## What is Mean Square Error?

Aliases: *MSE*

The mean square error (MSE) is a **measure of optimality** of a statistical estimator. There are two conflicting definitions for optimality in the two major inferential schools: frequentist and Bayesian ^{[1]}.

In frequentist inference an efficient estimator (optimal estimator) deviates as little as possible from the true value (θ*) one is trying to estimate: MSE(θ^_{n}(X);θ*) = E(θ^_{n}(X) - θ*)^{2}. Thus, the mean square error is defined with respect to the true state of nature, the true underlying data-generating mechanism (DGM) which in hypothesis testing is specified by defining a statistical model.

The importance of having frequentist statistics with low mean square error is then a natural extension of our desire to have as accurate estimation of the true data-generating mechanism so that we can make as informed decisions following an A/B test as possible.

In Bayesian inference the decision-theoretic definition of optimiality is used in which an optimal estimator is one that minimizes the loss function over every possible value of the parameter: MSE_{1}(θ^(X);θ)= E(θ^(X) - θ)^{2} = R(θ,θ^),∀θ∈Θ where R is a risk function. Unlike the frequentist notion, the Bayesian one has no reference to the true value of θ (θ*) and is in fact in direct contrast to the goal of minimizing the error with respect to it and of learning about the true data-generating mechanism. Instead, the objective is to minimize losses weighted by π(θ|x_{0}) the posterior distribution of θ given x_{0} for all values of θ in Θ.

References:

[1] Spanos A. (2017) "Why the Decision-Theoretic Perspective Misrepresents Frequentist Inference - Revisiting Steins Paradox and Admissibility"