Exchangeable random variables: Difference between revisions

Revision as of 06:18, 2 May 2009

Revision as of 03:31, 26 August 2009

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A sequence of [[independent and identically-distributed random variables]] (i.i.d.) is exchangeable, but so is [[simple random sample without replacement|sampling without replacement]], which is not independent.

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.