Maximum-likelihood estimator in a finite model

Mathematical statement

Every theta has a normalized finite law on the complete observation carrier. For an observed x construct every parameter likelihood and the full argmax set. A fixed deterministic parameter-independent tie order gives a total estimator on all observations. Ties and all-zero likelihoods need explicit witnesses. MLE maximizes likelihood, not posterior probability; uniqueness, unbiasedness, consistency and asymptotic optimality do not follow automatically.

Representative formula (exact TeX)

L(T,x)=h

Assumptions

• Parameter and observation carriers are finite and unique, with exact rational probabilities summing to1 under each theta.
• The tie order is a complete parameter permutation, independent of the unknown theta.

Symbol scope

• L — likelihood operator
• T — finite maximum-likelihood estimator
• x — observed sample
• h — selected parameter

Intuition, definition, and misconceptions

Qualify complete laws and the observation first.

Separate the argmax set from the tie-rule estimator.

Keep likelihood/posterior and asymptotic-claim boundaries explicit.

MLE is not automatically a posterior mode, and an argmax need not be unique.

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 complete parameter-indexed laws.
• Statistical estimator — Require a total estimator.
• Optimization problem — Retain all argmax parameters.

Verification boundary

An exact finite estimator does not establish unbiasedness, consistency or asymptotic efficiency. An all-zero observation column still has all theta as argmax under this convention, without creating positive evidence or a posterior. The English entry enforces the complete tie permutation premise; no original source row is changed.

not_formalized

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

Next obligation

Add finite likelihood ratios and risk comparisons.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.