this case we can conclude that the conditional IM is at least as efficient as the baseline IM. The independence condition holds in many examples; see Section \[S:more-examples\].

Remarks {#SS:remarks}
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\[re:separable\] More general decompositions in are possible. That is, one may replace “$H(x) = \eta(u)$” in with “$c(x,u) = 0$” for a function $c$. However, this more general “non-separable” case may not fit into the context of the conditional validity theorem; see Theorem \[thm:cond.valid\]. We will have more to say about this in Sections \[SS:condcred\] and \[S:gcim\].

\[re:hierarchy\] The decomposition boils down to a particular hierarchical representation of the sampling model for $X$. Indeed, for functions $H$ and $T$ as in , with $V_H=\eta(U)$, and $V_T = \tau(U)$, data $X \sim {\mathsf{P}}_{X|\theta}$ can be simulated as follows.

1.  Get $(V_T,V_H)$ by sampling $V_H \sim {\mathsf{P}}_{V_H}$ and $V_T \mid V_H \sim {\mathsf{P}}_{V_T|V_H}$;

2.  Obtain $X$ by solving the system $H(X)=V_H$ and $T(X)=b(V_T,\theta)$.

This hierarchical model representation also provides the following insight: when $X=x$ is observed, so too is the value of $V_H$, and this knowledge can be used to update the auxiliary variable distribution, analogous to Bayes’ theorem.

\[re:sufficiency\] There are close connections between the conditional IM and Fisher’s notion 