# Applying de Finetti's representation theorem to Dirichlet distribution

Let's begin from the de Finetti–Hewitt–Savage theorem: for an exchangeable sequence of random variables we can always write $$p(x_1, x_2,\cdots) = \int \prod p(x_i | L) P(dL)$$ where $$L$$ is a latent variable upon which the variables become conditionally independent. I've seen it applied to exchangeable but correlated Gaussian variables - they become independent when conditioned upon another Gaussian variable.

Now let's take a symmetric Dirichlet distribution, i.e. a Dirichlet with concentration parameters $$\alpha_i = \alpha$$ for all $$i$$. Thus the variables are exchangeable. This distribution only has support over $$\sum x_i = 1$$, which means that the $$x_i$$ are not independent.

I think I must be making a mistake because the theorem seems too good to be true, as it implies that there is a latent variable under which the $$x_i$$ are conditionally independent. But I just can't see how there could exist a single latent variable to enforce the fact that it only has support over $$\sum x_i = 1$$.

Maybe I am just missing a simple assumption for the theorem? or maybe I am just not seeing something.

• The de Finetti representation theorem applies only to infinite exchangeable sequences, so it does not bear on the finite-dimensional Dirichlet distribution. See this for more: Diaconis, Persi. "Finite forms of de Finetti's theorem on exchangeability." Synthese 36.2 (1977): 271-281. – a.arfe May 23 '19 at 3:58

The $$x_i$$ from a Dirichlet do not form a subset of a infinite sequence of exchangeable variables. a.arfe also noted that finite exchangable sequences were further discussed in Diaconis, Persi. "Finite forms of de Finetti's theorem on exchangeability." Synthese 36.2 (1977): 271-281.