A basic concept in Bayesian Statistics

Exchangeable

Let $p(y_1,…,y_n)$ be the joint density of $Y_1,…,Y_n$. If $p(y_1,…,y_n)=p(y_{\pi_1},…,y_{\pi_n})$ for all permutations $\pi$ of ${1,…,n}$, then $Y_1,…,Y_n$ are exchangeable.

de Finetti’s Theorem

The model can be written as
$$
P(y_1,…,y_{n})=\int \left{\prod_{i=1}^{n}p(y_i|\theta) \right}p(\theta)d\theta
$$
for some parameter $\theta$ if $Y_i$ are exchangeable.

  • $p(\theta)$ represents our beliefs about $\lim_{n\rightarrow\infty}\sum Y_i/n$ in the binary case.
  • $p(\theta)$ represents our beliefs about $\lim_{n\rightarrow\infty}\sum(Y_{i}\le c)/n$ for each $c$ in the general case.