The prediction-error floor of a perfect model

Question

Imagine a flawless model with zero bias and zero variance in its estimated response. The outcome it predicts has irreducible noise $\sigma^2 = 2.5$ around its true mean.

Prediction error for a new observation is $\text{bias}^2 + \text{variance} + \sigma^2$ .

What is the expected squared error when this perfect model predicts a single new observation?

Accepted Answer

The idea. A perfect model knows the true average outcome exactly, so its estimation error is zero. But knowing the average is not the same as knowing the next specific value. The outcome still scatters randomly around that average, and no model can predict that scatter, because it does not depend on the inputs. Compute. 	ext{Prediction MSE} = 0 + 0 + \sigma^2 = 2.5. So even a flawless model faces expected squared error 2.5 when predicting one new observation. That number is the irreducible floor: it is a property of the world, not of the model. This is the crucial difference between explaini…

The prediction-error floor of a perfect model

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