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).