# Exchangeable Processes over the Simplex

You are likely all familiar with Polya Urn process. I initially start with an urn containing $b$ black balls and $w$ white balls. At each step, I sample a black ball with probability $\frac{b}{b+w}$ (alternatively, a white ball with probability $\frac{w}{b+w}$), then put back the same ball and another of the same color. Equivalently, if $x_i$ is the $i$-ith ball drawn, I update $w \leftarrow w + \delta(x_i = w)$ and $b \leftarrow b +\delta(x_i = b)$

The Polya Urn process has some wonderful properties, first that the distribution of $\frac{b}{b+w}$ converges to a sample from a Beta distribution, and second that the probability of any sequence of draws is exchangeable.

I would like to use a similar process wherein each draw isn't a discrete random variable, but rather a continuous variable on $(0,1)$. My initial intuition was to replace each draw from the urn with a draw $p_i \sim Beta(b, w)$, then to update $b \leftarrow b + p_i$ and $w \leftarrow w + (1-p_i)$. While $\frac{b}{b+w}$ converges much the same way it does in the Polya urn process, the pdf of any sequence of draws is not exchangeable.

My question is, do there exist any urn-like processes over the probability simplex that are exchangeable?

• I don't quite understand the generalization you have in mind. You say "each draw isn't a discrete random variable, but rather a continuous variable", so I imagine the balls having a [0,1] continuum of greys, for example, rather than two colours. But then you write "$b \leftarrow b + p_i$ and $w \leftarrow w + (1-p_i)$", which seems to imply you still have balls with only two colours, not a continuum of colours. – pglpm Jul 14 '19 at 19:37