of sufficiency. At a high level, both theories provide a sort of dimension reduction. The key difference between the two is that sufficiency focuses on reducing the dimension of the data, while our approach focuses on reducing the dimension of the auxiliary variable. Although the conditional IM can, in some cases, correspond to a sufficient statistic-type of reduction, this is not necessary; see the remarks at the end of Section \[SS:student\]. Proper conditioning appears to be more important than the use of sufficient statistics. In fact, in some cases, it is possible, within the IM framework, to reduce the dimension further than that which is provided by sufficiency; see Section \[S:gcim\].

\[re:bayes\] As we mentioned previously, conditional IMs have some connections to Bayes’ theorem, in particular, in how information is combined or aggregated across samples. In fact, it can be shown that, in a certain sense, the Bayes solution is a special case of conditional IMs. To see this, consider a simple but generic example. The Bayes model, cast in terms of associations, is of the following form: $$\theta = U_0, \quad U_0 \sim {\mathsf{P}}_{U_0} \quad \text{and} \quad X=a(U_0,U_1), \quad U_1 \sim {\mathsf{P}}_{U_1},$$ where ${\mathsf{P}}_U$ for $U=(U_0,U_1)$ is such that 