Efficient score and efficient information

Let $\theta \in {\mathbb{R}}^p$, $p\ge 1$ finite, be a parameter of interest at a law $P$. Assume its score vector $S_{\theta }=(S_{\theta ,1},\dots ,S_{\theta ,p}{)}^T$ has components in $L_0^2(P)$; under a dominated differentiable parametric submodel these are the coordinate log-density scores of Score and Fisher information. Let ${\mathcal{T}}_{\eta }\subseteq L_0^2(P)$ be the closed nuisance tangent space of Nuisance tangent space. The ambient $L_0^2(P)$ is as in Mean-zero square-integrable functions. The score-density convention uses the Axiom of Choice, while Hilbert projection uses Countable Choice; both dependencies are retained separately.

The Hilbert decomposition Orthogonal decomposition by a closed subspace defines ${\mathrm{\Pi }}_{{\mathcal{T}}_{\eta }}f$ as the unique $g\in {\mathcal{T}}_{\eta }$ with $f-g\in {\mathcal{T}}_{\eta }^{\perp }$, where orthogonality and ${\mathcal{T}}_{\eta }^{\perp }$ have the meanings in Orthogonality and the orthogonal complement. The efficient score vector and efficient information matrix are

$S_{\mathrm{e}\mathrm{f}\mathrm{f}}=S_{\theta }-{\mathrm{\Pi }}_{{\mathcal{T}}_{\eta }}{S}_{\theta },I_{\mathrm{e}\mathrm{f}\mathrm{f}}=E_P[S_{\mathrm{e}\mathrm{f}\mathrm{f}}{S}_{\mathrm{e}\mathrm{f}\mathrm{f}}^T],$

where ${\mathrm{\Pi }}_{{\mathcal{T}}_{\eta }}$ is the Hilbert-space orthogonal projection applied coordinatewise. This projection exists and is unique because ${\mathcal{T}}_{\eta }$ is a closed linear subspace; the projection theorem used by Orthogonal decomposition by a closed subspace assumes Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

The formula does not assert nonsingularity. If $I_{\mathrm{e}\mathrm{f}\mathrm{f}}$ is singular, the inverse-information expression is not defined. The finite-dimensional canonical-gradient formula requires the full score-space decomposition and nonsingularity as additional hypotheses, established in the accompanying theorem on this page.