Bayesian predictive inference without a prior

Abstract

Let (Xn : n ≥ 1) be a sequence of random observations. Let σn(•) = P X n+1 ∈ • | X 1 , . . . , Xn be the n-th predictive distribution and σ 0 (•) = P (X 1 ∈ •) the marginal distribution of X 1 . In a Bayesian framework, to make predictions on (Xn), one only needs the collection σ = (σn : n ≥ 0). Because of 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. In this paper, σ is subjected to two requirements: (i) The resulting sequence (Xn) is conditionally identically distributed, in the sense of [4]; (ii) Each σ n+1 is a simple recursive update of σn. Various new σ satisfying (i)-(ii) are introduced and investigated. For such σ, the asymptotics of σn, as n → ∞, is determined. In some cases, the probability distribution of (Xn) is also evaluated.

A predictive approach to Bayesian forecasting

“…while Y satisfies condition (8). Furthermore, arguing as in [9,Sect. 4], the normal distribution could be replaced by any symmetric stable law.…”

“…NSA is also strictly connected to Dawid's prequential approach and to Pitman's treatment of species sampling sequences. A non exhaustive list of references is: [4], [8], [9], [15], [18], [19], [20], [21], [22], [23], [26], [28], [29], [31], [36], [37], [40], [44], [50], [51], [52].…”

“…sequences have been introduced in [5] and [43] and then investigated or applied in various papers; see e.g. [1], [2], [6], [7], [8], [9], [14], [26], [27], [30] and references therein.…”

“…To make the above argument concrete, the following problem is to be solved: For fixed f and µ, give conditions for the existence of ν satisfying equation (9). More importantly, give an explicit formula for ν provided it exists.…”

“…The strategies satisfying equation ( 4) are investigated in [9]. Under such strategies, X is usually not exchangeable but it is c.i.d.…”