Choosing Reporting Systems for Policy Learning

Conceptual map

I-15.01Operating and learning objectives
I-15.02Choosing reporting directions
I-15.03Dynamic adjustment
I-15.04Content-dependent selection
I-15.05Institutional interpretation

1. A reporting direction trades current and future losses

A reporting system determines an observation loading, its noise, and its compatibility with earlier records. Current operating value and later target value enter the same choice only after their weights and horizons are fixed.

Let $\theta \sim N(0,I_{2})$ and let one report be $y=a^{\top}\theta+\varepsilon$ with $\varepsilon \sim N(0,1)$ . Normalize $\lVert a\rVert=1$ and write $a=(\cos\phi,\sin\phi)$ , $\phi\in[0,\pi/2]$ . The sign and scale conventions prevent observationally equivalent loadings from being counted as distinct designs. A report reduces the posterior variance of target $c^{\top}\theta$ by $(c^{\top}a)^{2}/2$ .

Let the current target be $e_{1}$ , the future target be $e_{2}$ , and their weights be $\alpha,\beta \ge0$ . If the installed direction is $a_{0}=e_{1}$ , use adjustment cost $\kappa(1-\cos\phi)$ , $\kappa \ge0$ . The one-period net value is

\[J(\phi )=[\alpha \operatorname{cos}^{2}\phi +\beta \operatorname{sin}^{2}\phi ]/2-\kappa (1-\operatorname{cos} \phi ).\](1)

2. Adjustment cost creates interior and boundary regimes

Proposition 1 · Two-target direction choice

For $\alpha=1$ and $\beta=3$ , let $$t$$ =cos $\phi \in[0,1]$ . The maximizer of (1) is $t^{*}$ =min{ $\kappa/2$ ,1}. Thus $\phi ^{*}=\pi/2$ at $\kappa=0$ , $\phi ^{*}=\pi/3$ at $\kappa=1$ , and $\phi ^{*}=0$ for $\kappa \ge2$ .

Proof. Up to constants, (1) equals $-t^{2}+\kappa t$ on [0,1]. Its unconstrained maximizer is $\kappa/2$ . Projection onto the feasible interval gives the stated regimes. ∎

The endpoint $\kappa=0$ specializes entirely in the future target because $\beta$ exceeds $\alpha$ . At $\kappa=2$ , the interior optimum reaches the installed direction; larger adjustment costs leave the system there. An upper bound on reorientation speed would add another active boundary to the feasible interval.

3. A dynamic reporting problem needs a terminal contract

At date $$t$$ , a reporting rule can depend on inherited posterior covariance, the installed loading, current target weights, and a transition law for future targets. A finite-horizon Bellman recursion has state ( $\Sigma_{t},a_{t-1},z_{t}$ ), where $z_{t}$ indexes target demand. The action updates covariance by Gaussian conditioning and changes the installed loading.

The horizon, discount factor, terminal archive value, and treatment of reports after the terminal date determine the solution. With zero terminal archive value, the last-period report serves only the last-period objective. A continuation value can preserve diversified directions that current use alone would discard.

4. Content-dependent reporting changes the observation law

If $a_{t}$ is chosen from past observed content, likelihood evaluation conditions on the recorded selection history. If selection uses contemporaneous latent content or unrecorded discretion, the missingness and selection mechanism must be modeled or bounded.

Failure case · Adaptive fields treated as a fixed design

Suppose a system reports coordinate one when a preliminary signal is positive and coordinate two otherwise. The observed field label carries information about that signal. An analysis that conditions only on the reported value and discards the selection rule omits part of the likelihood and can distort both posterior learning and archive value.

Adjustment cost boundaries also require interpretation. $\kappa=0$ permits costless reorientation; $\kappa \to \infty$ freezes the installed system. A discrete field menu replaces the smooth first-order condition with finite comparisons and may generate ties. A tie-breaking rule is part of a deterministic policy.

5. Implementation, exercises, and sources

Normalize directions, record the installed system and selection history, solve the finite-horizon recursion backward from its declared terminal value, and simulate forward under the same target transition. Check all endpoint actions, discrete ties, covariance PSD residuals, and the sensitivity of policies to horizon and adjustment cost.

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Exercises

Verify every optimizer in Section 2 and compute $J(\phi ^{*})$ at $\kappa=0$ ,1,2.
Add a speed constraint $\lvert \phi -\phi _{0}\rvert \le \pi/6$ and solve the one-period problem.
Write a two-period recursion in which the future target switches between the coordinate axes.

Partial solutions

1. At $\kappa=0$ ,1,2 the maximizing cosines are 0,1/2,1, and direct substitution gives values 3/2,3/4,1/2. 2. Project the unconstrained angle onto [0, $\pi/6$ ] because $\phi _{0}=0$ . 3. The second-period value supplies the terminal covariance value in the first-period choice.

Thomas Kailath, Ali Sayed, and Babak Hassibi, Linear Estimation, chapters on Gaussian conditioning.Posterior value of a linear report.
Dimitri P. Bertsekas, Dynamic Programming and Optimal Control, Volume I.Finite-horizon Bellman recursion and terminal conditions.
Chae-Yeon Xon (2026), “Economica Obscura: Institutions, Archives, and Policy Learning.”Related dynamic reporting-system application.

6. Audit checkpoint

State the prior or inherited covariance, target process, loading normalization, noise, installed system, adjustment metric, feasible directions or field menu, horizon, discounting, terminal value, selection observability, tie-breaking rule, and endpoint policy checks.

7. Scope boundary

The chapter studies linear Gaussian reporting directions with adjustment and continuation value. Strategic disclosure, privacy, and political choice require additional models.

Prerequisites

I-14 · Bayesian Archives and Accumulated Information