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Exchangeable random variables

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In <a href="Statistics" title="Statistics">statistics</a>, an exchangeable sequence of random variables (also sometimes interchangeable)<a href="#cite_note-ChowTeicher-1">[1]</a> is a sequence X1, X2, X3, ... (which may be finitely or infinitely long) whose <a href="Joint_probability_distribution" title="Joint probability distribution">joint probability distribution</a> does not change when the positions in the sequence in which finitely many of them appear are altered. In other words, the joint distribution is invariant to finite permutation. Thus, for example the sequences

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<annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},X_{3},X_{4},X_{5},X_{6}\quad {\text{ and }}\quad X_{3},X_{6},X_{1},X_{5},X_{2},X_{4}}</annotation>

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</math><img src="./251087dbc3f2c2014f3d49f0bcb25959c52c6a51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:55.635ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},X_{3},X_{4},X_{5},X_{6}\quad {\text{ and }}\quad X_{3},X_{6},X_{1},X_{5},X_{2},X_{4}}" loading="lazy"></dd></dl>

both have the same joint probability distribution.

It is closely related to the use of <a href="Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">independent and identically distributed random variables</a> in statistical models. Exchangeable sequences of random variables arise in cases of <a href="Simple_random_sampling" class="mw-redirect" title="Simple random sampling">simple random sampling</a>.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

Formally, an exchangeable sequence of random variables is a finite or infinite sequence X1, X2, X3, ... of <a href="Random_variable" title="Random variable">random variables</a> such that for any finite <a href="Permutation" title="Permutation">permutation</a> σ of the indices 1, 2, 3, ..., (the permutation acts on only finitely many indices, with the rest fixed), the <a href="Joint_probability_distribution" title="Joint probability distribution">joint probability distribution</a> of the permuted sequence

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<annotation encoding="application/x-tex">{\displaystyle X_{\sigma (1)},X_{\sigma (2)},X_{\sigma (3)},\dots }</annotation>

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</math><img src="./7577de3143dc6f9ce3b0d0701f4823c9f3c7ecc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.419ex; height:3.009ex;" alt="{\displaystyle X_{\sigma (1)},X_{\sigma (2)},X_{\sigma (3)},\dots }" loading="lazy"></dd></dl>

is the same as the joint probability distribution of the original sequence.<a href="#cite_note-ChowTeicher-1">[1]</a><a href="#cite_note-2">[2]</a>

(A sequence E1, E2, E3, ... of events is said to be exchangeable precisely if the sequence of its <a href="Indicator_function" title="Indicator function">indicator functions</a> is exchangeable.) The distribution function FX1,...,Xn(x1, ..., xn) of a finite sequence of exchangeable random variables is symmetric in its arguments x1, ..., xn. <a href="Olav_Kallenberg" title="Olav Kallenberg">Olav Kallenberg</a> provided an appropriate definition of exchangeability for continuous-time stochastic processes.<a href="#cite_note-3">[3]</a><a href="#cite_note-Kallenberg-4">[4]</a>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

The concept was introduced by <a href="William_Ernest_Johnson" title="William Ernest Johnson">William Ernest Johnson</a> in his 1924 book Logic, Part III: The Logical Foundations of Science.<a href="#cite_note-5">[5]</a> Exchangeability is equivalent to the concept of <a href="Statistical_control" class="mw-redirect" title="Statistical control">statistical control</a> introduced by <a href="Walter_Shewhart" class="mw-redirect" title="Walter Shewhart">Walter Shewhart</a> also in 1924.<a href="#cite_note-6">[6]</a><a href="#cite_note-7">[7]</a>

<div class="mw-heading mw-heading2"><h2 id="Exchangeability_and_the_i.i.d._statistical_model">Exchangeability and the i.i.d. statistical model</h2></div>

The property of exchangeability is closely related to the use of <a href="Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">independent and identically distributed</a> (i.i.d.) random variables in statistical models.<a href="#cite_note-8">[8]</a> A sequence of random variables that are i.i.d, conditional on some underlying distributional form, is exchangeable. This follows directly from the structure of the joint probability distribution generated by the i.i.d. form.

Mixtures of exchangeable sequences (in particular, sequences of i.i.d. variables) are exchangeable. The converse can be established for infinite sequences, through an important <a href="De_Finetti's_theorem" title="De Finetti's theorem">representation theorem</a> by <a href="Bruno_de_Finetti" title="Bruno de Finetti">Bruno de Finetti</a> (later extended by other probability theorists such as <a href="Paul_Halmos" title="Paul Halmos">Halmos</a> and <a href="Leonard_Jimmie_Savage" title="Leonard Jimmie Savage">Savage</a>).<a href="#cite_note-9">[9]</a> The extended versions of the theorem show that in any infinite sequence of exchangeable random variables, the random variables are conditionally <a href="Independent_and_identically-distributed_random_variables" class="mw-redirect" title="Independent and identically-distributed random variables">independent and identically-distributed</a>, given the underlying distributional form. This theorem is stated briefly below. (De Finetti's original theorem only showed this to be true for random indicator variables, but this was later extended to encompass all sequences of random variables.) Another way of putting this is that <a href="De_Finetti's_theorem" title="De Finetti's theorem">de Finetti's theorem</a> characterizes exchangeable sequences as mixtures of i.i.d. sequences—while an exchangeable sequence need not itself be unconditionally i.i.d., it can be expressed as a mixture of underlying i.i.d. sequences.<a href="#cite_note-ChowTeicher-1">[1]</a>

This means that infinite sequences of exchangeable random variables can be regarded equivalently as sequences of conditionally i.i.d. random variables, based on some underlying distributional form. (Note that this equivalence does not quite hold for finite exchangeability. However, for finite vectors of random variables there is a close approximation to the i.i.d. model.) An infinite exchangeable sequence is <a href="Strictly_stationary" class="mw-redirect" title="Strictly stationary">strictly stationary</a> and so a <a href="Law_of_large_numbers" title="Law of large numbers">law of large numbers</a> in the form of <a href="Birkhoff%E2%80%93Khinchin_theorem" class="mw-redirect" title="Birkhoff–Khinchin theorem">Birkhoff–Khinchin theorem</a> applies.<a href="#cite_note-Kallenberg-4">[4]</a> This means that the underlying distribution can be given an operational interpretation as the limiting empirical distribution of the sequence of values. The close relationship between exchangeable sequences of random variables and the i.i.d. form means that the latter can be justified on the basis of infinite exchangeability. This notion is central to <a href="Bruno_de_Finetti" title="Bruno de Finetti">Bruno de Finetti's</a> development of <a href="Predictive_inference" class="mw-redirect" title="Predictive inference">predictive inference</a> and to <a href="Bayesian_statistics" title="Bayesian statistics">Bayesian statistics</a>. It can also be shown to be a useful foundational assumption in <a href="Frequentist_statistics" class="mw-redirect" title="Frequentist statistics">frequentist statistics</a> and to link the two paradigms.<a href="#cite_note-O'Neill-10">[10]</a>

The representation theorem: This statement is based on the presentation in O'Neill (2009) in references below. Given an infinite sequence of random variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} =(X_{1},X_{2},X_{3},\ldots )}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} =(X_{1},X_{2},X_{3},\ldots )}</annotation>

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</math><img src="./5be35f75388cd2f72e34fb218afa73a26b864b0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.688ex; height:2.843ex;" alt="{\displaystyle \mathbf {X} =(X_{1},X_{2},X_{3},\ldots )}" loading="lazy"> we define the limiting <a href="Empirical_distribution_function" title="Empirical distribution function">empirical distribution function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\mathbf {X} }}">

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<annotation encoding="application/x-tex">{\displaystyle F_{\mathbf {X} }}</annotation>

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</math><img src="./1a05c045f0196d825382a50f4f4bd9d2a7dd0e28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.155ex; height:2.509ex;" alt="{\displaystyle F_{\mathbf {X} }}" loading="lazy"> by

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<annotation encoding="application/x-tex">{\displaystyle F_{\mathbf {X} }(x)=\lim _{n\to \infty }{\frac {1}{n}}\sum _{i=1}^{n}I(X_{i}\leq x).}</annotation>

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</math><img src="./42c320d1d6fe2aff9b7ee30ec42eebaf5dfdf0f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.192ex; height:6.843ex;" alt="{\displaystyle F_{\mathbf {X} }(x)=\lim _{n\to \infty }{\frac {1}{n}}\sum _{i=1}^{n}I(X_{i}\leq x).}" loading="lazy"></dd></dl>

(This is the <a href="Ces%C3%A0ro_limit" class="mw-redirect" title="Cesàro limit">Cesàro limit</a> of the indicator functions. In cases where the Cesàro limit does not exist this function can actually be defined as the <a href="Banach_limit" title="Banach limit">Banach limit</a> of the indicator functions, which is an extension of this limit. This latter limit always exists for sums of indicator functions, so that the empirical distribution is always well-defined.) This means that for any vector of random variables in the sequence we have joint distribution function given by

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<annotation encoding="application/x-tex">{\displaystyle \Pr(X_{1}\leq x_{1},X_{2}\leq x_{2},\ldots ,X_{n}\leq x_{n})=\int \prod _{i=1}^{n}F_{\mathbf {X} }(x_{i})\,dP(F_{\mathbf {X} }).}</annotation>

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</math><img src="./811f1515e82cc43295bbe74f016a164cf0d8cedb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:61.316ex; height:6.843ex;" alt="{\displaystyle \Pr(X_{1}\leq x_{1},X_{2}\leq x_{2},\ldots ,X_{n}\leq x_{n})=\int \prod _{i=1}^{n}F_{\mathbf {X} }(x_{i})\,dP(F_{\mathbf {X} }).}" loading="lazy"></dd></dl>

If the distribution function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\mathbf {X} }}">

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<annotation encoding="application/x-tex">{\displaystyle F_{\mathbf {X} }}</annotation>

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</math><img src="./1a05c045f0196d825382a50f4f4bd9d2a7dd0e28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.155ex; height:2.509ex;" alt="{\displaystyle F_{\mathbf {X} }}" loading="lazy"> is indexed by another parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

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<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

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</math><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"> then (with densities appropriately defined) we have

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<annotation encoding="application/x-tex">{\displaystyle p_{X_{1},\ldots ,X_{n}}(x_{1},\ldots ,x_{n})=\int \prod _{i=1}^{n}p_{X_{i}}(x_{i}\mid \theta )\,dP(\theta ).}</annotation>

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</math><img src="./056df64403939ed9984a6484414841d16ea72687.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:47.053ex; height:6.843ex;" alt="{\displaystyle p_{X_{1},\ldots ,X_{n}}(x_{1},\ldots ,x_{n})=\int \prod _{i=1}^{n}p_{X_{i}}(x_{i}\mid \theta )\,dP(\theta ).}" loading="lazy"></dd></dl>

These equations show the joint distribution or density characterised as a mixture distribution based on the underlying limiting empirical distribution (or a parameter indexing this distribution).

Note that not all finite exchangeable sequences are mixtures of i.i.d. To see this, consider sampling without replacement from a <a href="Finite_set" title="Finite set">finite set</a> until no elements are left. The resulting sequence is exchangeable, but not a mixture of i.i.d. Indeed, conditioned on all other elements in the sequence, the remaining element is known.

<div class="mw-heading mw-heading2"><h2 id="Covariance_and_correlation">Covariance and correlation</h2></div>

Exchangeable sequences have some basic <a href="Covariance_and_correlation" title="Covariance and correlation">covariance and correlation</a> properties which mean that they are generally positively correlated. For infinite sequences of exchangeable random variables, the covariance between the random variables is equal to the variance of the mean of the underlying distribution function.<a href="#cite_note-O'Neill-10">[10]</a> For finite exchangeable sequences the covariance is also a fixed value which does not depend on the particular random variables in the sequence. There is a weaker lower bound than for infinite exchangeability and it is possible for negative correlation to exist.

Covariance for exchangeable sequences (infinite): If the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},X_{3},\ldots }">

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<annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},X_{3},\ldots }</annotation>

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</math><img src="./6c1dd59b3860ca16ef451f9000c769aaa3ce593b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.761ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},X_{3},\ldots }" loading="lazy"> is exchangeable, then

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {cov} (X_{i},X_{j})=\operatorname {var} (\operatorname {E} (X_{i}\mid F_{\mathbf {X} }))=\operatorname {var} (\operatorname {E} (X_{i}\mid \theta ))\geq 0\quad {\text{for }}i\neq j.}</annotation>

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</math><img src="./e16eb6b728b12b4fa27f75343e1a2b47dd7af191.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:64.05ex; height:3.009ex;" alt="{\displaystyle \operatorname {cov} (X_{i},X_{j})=\operatorname {var} (\operatorname {E} (X_{i}\mid F_{\mathbf {X} }))=\operatorname {var} (\operatorname {E} (X_{i}\mid \theta ))\geq 0\quad {\text{for }}i\neq j.}" loading="lazy"></dd></dl>

Covariance for exchangeable sequences (finite): If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},\ldots ,X_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},\ldots ,X_{n}}</annotation>

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</math><img src="./d67872301909a9d739e265252ad0c7339cead069.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.312ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},\ldots ,X_{n}}" loading="lazy"> is exchangeable with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}=\operatorname {var} (X_{i})}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}=\operatorname {var} (X_{i})}</annotation>

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</math><img src="./1d9c06df0d7e81b3b897a5b24f444887c177970d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.318ex; height:3.176ex;" alt="{\displaystyle \sigma ^{2}=\operatorname {var} (X_{i})}" loading="lazy">, then

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {cov} (X_{i},X_{j})\geq -{\frac {\sigma ^{2}}{n-1}}\quad {\text{for }}i\neq j.}</annotation>

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</math><img src="./1cfafc5e988c768cf94c8fbb7cc987423226ed30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.159ex; height:5.843ex;" alt="{\displaystyle \operatorname {cov} (X_{i},X_{j})\geq -{\frac {\sigma ^{2}}{n-1}}\quad {\text{for }}i\neq j.}" loading="lazy"></dd></dl>

The finite sequence result may be proved as follows. Using the fact that the values are exchangeable, we have

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<mtext>all ordered pairs</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}0&\leq \operatorname {var} (X_{1}+\cdots +X_{n})\\&=\operatorname {var} (X_{1})+\cdots +\operatorname {var} (X_{n})+\underbrace {\operatorname {cov} (X_{1},X_{2})+\cdots \quad {}} _{\text{all ordered pairs}}\\&=n\sigma ^{2}+n(n-1)\operatorname {cov} (X_{1},X_{2}).\end{aligned}}}</annotation>

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</math><img src="./9af27df1b51c3135258b35e740ae7a0eb45d5ef0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:53.096ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}0&\leq \operatorname {var} (X_{1}+\cdots +X_{n})\\&=\operatorname {var} (X_{1})+\cdots +\operatorname {var} (X_{n})+\underbrace {\operatorname {cov} (X_{1},X_{2})+\cdots \quad {}} _{\text{all ordered pairs}}\\&=n\sigma ^{2}+n(n-1)\operatorname {cov} (X_{1},X_{2}).\end{aligned}}}" loading="lazy"></dd></dl>

We can then solve the inequality for the covariance yielding the stated lower bound. The non-negativity of the covariance for the infinite sequence can then be obtained as a limiting result from this finite sequence result.

Equality of the lower bound for finite sequences is achieved in a simple urn model: An urn contains 1 red marble and n − 1 green marbles, and these are sampled without replacement until the urn is empty. Let Xi = 1 if the red marble is drawn on the i-th trial and 0 otherwise. A finite sequence that achieves the lower covariance bound cannot be extended to a longer exchangeable sequence.<a href="#cite_note-11">[11]</a>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<ul><li>Any <a href="Convex_combination" title="Convex combination">convex combination</a> or <a href="Mixture_distribution" title="Mixture distribution">mixture distribution</a> of <a href="Iid" class="mw-redirect" title="Iid">iid</a> sequences of random variables is exchangeable. A converse proposition is <a href="De_Finetti's_theorem" title="De Finetti's theorem">de Finetti's theorem</a>.<a href="#cite_note-12">[12]</a></li>

<li>Suppose an <a href="Urn_model" class="mw-redirect" title="Urn model">urn</a> contains <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> red and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> blue marbles. Suppose marbles are drawn without replacement until the urn is empty. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation>

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</math><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy"> be the indicator random variable of the event that the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">-th marble drawn is red. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{X_{i}\right\}_{i=1,\dots ,n+m}}">

<msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\{X_{i}\right\}_{i=1,\dots ,n+m}}</annotation>

</semantics>

</math><img src="./9bb80c9dce0fad112f8e73aa6eade351d0549c9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.497ex; height:3.176ex;" alt="{\displaystyle \left\{X_{i}\right\}_{i=1,\dots ,n+m}}" loading="lazy"> is an exchangeable sequence. This sequence cannot be extended to any longer exchangeable sequence.</li>

<li>Suppose an urn contains <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> red and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> blue marbles. Further suppose a marble is drawn from the urn and then replaced, with an extra marble of the same colour. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation>

</semantics>

</math><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy"> be the indicator random variable of the event that the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">-th marble drawn is red. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{X_{i}\right\}_{i\in \mathbb {N} }}">

<msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\{X_{i}\right\}_{i\in \mathbb {N} }}</annotation>

</semantics>

</math><img src="./e4bc0c389acd750cb78b913e6b413dcbc4694a7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.131ex; height:3.009ex;" alt="{\displaystyle \left\{X_{i}\right\}_{i\in \mathbb {N} }}" loading="lazy"> is an exchangeable sequence. This model is called <a href="Polya's_urn" class="mw-redirect" title="Polya's urn">Polya's urn</a>.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (X,Y)}</annotation>

</semantics>

</math><img src="./41f29b9537685f499713112d6802e811cbf51bba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.597ex; height:2.843ex;" alt="{\displaystyle (X,Y)}" loading="lazy"> have a <a href="Bivariate_normal_distribution" class="mw-redirect" title="Bivariate normal distribution">bivariate normal distribution</a> with parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu =0}</annotation>

</semantics>

</math><img src="./3753282c0ad2ea1e7d63f39425efd13c37da3169.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.663ex; height:2.676ex;" alt="{\displaystyle \mu =0}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{x}=\sigma _{y}=1}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{x}=\sigma _{y}=1}</annotation>

</semantics>

</math><img src="./fceaba4da86a3ee484e8de3fc865dbaa2de6cfb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.236ex; height:2.843ex;" alt="{\displaystyle \sigma _{x}=\sigma _{y}=1}" loading="lazy"> and an arbitrary <a href="Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">correlation coefficient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \in (-1,1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho \in (-1,1)}</annotation>

</semantics>

</math><img src="./d05d5ab2639e435b4027d8296a803b66aa746263.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.019ex; height:2.843ex;" alt="{\displaystyle \rho \in (-1,1)}" loading="lazy">. The random variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> are then exchangeable, but independent only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho =0}</annotation>

</semantics>

</math><img src="./2ba6310b27df5f9c9b0b1732e08cce27b99d68cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.463ex; height:2.676ex;" alt="{\displaystyle \rho =0}" loading="lazy">. The <a href="Density_function" class="mw-redirect" title="Density function">density function</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x,y)=p(y,x)\propto \exp \left[-{\frac {1}{2(1-\rho ^{2})}}(x^{2}+y^{2}-2\rho xy)\right].}">

<mrow>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(x,y)=p(y,x)\propto \exp \left[-{\frac {1}{2(1-\rho ^{2})}}(x^{2}+y^{2}-2\rho xy)\right].}</annotation>

</semantics>

</math><img src="./1b775ed0d79b8bd7ece01f18b919cee7985f067b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; margin-left: -0.089ex; width:54.75ex; height:6.343ex;" alt="{\displaystyle p(x,y)=p(y,x)\propto \exp \left[-{\frac {1}{2(1-\rho ^{2})}}(x^{2}+y^{2}-2\rho xy)\right].}" loading="lazy"></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

The <a href="Von_Neumann_extractor" class="mw-redirect" title="Von Neumann extractor">von Neumann extractor</a> is a <a href="Randomness_extractor" title="Randomness extractor">randomness extractor</a> that depends on exchangeability: it gives a method to take an exchangeable sequence of 0s and 1s (<a href="Bernoulli_trials" class="mw-redirect" title="Bernoulli trials">Bernoulli trials</a>), with some probability p of 0 and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=1-p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q=1-p}</annotation>

</semantics>

</math><img src="./22e387fe24ba3da5f9a0dc424923cdfc2c08990c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.34ex; height:2.509ex;" alt="{\displaystyle q=1-p}" loading="lazy"> of 1, and produce a (shorter) exchangeable sequence of 0s and 1s with probability 1/2.

Partition the sequence into non-overlapping pairs: if the two elements of the pair are equal (00 or 11), discard it; if the two elements of the pair are unequal (01 or 10), keep the first. This yields a sequence of Bernoulli trials with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=1/2,}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p=1/2,}</annotation>

</semantics>

</math><img src="./a23a9f66dbc9074ef2f2683ecd6b5c5721b759fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.492ex; height:2.843ex;" alt="{\displaystyle p=1/2,}" loading="lazy"> as, by exchangeability, the odds of a given pair being 01 or 10 are equal.

Exchangeable random variables arise in the study of <a href="U_statistic" class="mw-redirect" title="U statistic">U statistics</a>, particularly in the Hoeffding decomposition.<a href="#cite_note-13">[13]</a>

Exchangeability is a key assumption of the distribution-free inference method of <a href="Conformal_prediction" title="Conformal prediction">conformal prediction</a>.<a href="#cite_note-14">[14]</a>

<ul><li><a href="De_Finetti_theorem" class="mw-redirect" title="De Finetti theorem">De Finetti theorem</a></li>

<li><a href="Hewitt-Savage_zero-one_law" class="mw-redirect" title="Hewitt-Savage zero-one law">Hewitt-Savage zero-one law</a></li>

<li><a href="Resampling_(statistics)" title="Resampling (statistics)">Resampling</a></li>

<li><a href="Resampling_(statistics)#Permutation_tests" title="Resampling (statistics)">Resampling (statistics) § Permutation tests</a>, statistical tests based on exchanging between groups</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-ChowTeicher-1">^ <a href="#cite_ref-ChowTeicher_1-0">a</a> <a href="#cite_ref-ChowTeicher_1-1">b</a> <a href="#cite_ref-ChowTeicher_1-2">c</a> In short, the order of the sequence of random variables does not affect its joint probability distribution.

<ul><li>Chow, Yuan Shih and Teicher, Henry, Probability theory. Independence, interchangeability, martingales, Springer Texts in Statistics, 3rd ed., Springer, New York, 1997. xxii+488 pp. <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-98228-0</bdi></li></ul>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> Aldous, David J., Exchangeability and related topics, in: École d'Été de Probabilités de Saint-Flour XIII — 1983, Lecture Notes in Math. 1117, pp. 1–198, Springer, Berlin, 1985. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-15203-3</bdi> <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBFb0099421">10.1007/BFb0099421</a>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFDiaconis2009" class="citation journal cs1"><a href="Persi_Diaconis" title="Persi Diaconis">Diaconis, Persi</a> (2009). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0273-0979-09-01262-2">"Book review: Probabilistic symmetries and invariance principles (Olav Kallenberg, Springer, New York, 2005)"</a>. Bulletin of the American Mathematical Society. New Series. 46 (4): 691–696. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0273-0979-09-01262-2">10.1090/S0273-0979-09-01262-2</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2525743">2525743</a>.</cite>

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<li id="cite_note-Kallenberg-4">^ <a href="#cite_ref-Kallenberg_4-0">a</a> <a href="#cite_ref-Kallenberg_4-1">b</a> <a href="Olav_Kallenberg" title="Olav Kallenberg">Kallenberg, O.</a>, Probabilistic symmetries and invariance principles. Springer-Verlag, New York (2005). 510 pp. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-25115-4</bdi>.

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<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFZabell1992" class="citation journal cs1">Zabell, S. L. (1992). "Predicting the unpredictable". Synthese. 90 (2): 205. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf00485351">10.1007/bf00485351</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:9416747">9416747</a>.</cite>

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> Barlow, R. E. & Irony, T. Z. (1992) "Foundations of statistical quality control" in Ghosh, M. & Pathak, P.K. (eds.) Current Issues in Statistical Inference: Essays in Honor of D. Basu, Hayward, CA: Institute of Mathematical Statistics, 99-112.

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<li id="cite_note-7"><a href="#cite_ref-7">^</a> Bergman, B. (2009) "Conceptualistic Pragmatism: A framework for Bayesian analysis?", IIE Transactions, 41, 86–93

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<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFCordaniWechsler2006" class="citation book cs1">Cordani, L. K.; Wechsler, S. (2006). <a rel="nofollow" class="external text" href="https://iase-web.org/documents/papers/icots7/3I1_CORD.pdf?1402524964">"Teaching independence and exchangeability"</a> (PDF). Proceedings of the International Conference on Teaching Statistics. Den Haag: International Association for Statistical Education.</cite>

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<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFDiaconis1988" class="citation book cs1"><a href="Persi_Diaconis" title="Persi Diaconis">Diaconis, P.</a> (1988). "Recent Progress on de Finetti's Notions of Exchangeability". In <a href="Jos%C3%A9-Miguel_Bernardo" title="José-Miguel Bernardo">Bernardo, J. M.</a>; et al. (eds.). Bayesian Statistics. Vol. 3. Oxford University Press. pp. 111–125. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-19-852220-7</bdi>.</cite>

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<li id="cite_note-O'Neill-10">^ <a href="#cite_ref-O'Neill_10-0">a</a> <a href="#cite_ref-O'Neill_10-1">b</a> <cite id="CITEREFO'Neill2009" class="citation journal cs1">O'Neill, B. (2009). "Exchangeability, Correlation and Bayes' Effect". International Statistical Review. 77 (2): 241–250. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1751-5823.2008.00059.x">10.1111/j.1751-5823.2008.00059.x</a>.</cite>

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<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite id="CITEREFTaylorDafferPatterson1985" class="citation book cs1">Taylor, Robert Lee; Daffer, Peter Z.; Patterson, Ronald F. (1985). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6RaoAAAAIAAJ">Limit theorems for sums of exchangeable random variables</a>. Rowman and Allanheld. pp. 1–152. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780847674350</bdi>.</cite>

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<li id="cite_note-12"><a href="#cite_ref-12">^</a> Spizzichino, Fabio Subjective probability models for lifetimes. Monographs on Statistics and Applied Probability, 91. Chapman & Hall/CRC, Boca Raton, FL, 2001. xx+248 pp. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-58488-060-0</bdi>

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<li id="cite_note-13"><a href="#cite_ref-13">^</a> <cite id="CITEREFBorovskikh1996" class="citation book cs1">Borovskikh, Yu. V. (1996). "Chapter 10 Dependent variables". U-statistics in Banach spaces. Utrecht: VSP. pp. 365–376. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>90-6764-200-2</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1419498">1419498</a>.</cite>

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<li id="cite_note-14"><a href="#cite_ref-14">^</a> <cite id="CITEREFShaferVovk2008" class="citation journal cs1">Shafer, Glenn; Vovk, Vladimir (2008). <a rel="nofollow" class="external text" href="https://www.jmlr.org/papers/v9/shafer08a.html">"A Tutorial on Conformal Prediction"</a>. Journal of Machine Learning Research. 9: 371–421.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li>Aldous, David J., Exchangeability and related topics, in: École d'Été de Probabilités de Saint-Flour XIII — 1983, Lecture Notes in Math. 1117, pp. 1–198, Springer, Berlin, 1985. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-15203-3</bdi> <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBFb0099421">10.1007/BFb0099421</a></li>

<li>Chow, Yuan Shih and Teicher, Henry, Probability theory. Independence, interchangeability, martingales, Springer Texts in Statistics, 3rd ed., Springer, New York, 1997. xxii+488 pp. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-98228-0</bdi></li>

<li><cite id="CITEREFDawid2013" class="citation book cs1">Dawid, A. Philip (2013). "Exchangeability and its ramifications". In Damien, Paul; et al. (eds.). Bayesian Theory and Applications. Oxford University Press. pp. 19–30. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-969560-7</bdi>.</cite></li>

<li><a href="Olav_Kallenberg" title="Olav Kallenberg">Kallenberg, O.</a>, Probabilistic symmetries and invariance principles. Springer-Verlag, New York (2005). 510 pp. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-25115-4</bdi>.</li>

<li>Kingman, J. F. C., Uses of exchangeability, Ann. Probability 6 (1978) 83–197 <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0494344">0494344</a> <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2243211">2243211</a></li>

<li>O'Neill, B. (2009) Exchangeability, Correlation and Bayes' Effect. International Statistical Review 77(2), pp. 241–250. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-15203-3</bdi> <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1751-5823.2008.00059.x">10.1111/j.1751-5823.2008.00059.x</a></li>

<li><cite id="CITEREFTaylorDafferPatterson1985" class="citation book cs1">Taylor, Robert Lee; Daffer, Peter Z.; Patterson, Ronald F. (1985). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6RaoAAAAIAAJ">Limit theorems for sums of exchangeable random variables</a>. Rowman and Allanheld. pp. 1–152. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780847674350</bdi>.</cite></li></ul></div><div><div class="zim-footer">

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