Revision as of 03:24, 9 November 2022 edit Logicdavid (talk \| contribs) 89 edits →Proof of variance and bias relationship ← Previous edit		Revision as of 05:20, 15 December 2022 edit undo 2800:200:e840:25fc:e19f:4087:3109:b8e6 (talk) →Proof of variance and bias relationship Next edit →
Line 63: &= \operatorname{E}_{\theta}\left[\left(\hat{\theta}-\operatorname{E}_{\theta}[\hat\theta]\right)^2\right]+ 2 \left(\operatorname{E}_{\theta}[\hat\theta]-\theta\right) \operatorname{E}_{\theta}\left[\hat{\theta}-\operatorname{E}_{\theta}[\hat\theta] \right] + \left(\operatorname{E}_{\theta}[\hat\theta]-\theta\right)^2 && \operatorname{E}_{\theta}[\hat\theta]-\theta = \text{const.} \\ &= \operatorname{E}_{\theta}\left[\left(\hat{\theta}-\operatorname{E}_{\theta}[\hat\theta]\right)^2\right]+ 2 \left(\operatorname{E}_{\theta}[\hat\theta]-\theta\right) \left ( \operatorname{E}_{\theta}[\hat{\theta}]-\operatorname{E}_{\theta}[\hat\theta] \right )+ \left(\operatorname{E}_{\theta}[\hat\theta]-\theta\right)^2 && \operatorname{E}_{\theta}[\hat\theta] = \text{const.} \\ &= \operatorname{E}_{\theta}\left[\left(~~\hat{~~\theta}-\operatorname{E}_{\theta}[\hat\theta]\right)^2\right]+\left(\operatorname{E}_{\theta}[\hat\theta]-\theta\right)^2\\ &= \operatorname{Var}_{\theta}(\hat\theta)+ \operatorname{Bias}_{\theta}(\hat\theta,\theta)^2 \end{align}</math>

Mean squared error: Difference between revisions