Mean squared error vs. mean squared prediction error

What is the semantic difference between Mean Squared Error (MSE) and Mean Squared Prediction Error (MSPE)?

The difference is not the mathematical expression, but rather what you are measuring.

Mean squared error measures the expected squared distance between an estimator and the true underlying parameter:

$$\text{MSE}(\hat{\theta}) = E\left[(\hat{\theta} - \theta)^2\right].$$

It is thus a measurement of the quality of an estimator.

The mean squared prediction error measures the expected squared distance between what your predictor predicts for a specific value and what the true value is:

$$\text{MSPE}(L) = E\left[\sum_{i=1}^n\left(g(x_i) - \widehat{g}(x_i)\right)^2\right].$$

It is thus a measurement of the quality of a predictor.

The most important thing to understand is the difference between a predictor and an estimator. An example of an estimator would be taking the average height a sample of people to estimate the average height of a population. An example of a predictor is to average the height of an individual's two parents to guess his specific height. They are thus solving two very different problems.

• But the wiki page of MSE also gives an example of MSE on predictors,en.wikipedia.org/wiki/Mean_squared_error – avocado Dec 26 '13 at 13:09
• Not sure estimator vs predictor is meaningful here. Both are metrics that measure actual y vs f(x) where f(x) is meant to approximate y from feature vector x – Terence Parr Dec 10 '18 at 18:28
• This answer would be better if it addressed the possibility that MSE may be used to mean different things in different contexts. – eric_kernfeld Feb 10 at 21:12