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  "body": "\n## Summary\n\nA loss function assigns a number to each pair of a true state of nature and an action.\nThe number measures the cost of taking that action in that state.\nIn statistics the state is an unknown parameter and the action is often an estimate.\nLoss functions are the starting point of [[statistical decision theory|CPFSBBHXT5]]: averaging the loss over the data gives the risk, by which procedures are compared.\n\n## Explanations\n\n### L1 human\n\nA loss function says how bad a guess is when the truth is known.\nGuessing $7$ when the truth is $5$ costs $(7 - 5)^2 = 4$ under squared error loss, and $\\abs{7 - 5} = 2$ under absolute error loss.\nDifferent losses express different priorities: squared error punishes large mistakes heavily, while a $0$-$1$ loss only cares whether the answer is right.\nThe choice of loss changes which estimator is best.\n",
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