# Bias-variance decomposition

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

For each parameter value and its sampling law, scalar-estimator MSE equals variance plus squared bias.

mko-bias-variance-decomposition · theorem · v0.10.0



## Mathematical statement

For every theta supply a complete normalized finite sampling law. Under that same law separately compute the mean, bias=E_theta[T]−theta, variance=E_theta[(T−E_theta[T])^2] and MSE=E_theta[(T−theta)^2]. Expanding T−theta=(T−E_theta[T])+bias gives a cross term that vanishes because E_theta[T−E_theta[T]]=0. Thus MSE=variance+bias^2 pointwise in theta. Do not pool different theta laws. Unbiasedness gives MSE=variance, not zero MSE or consistency. One observed squared error is not the definition of expected loss, even if its numerical value coincides with MSE.





## Representative formula (exact TeX)

M(T,t)=V(T,t)+B(T,t)^2





## Assumptions


- Theta, listed estimator values and probabilities are exact rationals in a complete normalized law.
- This slice uses scalar estimators with finite sampling laws.





## Symbol scope


- M — mean squared error
- T — scalar estimator
- t — target parameter
- V — estimator variance
- B — estimator bias





## Intuition, definition, and misconceptions

Compute all four quantities under the same complete law.

Cross-term cancellation requires the explicit centering identity.

The biased/unbiased boundary is not a zero-error claim.

A single realized squared error is not MSE, and an unbiased estimator can still have positive variance.





## Adopted derivation guide

[Bias-variance derivation with the cross term visible](https://ocme.evemisslab.com/en/derivations/derivation-mko-bias-variance-decomposition-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Mean squared error of an estimator](https://ocme.evemisslab.com/en/mko/mko-estimator-mean-squared-error/) — Use its theta-specific definition.
- [Estimator bias](https://ocme.evemisslab.com/en/mko/mko-estimator-bias/) — Use the bias definition.
- [Estimator variance](https://ocme.evemisslab.com/en/mko/mko-estimator-variance/) — Use variance under the same law.





## Verification boundary

Check each theta separately. Finite tables do not establish consistency or asymptotic optimality.

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-model risks and estimator 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/bias-variance-decomposition.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-bias-variance-decomposition/index.md)