{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $\\set{P_\\theta : \\theta \\in \\Theta}$ be a statistical model with a loss function, where $\\Theta = \\set{\\theta_1, \\dots, \\theta_k}$ is finite.\nAllow randomized decision rules, and assume that every rule has finite [[risk|N26TSVRH2S]] at every $\\theta_i$.\nThen every [[admissible|DSTF6X8DRJ]] rule $\\delta$ is a [[Bayes rule|PP1A0JB47Q]] for some prior: there are weights $\\pi_1, \\dots, \\pi_k \\ge 0$ with $\\sum_i \\pi_i = 1$ such that\n$$\\sum_{i=1}^{k} \\pi_i R(\\theta_i, \\delta) = \\inf_{\\delta'} \\sum_{i=1}^{k} \\pi_i R(\\theta_i, \\delta').$$\n\n## Remarks\n\nConsider the risk set $S = \\set{(R(\\theta_1, \\delta), \\dots, R(\\theta_k, \\delta))}$ in $\\R^k$, over all randomized rules $\\delta$.\nMixing two rules mixes their risk points, so $S$ is a [[convex set|02AS45T3H9]].\nIf $\\delta$ is admissible, its risk point $x$ meets the convex set $\\set{y : y_i \\le x_i \\text{ for all } i}$ in $S$ only at points equal to $x$.\nA separating hyperplane through $x$ then has a normal vector with nonnegative entries, and normalizing it gives the prior $\\pi$.\nThe minimized quantity is the [[Bayes risk|VEBDNCJJTN]] of $\\pi$.\n\nA class $\\mathcal{C}$ of rules is called *complete* if every rule outside $\\mathcal{C}$ is dominated by some rule in $\\mathcal{C}$.\nComplete class theorems are results of this kind, which show that the Bayes rules, or their limits, form a complete class.\nFor infinite $\\Theta$, similar results hold under extra conditions, with limits of Bayes rules or generalized Bayes rules in place of Bayes rules.\n\nRandomization matters in the proof: without it, the risk set need not be convex.\n\n## Explanations\n\n### L1 human\n\nAny rule that cannot be beaten everywhere is, secretly, the best rule for some way of weighting the possible truths.\nSo when the set of possible truths is finite, a search for good rules can be restricted to Bayes rules.\n\n### L2 human\n\nThe theorem gives a frequentist reason to use Bayesian methods: admissibility, a purely frequentist property, forces a rule to be Bayes.\nIt reduces the search for admissible rules to the study of Bayes rules for all priors.\nThe proof is a direct application of the separating hyperplane theorem to the convex risk set.\nIn infinite-dimensional problems the analogues are more delicate, and admissible rules are typically limits of Bayes rules.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "DSTF6X8DRJ",
      "PP1A0JB47Q"
    ],
    "kind": "statement",
    "msc": [
      "62C07",
      "62C10",
      "62C15"
    ],
    "name": "Complete class theorem",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Admissible_decision_rule"
      }
    ],
    "tag": "M3PFV1BX9V",
    "uses": [
      "N26TSVRH2S",
      "DSTF6X8DRJ",
      "PP1A0JB47Q",
      "VEBDNCJJTN",
      "02AS45T3H9"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/M3PFV1BX9V/",
  "links": [
    {
      "field": "about",
      "target": "DSTF6X8DRJ"
    },
    {
      "field": "about",
      "target": "PP1A0JB47Q"
    },
    {
      "field": "uses",
      "target": "N26TSVRH2S"
    },
    {
      "field": "uses",
      "target": "DSTF6X8DRJ"
    },
    {
      "field": "uses",
      "target": "PP1A0JB47Q"
    },
    {
      "field": "uses",
      "target": "VEBDNCJJTN"
    },
    {
      "field": "uses",
      "target": "02AS45T3H9"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/M3PFV1BX9V.md",
  "msc": [
    "62C07",
    "62C10",
    "62C15"
  ],
  "name": "Complete class theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/M3P/M3PFV1BX9V.md",
  "status": null,
  "tag": "M3PFV1BX9V",
  "uri": "m2://M3PFV1BX9V",
  "url": "https://tamnd.github.io/math2/wiki/complete-class-theorem/",
  "writing": {
    "tag": "M3PFV1BX9V",
    "kind": "statement",
    "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
      }
    ]
  }
}