{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\set{P_\\theta : \\theta \\in \\Theta}$ be a [[statistical model|K3AMMSY8DD]] for an observation $X$ with values in a measurable space $\\mathcal{X}$.\nLet $\\mathcal{A}$ be a measurable space of actions and let $L \\colon \\Theta \\times \\mathcal{A} \\to [0, \\infty)$ be a [[loss function|3BTG6VVCGZ]], measurable in $a$ for each $\\theta$.\nA (non-randomized) *decision rule* is a measurable map $\\delta \\colon \\mathcal{X} \\to \\mathcal{A}$.\nThe *risk function* of $\\delta$ is the map $R(\\cdot, \\delta) \\colon \\Theta \\to [0, \\infty]$ given by\n$$R(\\theta, \\delta) = E_\\theta\\big[ L(\\theta, \\delta(X)) \\big] = \\int_{\\mathcal{X}} L(\\theta, \\delta(x)) \\, P_\\theta(dx).$$\n\n## Remarks\n\nThe risk is an [[expected value|0WQNRCNY3M]] over the data, for a fixed $\\theta$.\nUnder the squared error loss $L(\\theta, a) = (a - g(\\theta))^2$, the risk of an estimator is its [[mean squared error|05NZ1HHQMV]].\nFor randomized rules, one also averages over the randomization.\n\nExample: let $X_1, \\dots, X_n$ be independent [[normal|05AQE0QR0D]] with mean $\\theta$ and variance $1$, with squared error loss.\nFor $\\delta = \\bar X$, the risk is $R(\\theta, \\delta) = E_\\theta[(\\bar X - \\theta)^2] = \\operatorname{Var}_\\theta(\\bar X) = 1/n$ for every $\\theta$.\n\nNon-example: the realized loss $L(\\theta, \\delta(x))$ at the observed data $x$ is not the risk.\nIt depends on the data, while the risk is an average over all possible data.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "N26TSVRH2S",
    "kind": "definition",
    "msc": [
      "62C05"
    ],
    "name": "Risk function",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Loss_function"
      }
    ],
    "tag": "RNV041SNCT",
    "uses": [
      "K3AMMSY8DD",
      "3BTG6VVCGZ",
      "0WQNRCNY3M",
      "05NZ1HHQMV",
      "05AQE0QR0D"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/RNV041SNCT/",
  "links": [
    {
      "field": "defines",
      "target": "N26TSVRH2S"
    },
    {
      "field": "uses",
      "target": "K3AMMSY8DD"
    },
    {
      "field": "uses",
      "target": "3BTG6VVCGZ"
    },
    {
      "field": "uses",
      "target": "0WQNRCNY3M"
    },
    {
      "field": "uses",
      "target": "05NZ1HHQMV"
    },
    {
      "field": "uses",
      "target": "05AQE0QR0D"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/RNV041SNCT.md",
  "msc": [
    "62C05"
  ],
  "name": "Risk function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/RNV/RNV041SNCT.md",
  "status": null,
  "tag": "RNV041SNCT",
  "uri": "m2://RNV041SNCT",
  "url": "https://tamnd.github.io/math2/wiki/risk-function-definition/",
  "writing": {
    "tag": "RNV041SNCT",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}