Sufficient statistic in a finite model

Mathematical statement

A statistic is a total map on the full sample carrier and its fibers form a partition. For every (theta,t) first compute P_theta(T=t). Define and calculate the exact fiber conditional law only on a positive-mass cell, then compare all valid theta laws at each t. Leave zero-mass conditions undefined, encoded as null in these outputs. Separate explicit nonsufficiency witnesses, the identity statistic and the constant-statistic boundary. No factorization theorem or estimator-quality conclusion is introduced.

Representative formula (exact TeX)

S(T,t)=b

Assumptions

• Complete finite parameter-indexed sampling laws are normalized separately under each theta.
• The statistic is total and all fibers are complete.

Symbol scope

• S — sufficiency predicate
• T — finite statistic
• t — statistic value
• b — sufficiency Boolean

Intuition, definition, and misconceptions

Qualify the total statistic and fiber partition first.

Compute every positive-mass conditional table.

Separate theta invariance, nonsufficiency and zero-mass boundaries.

Do not fabricate a law on a zero-mass condition or infer estimator quality merely from data reduction.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Parametric statistical model — Use theta-indexed laws.
• Statistical estimator — Use the statistic mapping.
• Sampling distribution — Track the statistic law and fibers.
• Conditional probability in finite models — Define positive-mass conditional laws.

Verification boundary

This finite-model certificate does not use the factorization theorem. Null denotes an undefined conditional law, not a zero distribution.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

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