Prediction intervals and bias-variance tradeoff

I was looking for literature which connects prediction intervals with the bias-variance trade-off. Obviously both concepts deal with describing a mean squared deviation: the bias variance tradeoff deals with $E_{Y,x,D}[(Y-\hat{Y}(x)_D)^2]=E_{x}[E_{Y,D}[(Y-\hat{Y}(x)_D)^2]]$ , which takes the average over (potentially deterministic) x, different datasets D from the same distribution, and resulting variable y. Now $E_{Y,D}[(Y-\hat{Y}(x)_D)^2]$ looks quite related to the following quantity $Var_{Y,D}[Y-\hat{Y}(x)_D]=E_{Y,D}[((Y-\hat{Y}(x)_D)-E[Y-\hat{Y}(x)_D])^2]=E_{Y,D}[(Y-\hat{Y}(x)_D)^2]$ (assuming an unbiased estimator in the last equality) which is repeatedly used in the construction of prediction intervals. Both the prediction interval and the variance-bias tradeoff indicate a minimal mean squared error dictated by the measurement noise $Var(Y)=\sigma^2$ . So I really wonder why I cannot find papers/books discussing this close relationship. Could somebody please tell me a) literature or b) point out the weaknesses in my argument above? PS: I realize that the bias variance tradeoff typically has this additional average over x. Surely, this averaging will have an effect when talking about heteroscedastic noise, i.e. $\sigma(x)$ (this is not the scope of this question) I know that predicition intervals do not have to be symmetric, so higher moments than just the variance might be needed for constructing a prediction interval with correct coverage (but this is not the scope o…