establishes a desirable calibration property of the conditional IM, and facilitates a common interpretation across users and experiments.

Finding the dimension-reduced representation is sometimes a familiar task. For example, when the minimal sufficient statistic has dimension matching that of the parameter, the conditional IM is exactly that obtained by working directly with said statistic. In other cases, finding the lower-dimensional representation is not so simple. For this, in Section \[S:finding\], we propose a new differential equation-driven technique for identifying observed characteristics of the auxiliary variable. Two classical conditional inference problems are worked out in Section \[S:more-examples\], one showing how the proposed differential equation technique leads to an additional dimension reduction beyond what ordinary sufficiency provides. So, besides the development of conditional IMs, the proposed framework also casts new light on the familiar notion of sufficiency, as well as Fisher’s attractive but elusive ideas on ancillary statistics and conditional inference.

In some cases, however, it may not be possible to produce a valid conditional IM with these somewhat standard techniques. For this, we propose an extension of the conditional IM framework, in Section \[S:gcim\], which allows the lower-dimensional auxiliary variable representation to depend on $\theta$ in a certain sense. We refer 