content_image/307385.jpg
## 2. Setup 

Let S be a finite set of outcomes . Every period n = 0 , 1 , . . . an outcome is realized. Let Ω = S N be the set of realizations equipped with the product topology. A theory is a probability measure over Ω, representing the distribution of some stochastic process. The set Δ(Ω) of theories is a convex and compact metrizable subspace of the topological vector space of all signed measures equipped with the weak- ∗ topology. A paradigm is a set of theories. 

Let S < N = ⋃ n ≥ $_{0}$S n be the set of partial realizations including the empty sequence e . For every partial realization σ = ( s$_{0}$, . . . , s$_{n}$$_{−}$$_{1}$ ) let N ( σ ) ⊆ S N be the set of realizations x ∈ S N that extends σ , i.e. such that σ is the initial segment of x of length n . In particular N ( e ) = S $^{N}$. 

Definition 1. Let µ be a theory and x ∈ S N = ( s$_{0}$, s$_{1}$, . . . ) a realization. The forecast of µ over x at period n is the element p$_{n}$ ∈ Δ( S ) that is given by 

<img src='content_image/307385.jpg'>

for every outcome s ∈ S . That is, the forecast at period n is the forecast made at period n − 1 (based on all information available at period n − 1) on the outcome at period n . Here and later, when n = 0 the sequence ( s$_{0}$, . . . , s$_{n}$$_{−}$$_{1}$ ) refers to the empty sequence e . The forecast p$_{n}$ is undefined if µ ( N ( s$_{0}$, . . . , s$_{n}$$_{−}$$_{1}$ )) = 0. 

A prequential test is given by a Borel function T : (Δ( S ) × S ) N → { FAIL , PASS } : Bob passes the test if T ( p$_{0}$, s$_{0}$, p$_{1}$, s$_{1}$, . . . ) = PASS, where, for every k ∈ N , p$_{k}$ is the forecast on period k according to Bob’s theory and s$_{k}$ is the outcome of period k . This definition of test is based on Dawid’s (1984) prequential principle that assessment of a theory µ should depend on µ only through the sequence of forecasts that µ made over the realized sequence of outcomes, and not on forecasts conditional on data that has not been realized. More general tests allow the tester (Alice) to use forecasts of Bob’s theory after any finite history, including the histories that did not occur. This paper focuses on prequential tests. 

