= Ancillary conditioning that eliminates a nuisance parameter
For a <sufficient statistic> $(C,S)$ and parameter $(\psi,\lambda)$, a useful strong reduction requires the law of $S$ to be free of $\psi$ and the <conditional distribution> of $C$ given $S$ to be free of the <nuisance parameter> $\lambda$. In the discrete case it gives
$$
f(x;\psi,\lambda)=\phi_0(x)\phi_C(C(x),S(x);\psi)\phi_S(S(x);\lambda),
$$
where $\phi_0$ sums to one on each $(C,S)$ fiber, $\phi_S$ sums to one over $s$, and $\phi_C$ sums to one over $c$ separately for every fixed $s$. Ordinary <ancillarity>, or just the law of $S$ being free of $\psi$, does not by itself guarantee that conditioning removes $\lambda$.
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