Abstract: Let (X_n : n ≥ 1) be a sequence of random observations. Let σ_n (·) = P (X_n+1 ∈ · | X₁,…, X_n) be the nth predictive distribution and σ₀ (·) = P (X₁ ∈ ·) be the marginal distribution of X₁. To make predictions on (X_n), a Bayesian forecaster needs only the collection σ = (σ_n : n ≥ 0). From the Ionescu-Tulcea theorem, σ can be assigned directly, without passing through the usual prior/posterior scheme. One main advantage is that no prior probability has to be selected. This point of view is adopted in this paper. The choice of σ is subject to only two requirements: (i) the resulting sequence (X_n) must be conditionally identically distributed and (ii) each σ_n+1 must be a simple recursive update of σ_n. Various new σ satisfying (i) and (ii) are introduced and investigated. For such σ, we determine the asymptotics of σ_n as n → ∞. In some cases, we also evaluate the probability distribution of (X_n).

Key words and phrases: Asymptotics, Bayesian nonparametrics, conditional identity in distribution, exchangeability, predictive distribution, sequential prediction, total variation distance.