{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "7T98FQFGYY"
    },
    {
      "field": "uses",
      "target": "FB692YN0BZ"
    },
    {
      "field": "uses",
      "target": "M3PFV1BX9V"
    },
    {
      "field": "about",
      "target": "FB692YN0BZ"
    },
    {
      "field": "defines",
      "target": "FX1A4SN6CF"
    }
  ],
  "body": "\n## Summary\n\nThe Bayes risk of a decision rule is its [[risk function|N26TSVRH2S]] averaged over a [[prior distribution|00NBBSXRQS]] on the parameter.\nIt reduces the comparison of rules to the comparison of single numbers.\nThe smallest Bayes risk over all rules is the Bayes risk of the prior, and a rule attaining it is a [[Bayes estimator|PP1A0JB47Q]].\n\n## Explanations\n\n### L1 human\n\nThe risk of a rule depends on the unknown truth, so it is a whole curve rather than one number.\nIf one has an opinion about which values of the truth are likely, one can average the curve with those weights.\nThat weighted average is the Bayes risk.\nFor example, estimating a coin's heads probability $\\theta$ by the proportion of heads in $n$ tosses has risk $\\theta(1-\\theta)/n$, and its average over a uniform prior is $1/(6n)$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "62C10"
    ],
    "name": "Bayes risk",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Bayes_estimator"
      }
    ],
    "tag": "VEBDNCJJTN"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/VEBDNCJJTN/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/VEBDNCJJTN.md",
  "msc": [
    "62C10"
  ],
  "name": "Bayes risk",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/VEB/VEBDNCJJTN.md",
  "status": null,
  "tag": "VEBDNCJJTN",
  "uri": "m2://VEBDNCJJTN",
  "url": "https://tamnd.github.io/math2/wiki/bayes-risk/",
  "writing": {
    "tag": "VEBDNCJJTN",
    "kind": "concept",
    "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
      }
    ]
  }
}