# 37 Theory

## 37.1 de Finetti’s Theorem

Let $$X_1, X_2, \ldots$$ be an infinite exchangeable sequence of Bernoulli rv’s. There exists a random variable $$P \in [0, 1]$$ such that:

• $$X_1|P, X_2|P, \ldots$$ are conditionally independent
• $$X_1, X_2, \ldots | P=p \stackrel{{\rm iid}}{\sim} \mbox{Bernoulli}(p)$$

This theorem is often used to justify the assumption of exchangeability, which is weaker than iid, with a prior distribution on the parameter(s).

An estimator $$\tilde{\theta}$$ is admissible with respect to risk function $$R(\cdot, \theta)$$ if there is exists no other estimator $$\hat{\theta}$$ such that $$R(\hat{\theta}, \theta) < R(\tilde{\theta}, \theta)$$ for all $$\theta \in \Theta$$.