{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a [[probability space|0NFPJVJJHX]] and let $\\mathcal{G} \\subseteq \\mathcal{F}$ be a sub-[[σ-algebra|0SV8K0TDGY]].\nA *regular conditional probability* given $\\mathcal{G}$ is a map $\\nu \\colon \\Omega \\times \\mathcal{F} \\to [0, 1]$ such that:\n\n1. for each $\\omega \\in \\Omega$, the map $A \\mapsto \\nu(\\omega, A)$ is a [[probability measure|059TJN0NZ1]] on $(\\Omega, \\mathcal{F})$;\n2. for each $A \\in \\mathcal{F}$, the map $\\omega \\mapsto \\nu(\\omega, A)$ is $\\mathcal{G}$-measurable;\n3. for each $A \\in \\mathcal{F}$ and each $B \\in \\mathcal{G}$, $\\int_B \\nu(\\omega, A) \\, P(d\\omega) = P(A \\cap B)$.\n\n## Remarks\n\nClauses 2 and 3 say that $\\nu(\\cdot, A)$ is a version of the [[conditional probability|0990W5PN13]] $P(A \\mid \\mathcal{G}) = E[\\mathbf{1}_A \\mid \\mathcal{G}]$ for each $A$.\nThen $E[Z \\mid \\mathcal{G}](\\omega) = \\int Z(\\omega') \\, \\nu(\\omega, d\\omega')$ almost surely for every integrable $Z$.\nA regular conditional probability exists when $(\\Omega, \\mathcal{F})$ is a standard Borel space, for example a [[Polish space|584GXXY73M]] with its Borel σ-algebra.\nIt is then unique up to a $P$-null set of $\\omega$.\nIt need not exist in general.\nA related notion is the regular conditional distribution of a random variable $Y$ given $X$.\nIt is a [[Markov kernel|2AVNDP8BJP]] $\\kappa$ with $P(X \\in A, Y \\in B) = \\int_A \\kappa(x, B) \\, P_X(dx)$.\nIt exists whenever $Y$ takes values in a standard Borel space.\nExample: let $\\Omega = [0, 1]^2$ with Lebesgue measure, and let $\\mathcal{G}$ be generated by the first coordinate. Then $\\nu((x, y), A) = \\lambda(\\set{y' : (x, y') \\in A})$, with $\\lambda$ the Lebesgue measure on $[0, 1]$, is a regular conditional probability; clause 3 is [[Fubini's theorem|0R0T1VWH5X]].\nNon-example: if $\\mathcal{G} = \\set{\\emptyset, \\Omega}$ and $0 \u003c P(A) \u003c 1$ for some $A$, then $\\nu(\\omega, A) = \\mathbf{1}_A(\\omega)$ satisfies clause 1 but fails clause 2, since $\\mathbf{1}_A$ is not constant.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "PHHT5W3BEJ",
    "kind": "definition",
    "msc": [
      "60A10",
      "28A50"
    ],
    "name": "Regular conditional probability",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Regular_conditional_probability"
      }
    ],
    "tag": "AAT8PVDW23",
    "uses": [
      "0NFPJVJJHX",
      "0SV8K0TDGY",
      "059TJN0NZ1",
      "0TTDGTEFMN",
      "0990W5PN13"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/AAT8PVDW23/",
  "links": [
    {
      "field": "defines",
      "target": "PHHT5W3BEJ"
    },
    {
      "field": "uses",
      "target": "0NFPJVJJHX"
    },
    {
      "field": "uses",
      "target": "0SV8K0TDGY"
    },
    {
      "field": "uses",
      "target": "059TJN0NZ1"
    },
    {
      "field": "uses",
      "target": "0TTDGTEFMN"
    },
    {
      "field": "uses",
      "target": "0990W5PN13"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/AAT8PVDW23.md",
  "msc": [
    "60A10",
    "28A50"
  ],
  "name": "Regular conditional probability",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/AAT/AAT8PVDW23.md",
  "status": null,
  "tag": "AAT8PVDW23",
  "uri": "m2://AAT8PVDW23",
  "url": "https://math2.tamnd.com/wiki/regular-conditional-probability-definition/",
  "writing": {
    "tag": "AAT8PVDW23",
    "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
      }
    ]
  }
}