The previous two lessons both describe data reduction principles in the following way. A function T(x) of the sample is specified, and the principle states that if x and y are two sample points with T(x) = T(y), then the same inference about θ should be made whether x or y is observed. The function T(x) is a sufficient statistic when the Sufficiency Principle is used. The "value" of T(x) is the set of all likelihood functions proportions to L(θ|x) if the Likelihood Principle is used. The Equivariance Principle describes a data reduction technique in s slightly different way. In any application of the Equivariance Principle, a function T(x) is specified, but if T(x) = T(y), then the Equivariance Principle states that the inference made if x is observed should have a certain relationship to the inference made if y is observed, although the two inferences may not be the same. This restriction on the inference procedure sometimes leads to a simpler analysis, just as do the data reduction principles discussed in earlier lessons.
