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

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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.

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  • $\begingroup$ But the wiki page of MSE also gives an example of MSE on predictors,en.wikipedia.org/wiki/Mean_squared_error $\endgroup$ – avocado Dec 26 '13 at 13:09
  • $\begingroup$ 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 $\endgroup$ – Terence Parr Dec 10 '18 at 18:28
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    $\begingroup$ This answer would be better if it addressed the possibility that MSE may be used to mean different things in different contexts. $\endgroup$ – eric_kernfeld Feb 10 at 21:12

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