Information Budgets, Decision Loss, and Record Allocation

Conceptual map

I-13.01Information and decision loss
I-13.02Retention and acquisition
I-13.03Retention semidefinite programs
I-13.04Rounded rate releases
I-13.05Exact decision samples

1. Information is valued through a declared decision loss

A retention budget has no target-free optimum. The target derivative, loss curvature, baseline information, release likelihood, and acquisition cost jointly define the allocation problem.

Let $\theta \in \mathbb{R}^{p}$ , let the local target derivative be $C\in \mathbb{R}^{q\times p}$ , and let $W\succeq0$ weight target errors. Baseline Fisher information is $I_{0}\succeq0$ . Candidate source $$j$$ contributes information $J_{j}\succeq0$ at retention intensity $x_{j}\in[0,1]$ , with cost $c_{j}x_{j}$ . For a regular efficient estimator, local quadratic risk is summarized by $\operatorname{tr}[WCI(x)^{+}C^{\top}]$ , where $I(x)=I_{0}+\sum x_{j}J_{j}$ and $\operatorname{range}(C^{\top})\subseteq\operatorname{range}(I(x))$ .

Proposition 1 · Semidefinite epigraph

For fixed $$x$$ , the generalized Schur complement gives

\[\begin{bmatrix}I(x) & C^{\top} \\ C & V\end{bmatrix}\succeq0\quad\Longleftrightarrow\quad \operatorname{range}(C^{\top})\subseteq\operatorname{range}(I(x))\ \text{and}\ V\succeq CI(x)^{+}C^{\top}.\](1)

Minimizing $\operatorname{tr}(WV)$ subject to (1), $0\le x_{j}\le1$ , and $\sum c_{j}x_{j}\le B$ therefore represents the target-risk allocation problem.

Proof. Apply the generalized Schur-complement theorem to the positive-semidefinite upper-left block. The range condition is part of that theorem and also states target estimability. Since $$W$$ is positive semidefinite, the minimum over $$V$$ occurs at the covariance lower boundary on every weighted target direction. ∎

2. An allocation changes regime at the budget boundary

Take $I_{0}=I_{2}$ , two continuously scalable coordinate sources, unit costs, and target weights one and four. With $x_{1},x_{2}\ge0$ and $x_{1}+x_{2}\le B$ , the risk is

\[R(x)=1/(1+x_{1})+4/(1+x_{2}).\](2)

Ignoring upper intensity limits, the KKT solution is $x_{1}=0$ , $x_{2}=B$ for $0\le B\le1$ . For $B\ge1$ , the interior formulas are $x_{1}=(B-1)/3$ and $x_{2}=(2B+1)/3$ . At $$B=2$$ they give (1/3,5/3) and risk 2.25. If physical retention imposes $x_{j}\le1$ , source two binds at one for $1\le B\le2$ and source one receives $$B-1$$ ; both sources bind for $B\ge2$ , leaving any additional budget unused. The feasible-set boundary must therefore match the operational meaning of intensity.

3. Two release mechanisms have different likelihoods and costs

Proposition 2 · Expected reads for a fixed-decision release

Suppose independent records produce a decision indicator with probability $\pi\in(0,1]$ . If records are read until $$r$$ decisions occur, the number $$N$$ read has a negative-binomial law and $\mathbb{E}[N]=r/\pi$ .

Proof. Write $$N$$ as the sum of $$r$$ independent geometric waiting times with success probability $\pi$ . Each has mean 1/ $\pi$ . ∎

At $$r=40$$ and $\pi=0.5$ , expected records read equal 80. At $\pi=0$ , the stopping time is infinite almost surely. A finite archive cap changes the experiment by introducing a probability of failing to reach $$r$$ ; both that mass and the resulting release rule belong in the likelihood.

For a different mechanism, let $\bar{y}_{n}$ be a sample proportion released after rounding to grid width $\Delta _{n}$ . The rounding error is at most $\Delta _{n}/2$ . It is negligible on the root- $$n$$ scale when $\sqrt{n}\Delta _{n}\to0$ . A fixed grid or a grid with a positive root- $$n$$ limit changes the limiting experiment. At probabilities zero or one, the usual interior binomial normal approximation degenerates.

4. Release labels cannot substitute for experiment definitions

Failure case · Rounded rates analyzed as fixed-decision samples

A fixed-size binomial sample, a stopping-time sample with a fixed number of decisions, and a rounded conditional rate assign different probabilities to observed releases. Reusing one likelihood for another mechanism changes information, expected cost, and boundary behavior.

Sampling without replacement adds a finite-population correction. Dependence across records invalidates the independent geometric decomposition. Privacy noise adds a further convolution. These features enter before comparing designs through a common target loss.

5. Implementation, exercises, and sources

Store one contract per release: sampling frame, stopping rule, cap, rounding grid, added noise, record cost, and likelihood. In the allocation solver, verify PSD symmetry, block-matrix feasibility, target range inclusion, budget slackness, bound multipliers, and primal-dual residuals. Recompute risk directly from a factorization at the returned solution.

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Exercises

Derive both regimes of (2) and add the constraints $x_{j}\le1$ .
Compute the variance and cap-failure probability of a fixed-decision release for specified ( $r,\pi,N_{\operatorname{max}}$ ).
Compare root- $$n$$ rounding error under $\Delta _{n}=n^{-1}$ and $\Delta _{n}=n^{-1/2}$ .

Partial solutions

1. Equality of interior marginal gains gives $1/(1+x_{1})=2/(1+x_{2})$ . Nonnegativity binds below $$B=1$$ ; source-two capacity binds when its unconstrained allocation exceeds one. 2. Without a cap, $\operatorname{Var}(N)=r(1-\pi)/\pi ^{2}$ . The cap-failure probability is $P[\operatorname{Binomial}(N_{\max},\pi)<r]$ . 3. The scaled grid widths converge to zero and one.

Stephen Boyd and Lieven Vandenberghe, Convex Optimization, Appendix A.5.5 and Chapter 5.Generalized Schur complements and KKT conditions.
George Casella and Roger L. Berger, Statistical Inference, sections on negative-binomial sampling.Stopping-time likelihood and moments.
Chae-Yeon Xon (2026), “Almost a Policy: Transmission through Administrative Handoffs.”Related comparison of conditional release mechanisms.

6. Audit checkpoint

State the target loss, baseline and candidate information, range condition, budget and intensity bounds, release likelihood, stopping and cap rules, zero-probability behavior, finite-population design, rounding rate, privacy noise, KKT regime, and numerical feasibility residuals.

7. Scope boundary

The chapter covers local quadratic information allocation and two elementary release mechanisms. Strategic reporting, general optimal design, and privacy accounting require additional theory.