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  "body": "\n## Definition\n\nLet $\\Theta$ be a set of states of nature, for example the parameter set of a [[parametric model|MSC0MR8VMH]].\nLet $\\mathcal{A}$ be a set of actions.\nA *loss function* is a function $L \\colon \\Theta \\times \\mathcal{A} \\to [0, \\infty)$.\nThe value $L(\\theta, a)$ is the loss incurred by taking the action $a$ when the true state is $\\theta$.\n\n## Remarks\n\nFor estimating a real quantity $g(\\theta)$, the action set is $\\mathcal{A} = \\R$ and standard choices are the following.\nThe *squared error loss* is $L(\\theta, a) = (a - g(\\theta))^2$.\nThe *absolute error loss* is $L(\\theta, a) = \\abs{a - g(\\theta)}$.\nIn a test of $\\theta \\in \\Theta_0$ against $\\theta \\notin \\Theta_0$, with $\\mathcal{A} = \\set{0, 1}$, the *$0$-$1$ loss* is $1$ for a wrong decision and $0$ for a right one.\nSome authors allow real-valued losses that are only bounded below; this changes little in the theory.\n\nExample: $\\Theta = \\mathcal{A} = \\R$ and $L(\\theta, a) = (a - \\theta)^2$.\nIt is a function of the pair $(\\theta, a)$ with values in $[0, \\infty)$, and it is $0$ exactly when $a = \\theta$.\n\nNon-example: $L(\\theta, a) = a - \\theta$ on $\\R \\times \\R$.\nIt takes negative values, so it is not a loss function in the sense above.\nIt would also reward underestimating $\\theta$ without limit.\n",
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