= Non-uniqueness of maximal ancillary statistics
A maximal <ancillary statistic> has no strictly finer ancillary partition of the sample space, but different maximal ancillary partitions can exist. Let $(U,V)$ take values in $\{0,1\}^2$, with probabilities $\theta/2$ on each equal pair and $(1-\theta)/2$ on each unequal pair, for $0<\theta<1$. Both $U$ and $V$ have the <Bernoulli distribution> with success probability $1/2$ for every $\theta$. Their joint law depends on $\theta$, so combining them is not ancillary. Each partition is maximal: splitting one of its two-point cells singles out a point whose probability depends on $\theta$. There is no common ancillary refinement, and conditioning on one ancillary choice need not specify a unique conditioning principle for all models.
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