Hilbert Target Operators and Distribution States

Conceptual map

1. III-13.01Finite-to-integrated targets
2. III-13.02Integrability
3. III-13.03Spectra of target operators
4. III-13.04Distribution tangents
5. III-13.05Sieve approximation

1. A continuum of targets defines one operator

When a finite state perturbation changes an entire policy-response curve, the relevant object maps a direction into a function. Its Gram operator summarizes integrated sensitivity across the declared target measure.

Let (\(T,\mu\)) be a finite measure space and \(g:T\to \mathbb{R}^{d}\) a measurable response gradient with \(\int\lVert g(t)\rVert^{2}\,d\mu(t)<\infty\). Define \(B:\mathbb{R}^{d}\to L^{2}(\mu)\) by \((Bv)(t)=g(t)^{\top}v\). Its target Gram matrix is

2. A response curve on the unit interval

Take \(T\)=[0,1] with uniform measure and \(g(t)=(1,t)^{\top}\). Then

The eigenvalues are \((4\pm \sqrt{13})/6\), approximately 1.26759 and 0.06574. Their sum 4/3 equals \(\int _{0}^{1}(1+t^{2})\,dt\). The leading eigenvector identifies the state direction with the largest integrated squared response under the chosen uniform weighting. A different target measure changes that economic ranking.

Approximating (3) on a grid with positive quadrature weights gives \(G_{K}=\sum _{j}w_{j}g(t_{j})g(t_{j})^{\top}\). Positivity survives at every grid size because each summand is positive semidefinite.

3. Distribution states and score directions

Let \(P\) have density \(p\) and let {\(P_{\varepsilon}\)} be a regular dominated submodel whose density is differentiable in \(L^{1}\) at zero with derivative \(p s\). Then \(E_{P}[s]=0\); assume also \(s\in L^{2}(P)\). The score \(s\) is a tangent direction, and the Fisher metric is \(\langle s_{1},s_{2}\rangle=E_{P}[s_{1}s_{2}]\). For a square-integrable query \(q\),

The centered query is the Riesz representer of this derivative on the mean-zero tangent space. Restrictions on admissible scores replace that space by a subspace and project the representer accordingly.

4. Finite-rank approximation and spectral gaps

In a general Hilbert domain, a Hilbert–Schmidt response map \(B\) has a positive trace-class Gram operator \(B^{*}B\). If finite-rank maps \(B_{K}\) converge in operator norm, then

Eigenvalues then converge. Eigenspace stability additionally needs separation between the target eigenvalue cluster and the rest of the spectrum. A small gap makes the reported direction sensitive even when the Gram operator itself is estimated accurately.

5. Implementation, exercises, and sources

Normalize the target measure, use nonnegative quadrature weights, and check the trace identity against the summed gradient norm. Report spectral tails and eigenvalue gaps as the policy grid and state sieve expand.

Exercises

1. Derive the exact eigenvalues in (3) and compute normalized eigenvectors.
2. Approximate (3) with midpoint grids of sizes 5, 10, and 100; report operator-norm error and leading-direction error.
3. For a three-point distribution, write the mean-zero score space and calculate the Riesz representer of one event probability.

1. John B. Conway, A Course in Functional Analysis, Chapters II, IX.Compact positive operators and spectral representation.
2. Aad van der Vaart, Asymptotic Statistics, Chapter 25.Tangent spaces and pathwise derivatives.
3. Tosio Kato, Perturbation Theory for Linear Operators.Spectral perturbation and separated eigenspaces.