{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nA *statistical decision problem* consists of the following data.\n\n1. A [[statistical model|K3AMMSY8DD]] $\\set{P_\\theta : \\theta \\in \\Theta}$ on a sample space $\\mathcal{X}$, from which an observation $X$ is drawn.\n2. A measurable space $\\mathcal{A}$ of actions.\n3. A [[loss function|3BTG6VVCGZ]] $L \\colon \\Theta \\times \\mathcal{A} \\to [0, \\infty)$.\n\nA decision rule is a measurable map $\\delta \\colon \\mathcal{X} \\to \\mathcal{A}$, and its performance is measured by its [[risk function|N26TSVRH2S]] $R(\\theta, \\delta) = E_\\theta[L(\\theta, \\delta(X))]$.\n*Statistical decision theory* is the study of decision problems through the comparison of the risk functions of decision rules.\n\n## Remarks\n\nA rule $\\delta_1$ is at least as good as $\\delta_2$ if $R(\\theta, \\delta_1) \\le R(\\theta, \\delta_2)$ for all $\\theta$.\nThis is only a partial order, so the theory adds criteria: admissibility, the minimax principle and the Bayes principle.\nRandomized rules, which choose an action at random given the data, are often allowed; they make the set of risk functions convex.\n\nExample: point estimation of a normal mean is the decision problem with $P_\\theta$ the law of $n$ independent normal variables with mean $\\theta$ and variance $1$.\nHere $\\Theta = \\mathcal{A} = \\R$ and $L(\\theta, a) = (a - \\theta)^2$.\nThe rule $\\delta(X) = \\bar X$ is a measurable map from $\\R^n$ to $\\mathcal{A}$, with risk $1/n$.\n\nNon-example: a model and a set of estimators without a loss function do not form a decision problem.\nClause 3 is missing, so risks are undefined and rules cannot be compared.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "CPFSBBHXT5",
    "kind": "definition",
    "msc": [
      "62C05",
      "62A01"
    ],
    "name": "Statistical decision theory",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Decision_theory"
      }
    ],
    "tag": "969F3AWBKC",
    "uses": [
      "K3AMMSY8DD",
      "3BTG6VVCGZ",
      "N26TSVRH2S"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/969F3AWBKC/",
  "links": [
    {
      "field": "defines",
      "target": "CPFSBBHXT5"
    },
    {
      "field": "uses",
      "target": "K3AMMSY8DD"
    },
    {
      "field": "uses",
      "target": "3BTG6VVCGZ"
    },
    {
      "field": "uses",
      "target": "N26TSVRH2S"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/969F3AWBKC.md",
  "msc": [
    "62C05",
    "62A01"
  ],
  "name": "Statistical decision theory",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/969/969F3AWBKC.md",
  "status": null,
  "tag": "969F3AWBKC",
  "uri": "m2://969F3AWBKC",
  "url": "https://tamnd.github.io/math2/wiki/statistical-decision-theory-definition/",
  "writing": {
    "tag": "969F3AWBKC",
    "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
      }
    ]
  }
}