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I do not understand the intuition behind why the median is the best estimate if we are going to judge prediction accuracy using the Mean Absolute Error. Let's say you have a random variable X and you want to predict what the next X is. Let's denote your prediction as d.
Under Mean Squared Error, which is:
MSE = (X - d)^2
We know that expected MSE, or sum of MSEs, is minimized when d is equal to the mean or E[X]. This makes sense intuitively. The best predictor of a random variable is its mean.
However, under Mean Absolute Error, which is:
MAE = |X - d|
The expected MAE or sum of MAEs is minimized when d is equal to the median of the random variable. While the book I am reading has a fancy proof to show why this is the case, intuitively I don't understand why the median would be the best predictor. I also don't understand why the mean (or median) wouldn't be the best choice for both.