# Maximum-likelihood estimator in a finite model

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A finite-model MLE chooses a parameter maximizing the observed likelihood; ties use one fixed rule independent of the unknown parameter.

mko-finite-maximum-likelihood-estimator · definition · v0.10.0



## 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

[MLE with a positive-mass tie and an all-zero likelihood column](https://ocme.evemisslab.com/en/derivations/derivation-mko-finite-maximum-likelihood-estimator-01/)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](https://ocme.evemisslab.com/en/mko/mko-parametric-statistical-model/) — Use complete parameter-indexed laws.
- [Statistical estimator](https://ocme.evemisslab.com/en/mko/mko-statistical-estimator/) — Require a total estimator.
- [Optimization problem](https://ocme.evemisslab.com/en/mko/mko-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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/finite-maximum-likelihood-estimator.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-finite-maximum-likelihood-estimator/index.md)