$\theta$ is $\{\theta: {\mathsf{cpl}}_{T(x)|h}(\theta; {\mathcal{S}}) > \alpha\}$. Then, by , the conditional coverage probability is ${\mathsf{P}}_{X|\theta}\{{\mathsf{cpl}}_{T(X)|h}(\theta;{\mathcal{S}}) > \alpha \mid H(X)=h\} \geq 1-\alpha$. In Fisher’s mind, this is a more meaningful coverage probability since it is conditioned on a particular aspect of the observed data, namely, $H(x)=h$. In other words, the probability calculation focuses on a relevant subset $\{x: H(x)=h\}$ of the sample space. In some cases, though, conditional validity is the same as ordinary validity.

\[co:cond.valid\] Suppose that the predictive random set ${\mathcal{S}}$ does not depend on the observed $H(x)=h$, so that ${\mathsf{P}}_{{\mathcal{S}}|h} \equiv {\mathsf{P}}_{\mathcal{S}}$ and ${\mathsf{cbel}}_{T(x)|h} \equiv {\mathsf{cbel}}_{T(x)}$. Then under the conditions of Theorem \[thm:cond.valid\], the conditional IM is unconditionally valid, i.e., for any $A \subseteq \Theta$, $$\sup_{\theta \not\in A} {\mathsf{P}}_{X|\theta}\{{\mathsf{cbel}}_{T(X)}(A;{\mathcal{S}}) \geq 1-\alpha\} \leq \alpha, \quad \forall \; \alpha \in (0,1).$$

Two possible ways the condition of Corollary \[co:cond.valid\] may hold are as follows. First, in the P-step, the user may specify ${\mathcal{S}}$ directly without dependence on the observed $H(x)=h$; see Section \[SS:student\]. Second, it could happen that $V_T$ and $V_H$ are statistically independent, in which case the distribution ${\mathsf{P}}_{\mathcal{S}}$ for ${\mathcal{S}}$ is determined by the marginal distribution of $V_T$, which does not depend on $h$.

Finding conditional associations 