Stanford University, Spring 2016, STATS 205

## Exchangeablility

• The idea is that before we observe 20 flips of a coin
• and if we think of 17 heads and 3 tails as a possible outcome
• then exchangeablility means that we don't think of the positions that the 3 heads can occupy as being special

## Exchangeablility

• An infinite sequence $$X_1,\dots,X_n,\dots$$ of random variables is said to be exchangeable if for all $$n = 2,3,\dots$$, $X_1,\dots,X_n \overset{d}{=} X_{\pi(1)},\dots,X_{\pi(n)} \text{ for all } \pi \in S(n)$ where $$S(n)$$ is the group of permutations of $$\{1,\dots,n\}$$
• For example, for a binary sequence, we may have: $p(1,1,0,0,0,1,1,0) = p(1,0,1,0,1,0,0,1)$
• If $$X_1,\dots,X_n,\dots$$ are independent and identically distributed, they are exchangeable, but not conversely

## Polya's Urn

• Consider an urn with $$b$$ black balls and $$w$$ white balls
• Draw a ball at random and note its colour
• Replace the ball together with $$a$$ balls of the same colour
• Repeat the procedure ad infinitum
• Let $$X_i =1$$ if the $$i$$th draw yields a black ball and $$X_i = 0$$ otherwise
• The sequence $$X_1,\dots,X_n,\dots$$ is exchangeable: \begin{align} p(1,1,0,1) & = \frac{b}{b+w} \frac{b+a}{b+w+a} \frac{w}{b+w+2a} \frac{b+2a}{b+w+3a} \\ & = \frac{b}{b+w} \frac{w}{b+w+a} \frac{b+a}{b+w+2a} \frac{b+2a}{b+w+3a} = p(1,0,1,1) \end{align}
• $$X_1,\dots,X_n,\dots$$ are not independent