In my previous post, I gave an example that motivates the conditionality principle and presented Birnbaum’s formulation of that principle. In this post, I do likewise for the sufficiency principle. To recap the big picture: frequentists typically accept conditionality and sufficiency principles while rejecting the likelihood principle. The likelihood principle is a central tenant of Bayesianism that follows directly from using Bayes’ theorem as an update rule. Many of the features of Bayesianism that frequentists find objectionable follow from the likelihood principle alone, so it is a short step from accepting the likelihood principle to accepting Bayesianism. Birnbaum argues that the conditionality and sufficiency principles are equivalent to the likelihood principle, putting significant pressure on frequentists to justify their position.
The sufficiency principle appeals to the notion of a sufficient statistic. Roughly speaking, a sufficient statistic lumps together some outcomes of an experiment that have the following property: given that some outcome in the lumped-together set occurred, which one of those outcomes occurred is independent of the parameters of the experiment. For instance, consider the outcome of a series of two coin tosses, where the tosses are assumed to be independent and identically distributed Bernoulli trials with probability p of heads. The outcome space for this experiment is the set of possible sequences of outcomes of two coin tosses: HH, HT, TH, TT. A sufficient statistic for this experiment is the number of heads. This statistic is sufficient because, given the number of heads, the exact sequence of heads and tails is independent of p.
The sufficiency principle says, roughly, that a sufficient statistic summarizes the results of an experiment with no loss of information. In other words, given the value t(x) of a statistic T(X) that is sufficient for θ, you don’t learn any more about θ by learning x. This claim is very widely accepted and appears to be well-motivated. x is independent of θ conditional on t(x), and it’s hard to see how one quantity could provide information about another quantity of which it is independent. For instance, the sufficiency principle says that, given the number of heads obtained in a sequence of n tosses, you don’t learn any more about p by learning the exact sequence of heads and tails. (This application of the likelihood principle requires the assumption that the tosses are independent and identically distributed Bernoulli trials; if the possibility that the tosses are non-independent were on the table, for instance, then information about sequence would be relevant and the number of heads would not be a sufficient statistic.)
Birnbaum formulates the likelihood principle, like the sufficiency principle, as a claim about evidential equivalence. Take an experiment E with outcome x and a derived experiment E’ with outcome t=t(x), where T(X) is a sufficient statistic; then Ev(E, x)=Ev(E’,t). In other words, reduction to a sufficient statistic does not change the evidential meaning of an experiment.
My comments at the end of the previous post are also appropriate here: it is a good idea to be wary of general principles insofar as they are motivated merely by the fact that they seem to capture the intuitions at play in simple examples. However, it is worth keeping in mind that the sufficiency principle is not motivated only (or even, I think, primarily) by its intuitive appeal in simple cases, but also by the general claim that one quantity cannot provide information about another quantity of which it is independent. Similarly, the conditionality principle is motivated by the general claim that experiments that were not performed are irrelevant to the interpretation of the experiment that was performed. However, the conditionality principle is formulated to apply to experiments that are “mathematically equivalent” to mixture experiments, so it is not clear that this general claim is general enough to warrant the principle.