays with exchangeable rows and columns. I'd be happy about references that can help answering these questions. Unfortunately the references listed below didn't quite help, also because they often go beyond my technical understanding.

Background

De Finetti's representation theorem can be presented in a very intuitive way that makes very clear:

1. Its hierarchic information structure: Suppose for example that we have a sequence of $$n$$ binary quantities $$(X_i \in \{0,1\})$$. If the indices $$i$$ are completely uninformative, then, if we know the empirical frequencies of $$0$$s and $$1$$s in the sequence, its probability and the probability of any subsequence are, by symmetry, hypergeometric distributions divided by a multiplicity factor. If we don't know the empirical frequencies then the probability of the sequence is given by a mixture of such distributions, weighted by the probabilities of the possible empirical frequencies. This is just an application of the theorem of total probability.

2. Its connection to sufficient statistics: In the example above, the information of a subsequence about a disjoint subsequence is completely contained in its empirical frequencies, which are therefore sufficient statistics.

3. The connection between finite and infinite case: If we consider a longer and longer sequence and subsequences much shorter than the sequence itself, their distributions conditional on the empirical frequencies of the sequence become approximately iid (drawing with replacement; the bound of the error of this approximation is given by Diaconis & al 1980). Thus the form of the representation theorem obtains in the infinite limit.

4. The observational meaning of the parameter in the representation: For a longer and longer sequence, the empirical frequencies are "long-run" frequencies, and these are the parameter in the representation.

Questions

Now, a representation theorem also exists for a 2D array of quantities, if the probability for their values is invariant with respect to permutations of rows or columns: (separate) row-column exchangeability. The theorem was proven by Aldous (1981), Hoover (1979), and has been discussed in many works in the literature; a small sample is given in the references below.

If I understand it correctly, in the binary case above this theorem says (Diaconis & al 1981, pp. 118, 127) that the probability for a specific array of values is given by a mixture of product of identical distributions. The distributions in the product, one factor for each entry $$(i,j)$$, are Bernoulli (coin-tossing) with parameter $$p$$, where $$p=\phi(u_i,v_j)$$ for some map $$\phi \colon [0,1]^2 \to [0,1]$$, and every $$u_i \in [0,1]$$ and $$v_j \in [0,1]$$ has a uniform probability distribution. The mixture is over all possible maps $$\phi$$, weighted by a density for $$\phi$$.

• Question 1: Are there finite forms of this theorem?

• Question 2: Can the various parameters appearing in the theorem be given a more observational or "operational" meaning, similarly to the long-run frequencies in de Finetti's theorem?

• Question 3: Are the parameters of the theorem related to some sufficient statistics for the array?

Cheers!

References