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).

37.2 Admissibility

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 \(\theta \in \Theta\).

There’s a theoretical result that says all admissible estimators are Bayes estimates.