= Pathwise differentiability of a statistical functional
{title2=$D\psi_P(g)=P(\varphi g)$}
= Pathwise differentiable statistical functional
{synonym}
A statistical functional is pathwise differentiable relative to chosen <statistical paths> if its <derivative> along every path depends only on that path's <score function> and defines a <bounded linear functional> on their <statistical tangent space>. The <Riesz representation theorem> expresses this <derivative> as an <L2 inner product> with a unique element of the <statistical tangent space>, the <canonical gradient>. A path family and the associated <derivative> remainder conditions must both be specified; a formal <derivative> along one convenient family does not establish differentiability along all paths.
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