Definition of Term: Second-order Exchangeability

A sequence of random variables is described as exchangeable when our beliefs (in terms of a joint probability distribution) about that collection are unaffected by the order in which they appear.

A less-restrictive specification is that of second-order exchangeability. A sequence of random vectors is described as second-order exchangeable if our beliefs in terms of the mean, variance and covariance for that collection are unaffected by the permutation of the order of the vectors. In other words, this requires that every random vector has the same expectation and variance, and every pair of vectors have the same covariance.

Second-order exchangeability is a concept that is expressed in terms of expectations and variances rather than full probability distributions. It is therefore often exploited in Bayes linear analyses.