a most unusual definition of sufficiency

A most unusual definition (?) of sufficiency came up on X validated this morn, as stated in Koller and Friedman’s Probabilistic Graphical Models. But as reported, it is quite restrictive, apparently limited to the natural statistic of an exponential family with conditionally Uniform ancillary (since the likelihood functions are equal rather than proportional). Even more strangely, with this formulation, the Normal sample size n [typo on the last line of the question] appears as a component of the sufficient statistic (Example 17.4). While not being random.

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