Parametric statistical model

Mathematical statement

Validate the finite parameter set and every nonnegative normalized row of the complete law table. In L(theta;x)=P_theta(x), likelihood varies theta while the observation x is fixed. It is not a posterior; without a prior no Bayes update is performed. One observation does not establish identifiability.

Representative formula (exact TeX)

M(T)

Assumptions

• The finite parameter set and finite sample space are explicit and shared across all rows.
• Each theta has a complete exact rational law normalized to total mass 1.

Symbol scope

• M — parametric-model record
• T — parameter set

Intuition, definition, and misconceptions

All parameters share the same sample space.

Validate the complete law for each parameter.

Likelihood is not posterior, and parameter is not observation.

Do not treat likelihood as posterior or silently change the sample space with the parameter.

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

• Finite joint distribution — The finite law is complete.
• Probability on finite sample spaces — Every law is nonnegative and normalized.
• Function mapping — A parameter maps to its probability law.
• Set — Parameter and sample spaces are sets.

Verification boundary

Only complete finite tables are handled; identifiability and inferential optimality are not asserted.

not_formalized

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

Next obligation

Add estimators and sampling distributions.

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.