vaihtokkyys

Vaihtokkyys is a Finnish term that translates to "exchangeability" in English. It is a concept used in probability theory and statistics to describe a specific type of random variable relationship. When a set of random variables is exchangeable, the joint probability distribution of these variables remains unchanged if their order is permuted. In simpler terms, the order in which you observe the variables does not matter for their overall probability.

This property is important because it simplifies statistical inference. If a collection of random variables is exchangeable, then they behave as if they were drawn independently from the same underlying probability distribution. This assumption of exchangeability is weaker than independence but is often sufficient for many statistical methods.

Examples of exchangeable random variables include the outcomes of a sequence of coin flips, or the results of drawing balls from an urn with replacement. In these scenarios, any specific sequence of outcomes has the same probability as any other sequence with the same number of heads and tails, or the same count of each color of ball.

The concept of exchangeability was notably explored by Bruno de Finetti, a prominent Italian probabilist. He developed de Finetti's theorem, which states that any infinitely exchangeable sequence of random variables can be represented as a mixture of independent and identically distributed (i.i.d.) random variables. This theorem highlights the close relationship between exchangeability and the more commonly used i.i.d. assumption.