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  "body": "\n## Definition\n\nLet $\\hat{\\theta}$ be an [[estimator|0TTNRP5PFQ]] of a real parameter $\\theta$, with $E_\\theta[\\hat{\\theta}^2] \u003c \\infty$, where $E_\\theta$ is the expectation when $\\theta$ is the true parameter.\nThe mean squared error of $\\hat{\\theta}$ at $\\theta$ is\n$$\\operatorname{MSE}_\\theta(\\hat{\\theta}) = E_\\theta\\left[(\\hat{\\theta} - \\theta)^2\\right].$$\n\n## Remarks\n\nIt decomposes as $\\operatorname{MSE}_\\theta(\\hat{\\theta}) = \\operatorname{Var}_\\theta(\\hat{\\theta}) + \\operatorname{Bias}_\\theta(\\hat{\\theta})^2$, with the [[bias|01BD2TFFP4]] $\\operatorname{Bias}_\\theta(\\hat{\\theta}) = E_\\theta[\\hat{\\theta}] - \\theta$.\nFor an unbiased estimator, the mean squared error is the [[variance|007QED3E0Y]].\nFor the sample mean of $n$ i.i.d. variables with variance $\\sigma^2$, the mean squared error is $\\sigma^2/n$.\nFor $n$ predictions $\\hat{y}_i$ of observed values $y_i$, one also calls $\\frac{1}{n}\\sum_i (y_i - \\hat{y}_i)^2$ the mean squared error.\n\n## Explanations\n\n### L1 human\n\nThe mean squared error measures how far, on average, an estimator is from the truth.\nIt is large if the estimator spreads widely, or if it is systematically off, or both.\n",
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