Finite and Infinite Weighted Exchangeable Sequences

Motivated by recent interest in predictive inference under distribution shifts, we study the problem of approximating finite weighted exchangeable sequences by a mixture of finite sequences with independent terms. Two different approaches are developed: a combinatorial approach and a probabilistic approach. Various bounds are derived in terms of weight functions, extending previous results on finite exchangeable sequences. As a byproduct, we obtain a version of de Finetti’s theorem for infinite weighted exchangeable sequences, recovering an instance of Barber, Candes, Ramdas and Tibshirani 1 (Bernoulli, 30 (2024), pp. 3004–3028) in the Polish space with bounded and continuous weights.