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  "body": "\n## Statement\n\nLet $S$ be a [[Polish space|584GXXY73M]] with its [[Borel σ-algebra|0WQXY32Q8D]], and let $\\mathcal{P}(S)$ be the set of [[probability measures|059TJN0NZ1]] on $S$.\nGive $\\mathcal{P}(S)$ the smallest σ-algebra that makes the maps $m \\mapsto m(A)$ measurable for all Borel sets $A \\subseteq S$.\nLet $X_1, X_2, \\dots$ be an infinite [[exchangeable|4MR4YJS1ZD]] sequence of random variables with values in $S$.\nThen there is a unique probability measure $\\nu$ on $\\mathcal{P}(S)$ such that for every $n \\ge 1$ and all Borel sets $A_1, \\dots, A_n \\subseteq S$,\n$$P(X_1 \\in A_1, \\dots, X_n \\in A_n) = \\int_{\\mathcal{P}(S)} m(A_1) \\cdots m(A_n) \\, \\nu(dm).$$\n\n## Remarks\n\nIn words, the sequence is a mixture of [[i.i.d.|00MRN0FARV]] sequences: first draw a law $m$ from $\\nu$, then draw $X_1, X_2, \\dots$ i.i.d. with law $m$.\nFor $S = \\set{0, 1}$ the theorem says that there is a unique probability measure $\\mu$ on $[0, 1]$ with\n$$P(X_1 = x_1, \\dots, X_n = x_n) = \\int_0^1 p^k (1 - p)^{n - k} \\, \\mu(dp), \\qquad k = x_1 + \\dots + x_n,$$\nso the sequence is a mixture of [[Bernoulli|0386Y0Z7Y7]] sequences.\nIn this case the random parameter is the almost sure limit of the frequencies $(X_1 + \\dots + X_n)/n$, and its law is $\\mu$.\nOne proof shows that, given the σ-algebra of events invariant under finite permutations, the variables are conditionally i.i.d.; a reverse martingale argument gives the limit of the empirical measures.\nThe theorem fails for finite sequences: drawing both balls without replacement from an urn with one red and one blue ball gives an exchangeable pair $(X_1, X_2)$ with $P(X_1 = X_2 = 1) = 0$ and $P(X_1 = X_2 = 0) = 0$.\nA mixture would need $\\int p^2 \\, \\mu(dp) = 0$ and $\\int (1 - p)^2 \\, \\mu(dp) = 0$, which is impossible.\nIn the $\\set{0,1}$ case, $\\operatorname{Cov}(X_1, X_2) = \\int p^2 \\, d\\mu - \\left(\\int p \\, d\\mu\\right)^2 \\ge 0$, so infinite exchangeable sequences of this kind are never negatively correlated.\n\n## Explanations\n\n### L1 human\n\nSuppose you toss a coin of unknown bias many times, and you believe that the order of the tosses does not matter.\nDe Finetti's theorem says that this belief alone forces a simple picture.\nThere is a hidden bias $p$, drawn once from some law, and given $p$ the tosses are independent with success probability $p$.\nSo symmetry under reordering is the same as independence given an unknown parameter.\n\n### L2 human\n\nThe theorem is the main justification for the Bayesian model \"parameter with a prior, data i.i.d. given the parameter\": the measure $\\nu$ plays the role of the prior.\nThe [[Pólya urn model|DHPXDFEEMF]] gives an example: its colour sequence is exchangeable, and the mixing measure is a [[beta distribution|0EW824Z7G2]].\nFor general exchangeable sequences, the [[Dirichlet process|XT4SWG6J7N]] is a standard choice of $\\nu$ in nonparametric Bayesian statistics.\nThe [[Hewitt–Savage zero–one law|JTXNFQNQ4J]] is a related result: for an i.i.d. sequence, the exchangeable σ-algebra is trivial, which matches $\\nu$ being a point mass.\n",
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