Predicting the average versus predicting one draw

Question

A model's estimated response has squared bias 2 and variance 1. The outcome has irreducible noise $\sigma^2 = 6$ around its true mean. Estimating the mean response costs $\text{bias}^2 + \text{variance}$ ; predicting a single new observation costs $\text{bias}^2 + \text{variance} + \sigma^2$ .

How much larger is the error of predicting a single new observation than the error of estimating the mean response?

Accepted Answer

The idea. These are two different jobs. Estimating the mean response asks "what is the average outcome under these conditions?" Predicting a new observation asks "what will the next actual outcome be?" The second job is strictly harder, because on top of nailing the average you must also guess the random deviation of that one draw. Compare. - Mean response error: 	ext{bias}^2 + 	ext{variance} = 2 + 1 = 3. - New observation error: 	ext{bias}^2 + 	ext{variance} + \sigma^2 = 2 + 1 + 6 = 9. The difference is 9 - 3 = 6 = \sigma^2. The gap is exactly the irreducible noise. This is why a confide…

Predicting the average versus predicting one draw

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