Nuisance parameter

A nuisance parameter is a statistical parameter required to specify the distribution of observations but not itself the target of inference. A profile likelihood maximizes over it; Bayesian model evidence integrates over it using a proper prior distribution. These operations differ and need not give the same inference. A baseline hazard in a Cox proportional-hazards model is an infinite-dimensional example.

On a common positive support, a factorization $f={\varphi }_0(x){\varphi }_C(x;\psi ){\varphi }_S(x;\lambda )$ implies ${\partial }_{\psi }{\partial }_{\lambda }\log f=0$. Thus any nonzero mixed derivative prevents this exact separation. For $n$ independent gamma distributed observations with shape $a$ and rate $b$, this mixed derivative is $n/b$, so exact separation of shape and rate into such factors is impossible.