Bernstein operators and finite exchangeability

We present a method for proving finite versions of De Finetti-type theorems for general measurable spaces (). Bernstein operators on the one dimensional simplex [0, 1] are generalized to Bernstein operators on limits of simplices, representing distributions on (). The positivity of finite moment sequences allows the approximation, via the generalized Bernstein operators, of exchangeable or partially exchangeable distributions by mixtures of product measures or mixtures of certain subsets of product measures. Examples, including sequences of spherically invariant variables and sequences of separately rotatable random matrices, are given.