to these as *local conditional IMs*, and we describe their construction and prove a validity theorem. An important example of such a problem is the bivariate normal model with known means and variances but unknown correlation. For this example, we construct a local conditional IM based on a modification of our differential equations technique, and provide the results of a simulation study that shows that our conditional plausibility intervals outperform the classical $r^\star$-driven asymptotically approximate confidence intervals [@bn1986; @fraser1990] in both small and large samples. A local conditional IM analysis of the variance-components model, another benchmark problem, is also given.

Review of IMs {#S:ims}
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Notation and construction {#SS:notation}
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To fix notation, let $X$ be the observable data, taking values in a space ${\mathbb{X}}$, and let $\theta$ be the parameter of interest, taking values in the parameter space $\Theta$. The starting point of the IM framework is similar to that of fiducial, in the sense that an auxiliary variable, denoted by $U$ and taking values in a space ${\mathbb{U}}$ with probability measure ${\mathsf{P}}_U$, is associated with $X$ and $\theta$. It is this association, together with the distribution $U \sim {\mathsf{P}}_U$, that characterizes the sampling distribution $X \sim {\mathsf{P}}_{X|\theta}$. In 