Bayesian Archives and Accumulated Information

Conceptual map

I-14.01Information in a report
I-14.02Prior and data precision
I-14.03Cumulative excitation
I-14.04Aging records
I-14.05The meaning of fields

1. An archive accumulates directional precision

Observation count measures archive size. The sum of loading outer products determines which parameter directions the archive learns.

Let a static coefficient $\theta \in \mathbb{R}^{p}$ have Gaussian prior $N(m_{0},P_{0}^{-1})$ with $P_{0}\succ0$ . At date $$t$$ , the archive receives $y_{t}=a_{t}^{\top}\theta+\varepsilon _{t}$ , where conditionally independent $\varepsilon _{t}\sim N(0,r_{t})$ and $r_{t}>0$ .

Proposition 1 · Gaussian archive posterior

After $$T$$ reports, posterior precision and mean satisfy

\[\begin{aligned}P_{T}=P_{0}+\sum _{t=1}^{T}a_{t}a_{t}^{\top }/r_{t}, \\ m_{T}=P_{T}^{-1}[P_{0}m_{0}+\sum a_{t}y_{t}/r_{t}].\end{aligned}\](1)

Proof. Expand the Gaussian prior log density and all independent report log likelihoods. Collect their quadratic and linear terms in $\theta$ and complete the square. ∎

2. Equal archive sizes can leave different blind directions

Take $$p=2$$ , $P_{0}=I_{2}$ , unit report noise, and four reports. A concentrated archive uses $a_{t}=e_{1}$ every date, giving data information $\operatorname{diag}(4,0)$ , posterior covariance $\operatorname{diag}(1/5,1)$ , and smallest data-information eigenvalue zero.

A dispersed archive uses two reports in each coordinate, giving data information $\operatorname{diag}(2,2)$ and posterior covariance $\operatorname{diag}(1/3,1/3)$ . For target $c=(1,1)^{\top}$ , posterior variances are 6/5 and 2/3. Both archives contain four records; only the second persistently excites both target directions.

Proposition 2 · Directional learning criterion

For a sequence of archives with fixed prior precision, posterior variance in every unit direction converges to zero exactly when the smallest eigenvalue of cumulative data information diverges.

Proof. The largest eigenvalue of the posterior covariance is the reciprocal of the smallest eigenvalue of prior plus data precision. The prior eigenvalues stay bounded as archive length grows. ∎

3. A changing state requires prediction before updating

Let $\theta _{t}=F\theta_{t-1}+w_{t}$ with process covariance $Q\succeq0$ . If posterior covariance at $$t-1$$ is $\Sigma_{t-1|t-1}$ , the prior covariance for date $$t$$ is $F\Sigma_{t-1|t-1}F^{\top}+Q$ . The new report updates this predicted covariance through the Gaussian conditioning formula.

Failure case · Static precision addition under state drift

Adding all historical outer products as in (1) treats every report as a measurement of one unchanged $\theta$ . With $$Q$$ positive or time-varying coefficients, old records measure earlier states. Their value for the current target passes through $$F$$ and process noise.

The boundary $$Q=0$$ and $$F=I$$ returns the static model. Singular $$Q$$ permits deterministic and stochastic state directions; covariance propagation remains valid, while precision-form shortcuts require care at singular matrices.

4. A field name is part of the measurement contract

A repeated label identifies a common loading only when definitions, units, eligibility rules, and missing-value codes remain stable. If a field changes from $a^{\top}\theta$ to $\widetilde{a}^{\top}\theta$ , equation (1) must use the loading for its actual vintage. Silent field drift can create apparent excitation in a coordinate that the archive never measured consistently.

Initial conditions also matter: at $$T=0$$ the posterior equals the prior, and a diffuse-prior limit exists only on directions with sufficient data information. Zero noise variance describes an exact constraint and should be handled as a limiting or constrained Gaussian problem, outside the $r_{t}>0$ derivation above.

5. Implementation, exercises, and sources

Version each loading vector and field definition, accumulate information with symmetric rank-one updates, and monitor the full eigenvalue path. For changing states, use prediction and update steps with PSD factorizations. Report initial covariance, noise scales, missingness rules, eigenvalue tolerance, and target-weighted posterior loss.

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Exercises

Reproduce the two four-record posterior covariances and target variances.
For alternating loadings $e_{1},e_{2}$ , determine the smallest information eigenvalue after every date.
Compare static accumulation with a random-walk state for the same scalar report sequence.

Partial solutions

1. Add $\operatorname{diag}(4,0)$ or $\operatorname{diag}(2,2)$ to the identity prior and invert each diagonal matrix. 2. After $$T$$ dates, the two eigenvalues are $\lceil T/2\rceil$ and $\lfloor T/2\rfloor$ , so the smaller diverges. 3. In the random walk, add process variance before every measurement update; posterior precision is no longer the unweighted sum of all past report precisions.

Thomas Kailath, Ali Sayed, and Babak Hassibi, Linear Estimation, chapters on recursive least squares and Kalman filtering.Gaussian precision and covariance updates.
Roger A. Horn and Charles R. Johnson, Matrix Analysis, chapters on Hermitian eigenvalue inequalities.Directional excitation and eigenvalue bounds.
Chae-Yeon Xon (2026), “Economica Obscura: Institutions, Archives, and Policy Learning.”Related institutional archive application.

6. Audit checkpoint

State the latent coefficient or state process, initial prior, loading and units by vintage, report noise, conditional independence, missingness, archive horizon, cumulative eigenvalues, target, process transition and covariance, zero-noise handling, diffuse-prior limit, and PSD residuals.

7. Scope boundary

The chapter covers linear Gaussian archives for static coefficients and the covariance recursion for changing states. Adaptive reporting policies and endogenous field definitions are treated in the next chapter.