{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "9RMS0ZJ618"
    },
    {
      "field": "defines",
      "target": "AAT8PVDW23"
    },
    {
      "field": "about",
      "target": "N8NXNCMSK5"
    }
  ],
  "body": "\n## Summary\n\nA regular conditional probability is a version of conditional probability given a σ-algebra that is a genuine probability measure for each outcome.\nConditional probabilities $P(A \\mid \\mathcal{G})$ are defined one event at a time and only up to null sets.\nSo in general they need not fit together into a measure.\nA regular version is a [[Markov kernel|2AVNDP8BJP]] that does fit them together, and it lets one compute conditional expectations as integrals.\nIt exists when the underlying space is nice enough, for example a [[Polish space|584GXXY73M]] with its Borel σ-algebra.\n\n## Explanations\n\n### L1 human\n\nGiven some partial information, one wants a whole new probability law that describes the remaining uncertainty.\nFor a point $(x, y)$ chosen uniformly in the unit square, knowing $x$ leaves $y$ uniform on $[0, 1]$, and that uniform law is the regular conditional probability.\nThe difficulty is that the information may be an event of probability zero, like \"$x = 0.3$\".\nA regular conditional probability fixes one law for each value of the information at once, so that all of them together reproduce the original probabilities.\nThe [[Borel–Kolmogorov paradox|9RMS0ZJ618]] shows why the family must be chosen together, not one null event at a time.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "60A10",
      "28A50"
    ],
    "name": "Regular conditional probability",
    "same_as": {
      "wikidata": "Q7309594",
      "wikipedia": "Regular conditional probability"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Regular_conditional_probability"
      }
    ],
    "tag": "PHHT5W3BEJ"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/PHHT5W3BEJ/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/PHHT5W3BEJ.md",
  "msc": [
    "60A10",
    "28A50"
  ],
  "name": "Regular conditional probability",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PHH/PHHT5W3BEJ.md",
  "status": null,
  "tag": "PHHT5W3BEJ",
  "uri": "m2://PHHT5W3BEJ",
  "url": "https://math2.tamnd.com/wiki/regular-conditional-probability/",
  "writing": {
    "tag": "PHHT5W3BEJ",
    "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
      }
    ]
  }
}