The framework of Inferential Models (IMs) has recently been developed in search of what is referred to as the holy grail of statistical theory, that is, prior-free probabilistic inference. Its method of Conditional IMs (CIMs) is a critical component in that it serves as a desirable extension of the Bayes theorem for combining information when no prior distribution is available. The general form of CIMs is defined by a system of first-order homogeneous linear partial differential equations (PDEs). When admitting simple solutions, they are referred to as regular, whereas when no regular CIMs exist, they are used as the so-called local CIMs. This paper provides conditions for regular CIMs, which are shown to be equivalent to the existence of a group-theoretical representation of the underlying statistical model. It also establishes existence theorems for CIMs, which state that under mild conditions, local CIMs always exist. Finally, the paper concludes with a simple example and a few remarks on future developments of CIMs for applications to popular but inferentially nontrivial statistical models.

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