Define the Research Object as a Contract

Conceptual map

I-01.01Structure, observations, and targets
I-01.02The state-design contract
I-01.03Units of counterfactual comparison
I-01.04Quantifiers and domains
I-01.05Types of evidence

1. The research object is a typed contract

A sufficiency claim has meaning only relative to a state model, an observation, and a target. Changing the target can change the answer while the observed data remain fixed.

Let $$X$$ be a nonempty state set, $$Y$$ an observation set, and $$T$$ a target set. A research contract specifies an admissible state class $\mathcal{X}\subseteq X$ , an observation map $h:\mathcal{X}\to Y$ , a target $\tau:\mathcal{X}\to T$ , and the interventions or counterfactual environments under which these maps are evaluated.

Definition 1 · Observational equivalence

States $$x$$ and $x^{\prime}$ are observationally equivalent, written $x\sim _{h}x^{\prime}$ , when $h(x)=h(x^{\prime})$ . The fiber at report $$y$$ is $h^{-1}(\{y\})$ .

Definition 2 · Exact target sufficiency

The report $$h$$ is exactly sufficient for $\tau$ on $\mathcal{X}$ when a decoder $\delta:h(\mathcal{X})\to T$ exists such that $\tau=\delta\circ h$ on $\mathcal{X}$ .

2. Fiber constancy is the exact criterion

Proposition 1 · Factorization through an observation

The report $$h$$ is exactly sufficient for $\tau$ if and only if $\tau$ is constant on every observational fiber:

\[h(x)=h(x^{\prime}) \Rightarrow \tau (x)=\tau (x^{\prime}).\](1)

Proof. If $\tau=\delta\circ h$ , equal reports give equal decoded targets. Conversely, assume (1). For each attained report $$y$$ , select any $$x$$ with $$h(x)=y$$ and define $\delta(y)=\tau(x)$ . Fiber constancy makes this definition independent of the selected state and gives the factorization. ∎

The decoder is defined on the attained report set $h(\mathcal{X})$ . Extension to reports outside that set is a separate extrapolation choice.

3. One report, two target conclusions

Take four states $\mathcal{X}=\{a,b,c,d\}$ and define $$h(a)=h(b)=0$$ and $$h(c)=h(d)=1$$ . The fibers are $\{a,b\}$ and $\{c,d\}$ .

For target $\tau_{1}=h$ , the within-fiber target diameter is zero and decoder $\delta_{1}(y)=y$ works. Define a second scalar target by

\[\tau _{2}(a)=0, \tau _{2}(b)=2, \tau _{2}(c)=1, \tau _{2}(d)=4.\](2)

The two fiber diameters are 2 and 3. Equation (1) fails in each fiber, so the same report loses the second target. Under squared loss and a uniform distribution within each fiber, the conditional-mean decoder gives 1 at report 0 and 5/2 at report 1. Its conditional mean-squared errors are 1 and 9/4.

4. What must remain fixed

A complete contract records the state class, observation, target, environment, admissible intervention, loss, and claim level. For a probabilistic statement it also specifies a law or a class of laws. For an approximate decoder it states the norm or risk under which approximation is measured.

Failure case · Target-free sufficiency language

Calling $$h$$ “sufficient” without naming $\tau$ suppresses the quantifier that determines the claim. In the finite example, the statement is true for $\tau_{1}$ and false for $\tau_{2}$ .

Changing the admissible state set can also change the fibers. If state $$b$$ is excluded, report 0 becomes sufficient for $\tau_{2}$ on that restricted fiber. The restriction must be justified by the model or design.

5. Implementation, exercises, and sources

For a finite model, group states by the exact report key and compute each target range. With floating reports, define a tolerance only after the measurement scale is fixed; tolerance-based grouping changes the equivalence relation and should be reported as an approximation.

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Exercises

Construct both decoders for the finite example wherever they exist and verify Proposition 1.
Under a nonuniform law with probabilities (0.1,0.4,0.2,0.3), compute the conditional-mean decoder and total squared-error risk for $\tau_{2}$ .
Give a target that is sufficient on { $$a,b,c$$ } and loses sufficiency when $$d$$ is added.

Partial solutions

1. For $\tau_{1}$ , set $\delta(0)=0$ and $\delta(1)=1$ . For $\tau_{2}$ , report 0 would require one decoder value to equal both 0 and 2, so no decoder exists. 2. The conditional means are 8/5 and 14/5. The within-fiber variances are 16/25 and 54/25; weighting each by its fiber probability 1/2 gives total risk 7/5.

Paul R. Halmos, Naive Set Theory, sections on equivalence relations and quotient sets.Fibers and induced quotient maps.
David Blackwell (1951), “Comparison of Experiments,” Proceedings of the Second Berkeley Symposium, 93–102.Decision-relative comparison of information.
Thomas M. Cover and Joy A. Thomas, Elements of Information Theory, §2.8.Data processing and information loss under mappings.

6. Audit checkpoint

State the admissible states, observation, target, attained report set, fibers, decoder domain, intervention environment, probability law when used, loss, and exact or approximate claim.

7. Scope boundary

The chapter treats deterministic factorization through an observation and a finite risk illustration. Statistical sufficiency for a parameterized family and experiment comparison receive separate treatment.