# Partial exchangeability - Theory and results

De Finetti has defined in 1938 the concept of partial exchangeability as a weaker form of exchangeability in order to deal with situations in which the symmetry among all observations is not desirable.

That's what I know and mere googling does not help this time because I've just found a bunch of papers that are a bit too advanced for me.

My questions are: what is the formal definition of partial exchangeability and does the results holding with exchangeability still holding (e.g. De Finetti representation theorem)? Can you give me an example of a situation in which partial exchangeability is meaningful?

## 2 Answers

This is described by Diaconis (1988; see also Diaconis and Freedman, 1980):

In 1938 de Finetti broaden the concept of exchangeability. Consider first the special case with two observations $X_1,X_2,\dots;Y_1,Y_2,\dots$. The $X_i$ might represent binary outcomes for a group of men and $Y_i$ might represent binary outcomes for a group of women. If it were judged that the observable covariate men/women did not matter, all of the variables would be judged exchangeable. Often, the covariate is judged as potentially meaningful, the $X_i$'s are judged exchangeable between themselves and the $Y_i$'s are judged exchangeable between themselves. Mathematically, the joint law must be invariant under permutations within the $X$'s and $Y$'s:

$$\mathcal{L}(X_1,\dots,X_n;Y_1,\dots,Y_m) = \mathcal{L}(X_{\pi(1)},\dots,X_{\pi(n)};Y_{\sigma(1)},\dots,Y_{\sigma(m)})$$

This must hold for all $n$ and $m$, and permutations $\pi$ and $\sigma$.

You could check as well the original paper by de Finetti (1938) for discussion and multiple examples as he has very light and clear style of writing, what makes his papers easy to read.

Diaconis, P. (1988). Recent progress on de Finetti’s notions of exchangeability. Bayesian statistics, 3, 111-125.

Diaconis, P., & Freedman, D. (1980). De Finetti’s generalizations of exchangeability. Studies in inductive logic and probability, 2, 233-249.

De Finetti, B. (1938/1980). On the condition of partial exchangeability. Studies in inductive logic and probability, 2, 193-205.

To complete Tim's answer, from the useful references given there:

Suppose the quantities $\{X_i\} := \{X_1, X_2, \dotsc\}$ and $\{Y_j\}$ can have values $\{x\}$ and $\{y\}$ from two discrete sets. The assumption of infinite partial exchangeability for their joint probabilities means $$\mathrm{P}(X_1=x_1, \dotsc, X_n=x_n,\; Y_1=y_1, \dotsc, Y_m=y_m) ={}\\ \mathrm{P}(X_1=x_{\pi(1)}, \dotsc, X_n=x_{\pi(n)},\; Y_1=y_{\sigma(1)}, \dotsc, Y_m=y_{\sigma(m)}),\\ \text{for all m,n, and all permutations \pi,\sigma of \{1,\dotsc,n\} and \{1,\dotsc,m\}.}$$

Then the probability above can be expressed in this integral form: $$\mathrm{P}(X_1=x_1, \dotsc, X_n=x_n,\; Y_1=y_1, \dotsc, Y_m=y_m) ={}\\ \iint \prod_{i=1}^n p_{x_i}\; \prod_{j=1}^m q_{y_j}\; f(\mathbf{p},\mathbf{q})\;\mathrm{d}\mathbf{p}\,\mathrm{d}\mathbf{q},$$ where $\mathbf{p} := \{p_x\}$ and $\mathbf{q} := \{q_y\}$ are distributions over the possible values $\{x\}$ and $\{y\}$, and $f$ is a probability density over all such pairs of distributions.

Another useful reference for partial exchangeability, with examples, is Bernardo & Smith's Bayesian Theory (Wiley 2000), especially § 4.6.