Does anyone know of a metric that quantifies the bias-variance tradeoff of a given fitted model?

I'm not talking about measuring the MSE in cross validation, I'm interested in a single generic or model-specific metric (or statistical test) that measures the degree of bias and/or variance of a fitted model.

• Knowing the bias would mean knowing the truth, in which case you would not need a model. You can never measure the bias without knowing the truth. – Matthew Drury Apr 9 '17 at 5:36
• This seems like a perfectly good, & clear, question to me. I'd be interested in reading some good answers, too. I'm voting to leave open. – gung Apr 9 '17 at 11:58
• @MatthewDrury considering that the answer talks about the MSE as well, I think the OP may have data on the truth, like in regression analysis. In many cases you may still want to model it for forecasting purposes – KenHBS Jul 25 '17 at 11:40

$E(y_0-\hat{f}(x_0))^2 = Var(\hat{f}(x_0)) + Bias(\hat{f}(x_0))^2 + Var(\epsilon)$
- $E(y_0-\hat{f}(x_0))^2$ is the expected test MSE.
- $Var(\hat{f}(x_0))$ is variance of fitted model.
- $Bias(\hat{f}(x_0))^2$ is squared biased of model.
- $Var(\epsilon)$ is variance of error terms