{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(X, \\mathcal{A})$ be a measurable space, where $\\mathcal{A}$ is a [[sigma-algebra|04WSW8T3SY]] on $X$.\nLet $H$ be a [[probability measure|0VGMEBVW1M]] on $(X, \\mathcal{A})$ and let $\\alpha \u003e 0$ be a real number.\nA random probability measure $G$ on $(X, \\mathcal{A})$ is a *Dirichlet process* with concentration parameter $\\alpha$ and base measure $H$ if the following holds.\nFor every finite partition $A_1, \\ldots, A_r$ of $X$ into sets of $\\mathcal{A}$, the random vector $(G(A_1), \\ldots, G(A_r))$ has the [[Dirichlet distribution|RX4CVSSRRJ]] with parameters $(\\alpha H(A_1), \\ldots, \\alpha H(A_r))$.\nHere a parameter equal to $0$ means that the corresponding coordinate is $0$ almost surely.\nOne writes $G \\sim \\mathrm{DP}(\\alpha, H)$.\n\n## Remarks\n\nExample: take $A \\in \\mathcal{A}$ with $0 \u003c H(A) \u003c 1$, and the partition $A, X \\setminus A$.\nThen $G(A)$ has the [[Beta distribution|0EW824Z7G2]] with parameters $\\alpha H(A)$ and $\\alpha (1 - H(A))$.\nHence $E[G(A)] = H(A)$ and $\\mathrm{Var}(G(A)) = H(A)(1 - H(A))/(\\alpha + 1)$.\n\nNon-example: let $H$ be uniform on $[0, 1]$ and let $G = H$ be constant.\nThen $G$ is not a Dirichlet process for any $\\alpha$.\nFor $A = [0, 1/2]$ the variance of $G(A)$ is $0$, while a Dirichlet process gives the variance $1/(4(\\alpha + 1)) \u003e 0$.\n\nA Dirichlet process exists for every $\\alpha$ and $H$ as above.\nOne construction is stick-breaking.\nLet $V_1, V_2, \\ldots$ be independent with the Beta distribution with parameters $1$ and $\\alpha$, and let $\\theta_1, \\theta_2, \\ldots$ be independent with law $H$ and independent of the $V_i$.\nPut $w_k = V_k \\prod_{i \u003c k} (1 - V_i)$.\nThen $G = \\sum_{k \\ge 1} w_k \\delta_{\\theta_k}$ is a Dirichlet process with parameters $\\alpha$ and $H$, where $\\delta_\\theta$ is the [[Dirac measure|099SXAG014]] at $\\theta$.\nIn particular a Dirichlet process is a discrete measure almost surely.\n\nConjugacy: suppose $G \\sim \\mathrm{DP}(\\alpha, H)$ and, given $G$, the points $x_1, \\ldots, x_n$ are independent with law $G$.\nThen, given $x_1, \\ldots, x_n$, the measure $G$ is a Dirichlet process with concentration $\\alpha + n$ and base measure $(\\alpha H + \\sum_{i=1}^n \\delta_{x_i})/(\\alpha + n)$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "XT4SWG6J7N",
    "kind": "definition",
    "msc": [
      "60G57",
      "62G05",
      "62F15"
    ],
    "name": "Dirichlet process",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Dirichlet_process"
      }
    ],
    "tag": "7174H28FT6",
    "uses": [
      "0VGMEBVW1M",
      "04WSW8T3SY",
      "RX4CVSSRRJ",
      "099SXAG014",
      "0EW824Z7G2"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/7174H28FT6/",
  "links": [
    {
      "field": "defines",
      "target": "XT4SWG6J7N"
    },
    {
      "field": "uses",
      "target": "0VGMEBVW1M"
    },
    {
      "field": "uses",
      "target": "04WSW8T3SY"
    },
    {
      "field": "uses",
      "target": "RX4CVSSRRJ"
    },
    {
      "field": "uses",
      "target": "099SXAG014"
    },
    {
      "field": "uses",
      "target": "0EW824Z7G2"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/7174H28FT6.md",
  "msc": [
    "60G57",
    "62G05",
    "62F15"
  ],
  "name": "Dirichlet process",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/717/7174H28FT6.md",
  "status": null,
  "tag": "7174H28FT6",
  "uri": "m2://7174H28FT6",
  "url": "https://math2.tamnd.com/wiki/dirichlet-process-definition/",
  "writing": {
    "tag": "7174H28FT6",
    "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
      }
    ]
  }
}