Vahetuvus

Vahetuvus, or exchangeability, is a concept in probability theory that describes a property of a sequence of random variables. A sequence X1, X2, X3, ... is exchangeable if its joint distribution is unchanged by finite permutations of the indices. In practical terms, any finite initial segment (X1, ..., Xn) has the same joint distribution as (X_{pi(1)}, ..., X_{pi(n)}) for any permutation pi of {1, ..., n}.

Exchangeability is weaker than independence. Every independent and identically distributed (i.i.d.) sequence is exchangeable, but an exchangeable sequence need not be independent. A central result is de Finetti's theorem: an infinite exchangeable sequence is conditionally i.i.d. given some latent random parameter; equivalently, conditioned on a random distribution, the Xi are i.i.d.

Examples help illustrate the concept. A sequence obtained by drawing cards without replacement from a well-mixed finite deck is exchangeable—the order among drawn cards does not affect the joint distribution. A Polya urn scheme yields exchangeable sequences with dependence between terms, yet still invariant under permutations of indices.

Applications of exchangeability abound in Bayesian statistics and nonparametric modeling. It provides a foundational justification for using i.i.d. assumptions in many practical settings while offering a framework to reason about data that may not be strictly independent. In theoretical work, exchangeability helps connect finite-sample behavior to asymptotic descriptions and informs modeling choices when the underlying data-generating process is only partially understood.