{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $G = (V, E)$ be a finite undirected [[graph|00C0BR7D7H]] and let $X = (X_v)_{v \\in V}$ be [[random variables|0PNRKMRC55]].\nFor $A \\subseteq V$ write $X_A = (X_v)_{v \\in A}$.\nA set $S \\subseteq V$ *separates* $A$ from $B$ if every path from a vertex of $A$ to a vertex of $B$ passes through $S$.\nThen $X$ is a *Markov random field* with respect to $G$ if it has the global Markov property.\nThis means: whenever $A, B, S \\subseteq V$ are disjoint and $S$ separates $A$ from $B$, $X_A$ and $X_B$ are [[conditionally independent|0THJ7416DX]] given $X_S$.\n\n## Remarks\n\nTwo weaker properties are also used.\nIn the local Markov property, each $X_v$ is conditionally independent of all non-neighbours given its neighbours $X_{N(v)}$.\nIn the pairwise Markov property, $X_u \\perp X_v \\mid X_{V \\setminus \\set{u, v}}$ for all non-adjacent $u \\ne v$.\nGlobal implies local implies pairwise, and all three are equivalent when the joint density is strictly positive.\nExample: on the path $1 - 2 - 3$, a Markov chain $X_1, X_2, X_3$ is a Markov random field, since $\\set{2}$ separates $1$ from $3$ and $X_1 \\perp X_3 \\mid X_2$.\nNon-example: let $X_1, X_2$ be independent fair coins and $X_3 = X_1$.\nThen $X_1$ and $X_3$ are dependent given $X_2$, so $X$ is not a Markov random field on this path.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "6W69H3A3G5",
    "kind": "definition",
    "msc": [
      "62H22",
      "60G60"
    ],
    "name": "Markov random field",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Markov_random_field"
      }
    ],
    "tag": "4JX3WDYJ83",
    "uses": [
      "Q3FQAD1BXG",
      "00C0BR7D7H",
      "0THJ7416DX",
      "0PNRKMRC55"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/4JX3WDYJ83/",
  "links": [
    {
      "field": "defines",
      "target": "6W69H3A3G5"
    },
    {
      "field": "uses",
      "target": "Q3FQAD1BXG"
    },
    {
      "field": "uses",
      "target": "00C0BR7D7H"
    },
    {
      "field": "uses",
      "target": "0THJ7416DX"
    },
    {
      "field": "uses",
      "target": "0PNRKMRC55"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/4JX3WDYJ83.md",
  "msc": [
    "62H22",
    "60G60"
  ],
  "name": "Markov random field",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/4JX/4JX3WDYJ83.md",
  "status": null,
  "tag": "4JX3WDYJ83",
  "uri": "m2://4JX3WDYJ83",
  "url": "https://tamnd.github.io/math2/wiki/markov-random-field-definition/",
  "writing": {
    "tag": "4JX3WDYJ83",
    "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
      }
    ]
  }
}