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Bias–variance decomposition

Statement

In a statistical experiment, let $ψ : Θ \to R$ and let $T$ be a real estimator with $E_{θ} [T^{2}] < \infty$ . For every $θ$ , ${MSE}_{θ} (T) = {Var}_{θ} (T) + {Bias}_{θ} (T)^{2} .$

Facts & Assumptions

Given: A statistical experiment, a real estimand $ψ$ , and an estimator $T$ that is square-integrable under each parameter law considered.

[F1]

Bias is $E_{θ} [T] - ψ (θ)$ and MSE is $E_{θ} [(T - ψ (θ))^{2}]$ . (Bias and mean squared error)

[F2]

For square-integrable real $T$ , ${Var}_{θ} (T) = E_{θ} [(T - E_{θ} T)^{2}]$ . (Moments, variance, and covariance on a probability space)

[F3]

The Lebesgue integral is linear on finite linear combinations of integrable functions. (The Lebesgue integral is linear on $L^{1} (μ)$ )

[F4]

Cauchy–Schwarz gives $\int ∣ f g ∣ d μ \leq ∥ f ∥_{2} ∥ g ∥_{2}$ for square-integrable $f, g$ . (Cauchy-Schwarz inequality for $L^{2}$ )

Proof

technique · direct

1.1givenF1F4

Fix $θ$ . Since $P_{θ}$ is a probability measure, the constant $1$ has $L^{2} (P_{θ})$ norm $1$ . Applying [F4] to $T$ and $1$ gives $E_{θ} ∣ T ∣ \leq (E_{θ} [T^{2}])^{1 / 2} < \infty$ . Put $μ = E_{θ} [T]$ and $b = μ - ψ (θ)$ . Then [F1] gives $b = {Bias}_{θ} (T)$ and $T - ψ (θ) = (T - μ) + b$ .

2.1step 1.1F1F2F3algebra∎

Since $T, μ \in L^{1} (P_{θ})$ by step 1.1 and $P_{θ} (X) = 1$ , [F3] gives $E_{θ} [T - μ] = E_{θ} [T] - μ E_{θ} [1] = 0$ . For every finite $c$ , $(T - c)^{2} \leq 2 T^{2} + 2 c^{2}$ , so the centered square and squared error are integrable. Expanding $(T - ψ (θ))^{2} = ((T - μ) + b)^{2}$ now gives ${MSE}_{θ} (T) = E_{θ} [(T - μ)^{2}] + b^{2} = {Var}_{θ} (T) + {Bias}_{θ} (T)^{2}$ by [F1] and [F2].

Depends on

Used by

The binomial sample proportion Example

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

George Deligiannidis, SB2.1 Foundations of Statistical Inference, Chapter 5, PDF pp. 29–31 (standard reference, not scraped)
Karl B. Gregory, STAT 713 Lecture 6: Best Unbiased Estimators, Rao-Blackwell Theorem, PDF pp. 2–8 (standard reference, not scraped)