# 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 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 of this question)
• Typically, we estimate prediction intervals, but let us assume correctly calibrated prediction intervals, having close to correct coverage.

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.

The reason might be that the relationship is not that special. If both concepts use 'variance' in some way, then there's not yet a meaningful relationship. To take an extreme example, are two concepts related when they both use 'addition'?

• The bias-variance trade-off deals with the range of the prediction error, and how bias and variance of the estimate have competing roles.

• The prediction deals with the range of the prediction error, and how we can describe an interval with a certain percentage of coverage.

Each can be discussed independently from the other.

There's not much reason to explicitly combine them, e.g. discussing the bias-variance tradeoff in terms of the size of prediction intervals.

Although, in a way there are indirectly discussions about this when different types of (prediction) intervals are compared. An example is the following discussion about the use of unbiased (no prior) versus biased (including a Bayesian prior) intervals: Are there any examples where Bayesian credible intervals are obviously inferior to frequentist confidence intervals .