Content deleted Content added
m Em-dash for em-dash. |
mv properties to one place |
||
Line 4:
The notion is central to [[Bruno de Finetti|Bruno de Finetti's]] development of [[predictive inference]] and to [[Bayesian statistics]] — where [[frequentist statistics]] uses i.i.d. variables (samples from a population), Bayesian statistics more frequently uses exchangeable sequences. They are a key way in which Bayesian inference is "data-centric" (based on past and future observations), rather than "model-centric", as exchangeable sequences that are not i.i.d. cannot be modeled as "sampling from a fixed population".
[[de Finetti's theorem]] characterizes exchangeable sequences as "mixtures" of i.i.d. sequences — while an exchangeable sequence need not itself be i.i.d., it can be expressed as a "mixture" of underlying i.i.d. sequences.▼
== Definition ==
Line 29 ⟶ 27:
* Let <math>(X, Y)</math> have a [[bivariate normal distribution]] with parameters <math>\mu = 0</math>, <math>\sigma_x = \sigma_y = 1</math> and an arbitrary [[correlation coefficient]] <math>\rho\in (-1, 1)</math>. The random variables <math>X</math> and <math>Y</math> are then exchangeable, but independent only if <math>\rho=0</math>. The [[density function]] is <math>p(x, y) = p(y, x) \propto \exp(-.5(1-\rho^2)^{-1}(x^2+y^2-2\rho xy))</math>.
==
;[[von Neumann extractor]]:▼
The [[von Neumann extractor]] is a [[randomness extractor]] that depends on exchangeability: it gives a method to take an exchangeable sequence of 0s and 1s ([[Bernoulli trials]]), with some probability ''p'' of 0 and <math>q=1-p</math> of 1, and produce a (shorter) exchangeable sequence of 0s and 1s with probability 1/2.▼
▲* [[de Finetti's theorem]] characterizes exchangeable sequences as
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>p=1/2,</math> as, by exchangeability, the odds of a given pair being 01 or 10 are equal.▼
* An infinite exchangeable sequence is [[strictly stationary]], thus a [[law of large numbers]] in the form of [[Ergodic theory|Birkhoff-Khinchin theorem]] applies.▼
* [[Covariance]]: for a finite exchangeable sequence { ''X''<sub>''i''</sub> }<sub>''i'' = 1, 2, 3, ...</sub> of length ''n'':
Line 63 ⟶ 59:
:: <math> \operatorname{cov} (X_i,X_j) = \text{constant} \ge 0.\,</math>
== Applications ==
▲;[[von Neumann extractor]]:
▲* An infinite exchangeable sequence is [[strictly stationary]], thus a [[law of large numbers]] in the form of [[Ergodic theory|Birkhoff-Khinchin theorem]] applies.
▲The [[von Neumann extractor]] is a [[randomness extractor]] that depends on exchangeability: it gives a method to take an exchangeable sequence of 0s and 1s ([[Bernoulli trials]]), with some probability ''p'' of 0 and <math>q=1-p</math> 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>p=1/2,</math> as, by exchangeability, the odds of a given pair being 01 or 10 are equal.
==See also==
| |||