Target-Directed Record Selection

Conceptual map

1. I-08.01Fixing source records
2. I-08.02Conditional covariance
3. I-08.03Free coordinates and field dictionaries
4. I-08.04Selection algorithms
5. I-08.05Ex ante collection rules

1. One record has target-specific value

A record is valuable when it reduces uncertainty in the chosen target direction. Its raw variance or rank can be large while its target covariance is zero.

Let the current posterior for a latent state be \(x\sim N(m,P)\), with \(P\) positive semidefinite. The scalar target is \(c^{\top}x\). A candidate record is \(z=a^{\top}x+\varepsilon\), where \(\varepsilon\) is independent with variance \(r>0\).

2. Standalone rankings can select redundant records

Take \(P=I_{2}\) and \(c=(1,1)^{\top}\). Records 1 and 2 both measure \(e_{1}\) with noise variance 0.1. Record 3 measures \(e_{2}\) with noise variance 0.2. Their standalone reductions are 1/\(1.1\approx0.90909\), 1/\(1.1\approx0.90909\), and 1/\(1.2\approx0.83333\).

Selecting the top two standalone scores chooses the duplicate pair. Two independent measurements of the first coordinate yield posterior target variance

The diversified pair {1,3} reduces prior target variance two by about 1.74242, while the duplicate pair reduces it by about 0.95238. Recomputing conditional marginal values after selecting record 1 assigns only about 0.04329 additional reduction to its duplicate and 0.83333 to record 3.

3. Exact and algorithmic selection claims

For a finite dictionary and budget \(k\), define \(V(S)=c^{\top}P_{S}c\), where independent record precisions are accumulated before inversion. Exhaustive enumeration gives the exact optimum for small dictionaries. A greedy algorithm chooses the largest conditional marginal reduction at each step. Its approximation quality requires a property such as submodularity under the stated covariance and noise model.

Formula (1) gives a one-step marginal value conditional on the current covariance. It does not by itself prove that greedy selection is globally optimal.

4. Estimation and evaluation after selection

In applications, \(P\), \(c\), and record noise are estimated. Selecting and evaluating records on the same sample makes the reported improvement optimistic. Sample splitting or cross-fitting can separate dictionary selection from out-of-sample target-risk evaluation. The split rule and all candidate records remain part of the design.

5. Implementation, exercises, and sources

Update precision or covariance with stable rank-one formulas, recompute conditional values after each selection, and verify the final subset against exhaustive search when the dictionary is small. Report PSD residuals and out-of-sample target loss.

Exercises

1. Reproduce every variance and reduction in (2).
2. Change record 3 noise variance to a parameter \(r_{3}\) and find when standalone ranking chooses it ahead of a duplicate.
3. Construct a three-record example in which greedy conditional selection and the global two-record optimum differ.

1. Thomas Kailath, Ali Sayed, and Babak Hassibi, Linear Estimation, chapters on Gaussian conditioning.Rank-one posterior covariance.
2. Andreas Krause and Carlos Guestrin (2007), “Near-optimal Observation Selection Using Submodular Functions,” AAAI.Conditions for greedy information selection.
3. III-17 · Matrix Optimization and Information Allocation.PSD epigraphs and constrained precision design.
4. Chae-Yeon Xon (2026), “It’s a Price! It’s a Value!: Economic Measurement Across Environments.”Related target-directed field-selection application.