# Exchangeable Probability Distribution

Let {`⚠ $Z$`

} be a measurable space. A probability distribution {`⚠ $P$`

} on the measurable space {`⚠ $Z^n$`

} of sequences of length {`⚠ $n$`

}, where {`⚠ $n \in {1, 2, \ldots}$`

}, is *exchangeable* if

{`⚠ $P(E) = P\{z_1, \ldots, z_n: z_{\pi(1)}, \ldots, z_{\pi(n)}\in E\}$`

}

for any measurable {`⚠ $E \subseteq Z^n$`

} and any permutation {`⚠ $\pi$`

} of the set {`⚠ $\{1, \ldots, n\}$`

} (i.e., if the distribution of the sequence {`⚠ $z_1, \ldots, z_n$`

} is invariant under any permutation of the indices).

A probability distribution {`⚠ $P$`

} on the power measurable space {`⚠ $Z^{\infty}$`

} is *exchangeable* if the marginal distribution {`⚠ $P_n$`

} of {`⚠ $P$`

} on {`⚠ $Z^n$`

} (defined by

{`⚠ $P_n(E) := P\{(z_1, z_2, \ldots) \in Z^{\infty}: z_1, \ldots, z_n\in E\}$`

}

for all events {`⚠ $E \subseteq Z^n$`

}) is exchangeable for each {`⚠ $n = 1, 2, \ldots$`

} (i.e., if the distribution of the sequence {`⚠ $z_1, z_2, \ldots$`

} is invariant under any permutation of the finite number of the indices).