1.4 Definetti’s Representation Theorem

Example 1.54

■

Theorem 1.56

• \(\{X_{i}\}_{i \in \mathbb{N}}\),
  • exchangeable Bernoulli random variables
  • \(X_{i} \in \{0, 1\}\),
  • 1 means success
• $\Theta: S \rightarrow [0, 1]$,

\[\Theta := \lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{X_{i}}{n}\]

• $\mu_{\Theta}:\mathcal{B} \rightarrow [0, 1]$
  • distribution of $\Theta$,
• $n^{*} \in \mathbb{N}$
  • the number of trials
  • given
• $k^{*} \in \mathbb{Z}_{\ge 0}$,
  • the number of success
  • $k^{*} := \sum_{i=1}^{n}X_{i}$,
• $F^{*}$
  • c.d.f. of $\Theta$ conditional on $\sum_{i=1}^{n}X_{i} = k$.

\[F^{*}(t \mid \sum_{i=1}^{n}X_{i} = k) = \frac{ \int_{[0, t]} \theta^{k} (1 - \theta)^{n^{*} - k} \ \mu_{\Theta}(d\theta) }{ \int_{[0, 1]} \psi^{k} (1 - \psi)^{n^{*} - k} \ \mu_{\Theta}(d \psi) }\]

proof

$X_{i}$ is conditionally I.I.D with Ber($\Theta$) distirbuiton given $\Theta = \theta$. Let \(k \in \{0, 1, \ldots, n\}\) be fixed

\[\begin{eqnarray} \mathrm{Pr} \left( \sum_{i=1}^{n} X_{i} = k, \Theta \in B \right) & = & \int_{B} \left( \begin{array}{c} n^{*} \\ k \end{array} \right) \theta^{k} (1 - \theta)^{n^{*} - k} \ \mu_{\Theta}(\theta) \nonumber \end{eqnarray}\]

Divide this by

\[\mathrm{Pr} \left( \sum_{i=1}^{n} X_{i} = k \right) = \int_{[0, 1]} \theta^{k} (1 - \theta)^{n^{*} - k} \ \mu_{\Theta}(\theta)\]

$\Box$