$\tau(U)$, of the auxiliary variable $U$ to $\theta$ and some function $T(X)$ of the data $X$.

The second step is familiar, as it relates to working with, say, a minimal sufficient statistic. The first step, however, is less familiar and generally more difficult; see Section \[S:finding\].

To make this formal, suppose that $x \mapsto (T(x),H(x))$ and $u \mapsto (\tau(u),\eta(u))$ are one-to-one functions. Suppose that the relationship $x=a(u,\theta)$ in the baseline association can be decomposed as

\[eq:decomposition\] $$\begin{aligned}
H(x) & = \eta(u), \label{eq:condition} \\
T(x) & = b(\tau(u), \theta). \label{eq:condition0} \end{aligned}$$

This decomposition immediately suggests an alternative association. Let $(V_T,V_H) \in {\mathbb{V}}_T \times {\mathbb{V}}_H$ be the image of $U$ under $(\tau,\eta)$, and let ${\mathsf{P}}_{V_T|h}$ be the conditional distribution of $V_T$, given $V_H=h$, where $h \in H({\mathbb{X}})$. Since $H(x)$ provides no information about $\theta$, we can take a new association $$\label{eq:cond-aeqn}
T(X) = b(V_T,\theta), \quad V_T \sim {\mathsf{P}}_{V_T | H(x)}.$$ We shall refer to this as a *conditional association*. This alternative association can be understood via a certain hierarchical representation of the sampling model; see Remark \[re:hierarchy\]. The important point is that $\tau$ can often be chosen so that $V_T$ is of lower dimension than $U$. In fact, $V_T$ will often have 