Yonsei University
Chae-Yeon Xon
Yonsei University, School of Economics
Home/Resources/Identification, Measurement, and Counterfactual Reasoning/I-11
Information Transmission and Learning · Chapter I-11

Administrative Handoffs as Statistical Experiments

Policies pass through channels that transform an upstream experiment and may generate new information downstream. Score projection describes information contraction under a Markov handoff. Delay, arrival, measurement, and causal mediation remain separate objects in the resulting stage ledger.

Conceptual map

  1. I-11.01Models of administrative handoffs
  2. I-11.02Score projection
  3. I-11.03Information loss and acquisition
  4. I-11.04Delay and arrival rates
  5. I-11.05Policy pathways

1. A parameter-free handoff projects the score

An administrative stage is a statistical channel only after its conditional law is specified. A parameter-free channel contracts Fisher information through conditional expectation.

Let \(X\) have dominated density \(p_{\theta}(x)\) and score \(s_{X}(x)=\partial _{\theta}\log p_{\theta}(x)\in \mathbb{R}^{p}\). A handoff draws \(Y\) from a Markov kernel \(Q(dy\mid x)\) that is independent of \(\theta\). Assume differentiation can pass under the integral and the scores are square-integrable.

Proposition 1 · Score projection and information contraction

For almost every downstream outcome,

\[s_{Y}(Y)=E_{\theta }[s_{X}(X)|Y], \quad I_{Y}\preceq I_{X}.\](1)

Proof. Differentiate \(p_{\theta}^{Y}(y)=\int Q(dy\mid x)p_{\theta}(x)\) and divide by the downstream density to obtain the conditional expectation. For every vector \(a\), conditional Jensen gives \(\mathbb{E}[(a^{\top}\mathbb{E}[s_{X}\mid Y])^{2}]\le\mathbb{E}[(a^{\top}s_{X})^{2}]\), which is the Loewner inequality. ∎

2. A binary handoff calculation

Let \(X\sim\operatorname{Bernoulli}(p)\) with \(p\in(0,1)\), and let a parameter-free channel flip its input with probability \(\varepsilon\in(0,1/2)\). Then \(q=P(Y=1)=\varepsilon+(1-2\varepsilon)p\). The upstream and downstream Fisher informations for \(p\) are

\[I_{X}=1/[p(1-p)], \quad I_{Y}=(1-2\varepsilon )^{2}/[q(1-q)].\](2)

At \(p=0.4\) and \(\varepsilon=0.2\), \(q=0.44\), \(I_{X}=4.16667\), and \(I_{Y}=1.46104\). For \(Y=1\), the downstream score is \(0.6(1-0.44)/(0.44\cdot0.56)=1.36364\). Direct conditioning gives the same value from upstream scores 2.5 and \(-1.66667\) with posterior weights 0.32/0.44 and 0.12/0.44.

3. Loss and acquisition occupy separate ledger entries

Let \(\theta=(p,r)\). The upstream observation \(X\) depends only on \(p\), so its information matrix has a zero \(r\) direction. Passing \(X\) through the binary channel contracts the \(p\) entry. If the downstream office additionally collects \(Z\sim\)Bernoulli(\(r\)) independently conditional on \(\theta\), the joint downstream information is

\[I_{(Y,Z)}=\operatorname{diag}((1-2\varepsilon )^{2}/[q(1-q)], 1/[r(1-r)]).\](3)

The first entry records transmitted upstream information; the second comes from a new measurement. Comparing only the total trace would conceal those origins. Arrival, elapsed time, recording error, and causal effects of passage through the office are additional random objects with their own likelihoods or identification assumptions.

4. Parameter dependence changes the score identity

Failure case · A parameter-dependent stage treated as pure post-processing

If the handoff kernel is \(Q_{\theta}\), differentiating the downstream density produces a conditional expectation of the sum of the upstream score and the channel score \(\partial _{\theta}\)log \(q_{\theta}(Y|X)\). Inequality (1) against \(I_{X}\) alone then has no general basis. The stage can encode a new parameter-sensitive measurement or selection mechanism.

Conditioning on post-handoff arrival can also change the experiment through selection. A reported arrival rate identifies a marginal probability under its sampling design; it does not by itself identify the causal effect of an administrative stage.

5. Implementation, exercises, and sources

Enumerate the upstream law and transition matrix, verify that each transition column or row follows the declared convention, compute scores from normalized probabilities, and check their means are numerically zero. Compare information matrices through eigenvalues of \(I_{X}-I_{Y}\). Add acquired measurements as separate likelihood factors.

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Exercises

  1. Reproduce every probability, score, and information value in Section 2.
  2. Take \(\varepsilon=1/2\) and explain the downstream score and information.
  3. Construct two two-parameter channels with the same final arrival probability and different Fisher-information ranks.
Partial solutions

1. Joint masses for (\(X,Y\)) are 0.48, 0.12, 0.08, and 0.32 under a consistent ordering; summing by \(Y\) gives 0.56 and 0.44. 2. At \(\varepsilon=1/2\), \(q=1/2\) for every \(p\), so the downstream score and Fisher information for \(p\) are zero.

  1. A. W. van der Vaart, Asymptotic Statistics, Chapter 7.Scores, differentiability, and Fisher information.
  2. Erik Torgersen, Comparison of Statistical Experiments.Markov transformations of experiments.
  3. David Blackwell (1951), “Comparison of Experiments,” Proceedings of the Second Berkeley Symposium, 93–102.Experiment comparison through randomization.
  4. Chae-Yeon Xon (2026), “Almost a Policy: Transmission through Administrative Handoffs.”Related administrative-handoff application.

6. Audit checkpoint

State the parameter, upstream likelihood, channel orientation, parameter dependence, downstream likelihood, score regularity, information matrices, contraction residual, acquired measurements, arrival and delay variables, and the separate assumptions behind any causal claim.

7. Scope boundary

The chapter treats regular dominated experiments and finite administrative channels. Communication capacity, strategic administration, and causal mediation need separate models.

Prerequisites