In my previous post I gave a few examples of cross-embedded models with multiple ancillaries. In this post, I’ll present the Evans et al. proof that (C) entails (L), which involves constructing a cross-embedded model.
Evans et al. start with an arbitrary experimental outcome (E,x0) and construct by stipulation a hypothetical outcome of a hypothetical Bernoulli experiment (B,h) that has the same likelihood function. They then build a cross-embedded experiment out of E and B. To preview where this is going, they invoke (C) twice to establish that the outcome (x0,h) of the cross-embedded experiment has the same evidential meaning as (E,x0) and as (B,h), and thus that (E,x0) and (B,h) have the same likelihood function as one another. They then repeat the process with an arbitrary experimental outcome (E’,y0) that has the same likelihood function as (E,x0) to establish that (E’,y0) has the same evidential meaning as (B,h), and thus that it has the same evidential meaning as (E,x0). The likelihood principle follows immediately.
Let f(X;θ) be the likelihood function for experiment E at sample point x. Evans et al. construct the following cross-embedded experiment:
V\X | x0 | x1 | … | … | xi | … |
h | ½ f(x0;θ) | ½ f(x1;θ) | … | … | ½ f(xi;θ) | … |
t | ½ - ½f(x0;θ) | ½ f(x0;θ) | 0 | … | 0 | … |
The indicator variable for h is ancillary: its distribution is Bernoulli(1/2) independent of θ. The indicator variable for X=x0 (as opposed to X itself) is ancillary: its distribution is also Bernoulli(1/2) independent of θ. Thus, (C) says that one can conditionalize on these variables without changing evidential meaning.
Call this hypothetical cross-embedded experiment E*. By (C) and the ancillarity of the indicator for h, Ev(E*,(h,x0))=Ev(E,x0). By (C) and the ancillarity of the indicator for x0, Ev(E*,(h,x0))=Ev(B,h).
By the same series of steps, replacing x’s with y’s and Es with E’s, one can show that Ev(E’,y0)=Ev(B,h) for any (E’,y0) such that y0 has the same likelihood function in E’ as x0 has in E. The likelihood principle follows immediately.
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