A Clarification of (C)

Birnbaum proves in his (1962) that (S) and (C) jointly entail (L).  He claims that (C) entails (S), which would mean that (C) entails (L) by itself, but he does not prove it.  In his (1964), he clarifies that he can only prove that (C) entails (S) by using a principle (M) that is strictly weaker than (S) but does not follow from (C).  Thus, the strongest result he can claim is that (C) and (M) jointly entail (L).  In their (1986), however, Evans et al. prove that in fact (C) alone does entail (L).  In this post I begin the task of reconstructing the Evans et al. proof.  First I need to clarify a point of obscurity in Birnbaum’s 1962 formulation of (C).

In 1962, Birnbaum formulates (C) as follows:

The Principle of Conditionality (C): If an experiment E is (mathematically equivalent to) a mixture G of components {E_h}, with possible outcomes (E_h, x_h), then Ev(E,(E_h, x_h)) = Ev(E_h, x_h).

The parenthetical phrase “mathematically equivalent to” here turns out to be essential.  (C) applies to any experiment that contains an ancillary statistic.  In a mixture experiment, the outcome of the random process that determines which component experiment to perform is an ancillary statistic.  However, non-mixture experiments can have ancillary statistics as well.  These experiments are “mathematically equivalent to” mixture experiments, but they do not involve an actual two-stage process that consists of first using a random process to choose a component experiment and then performing that component experiment.

In his (1972), Birnbaum formulates the notion of an ancillary statistic as follows:

h = h(x) is called an ancillary statistic if it admits the factored form f(x;θ) = g(h) f(x|h; θ) where g = g(h) = Prob (h(X)=h) is independent of θ.  

In other words, an ancillary statistic is a statistic independent of the parameters whose probability distribution that can be factored out of the likelihood function f(x;θ) to yield that conditional likelihood function f(x|h; θ).  A generic mixture experiment yields a simple example.  Suppose you flip a coin to decide whether to perform experiment E₁ or E₂, where information about the bias of that coin is not informative about the data-generating process in E₁ or E₂.  There is an unconditional likelihood function f(x;θ) for this mixture experiment as a whole.  However, if a frequentist knows which way the flip turns out and thus which of E₁ or E₂ is performed he or she will typically take that information into account and use the conditional likelihood function f(x|h; θ) that takes this information into account.  He or she will thus neglect the mixture structure of the experiment, acting as if it had been known all along that the experiment which is actually performed would be performed.

In his (1972), Birnbaum formulates (C) in terms of the notion of an ancillary statistic.  He first defines some notation:

(E_h,x) denotes a model of evidence determined by an outcome x of the experiment E_h: (Ω,S_h,f_h)  where S={x: h(x)=h}.  E may be called a mixture experiment, with components E_h having respective probabilities g(h).

He then reformulates (C):

Conditionality (C): If h(x) is an ancillary statistic, then Ev(E,x)=Ev(E_h,x), where h=h(x).

This formulation is not different in substance from Birnbaum’s 1962 formulation; it is merely more explicit that being mathematically equivalent to a mixture experiment means having an ancillary statistic.