{
  "id": "2105.03248",
  "version": "v1",
  "published": "2021-05-05T18:01:11.000Z",
  "updated": "2021-05-05T18:01:11.000Z",
  "title": "Parameter Priors for Directed Acyclic Graphical Models and the Characterization of Several Probability Distributions",
  "authors": [
    "Dan Geiger",
    "David Heckerman"
  ],
  "comment": "Annals October 2002 version with corrections and updates made May 2021",
  "journal": "The Annals of Statistics, 30: 1412-1440, 2002",
  "categories": [
    "stat.ML",
    "cs.LG",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "We develop simple methods for constructing parameter priors for model choice among Directed Acyclic Graphical (DAG) models. In particular, we introduce several assumptions that permit the construction of parameter priors for a large number of DAG models from a small set of assessments. We then present a method for directly computing the marginal likelihood of every DAG model given a random sample with no missing observations. We apply this methodology to Gaussian DAG models which consist of a recursive set of linear regression models. We show that the only parameter prior for complete Gaussian DAG models that satisfies our assumptions is the normal-Wishart distribution. Our analysis is based on the following new characterization of the Wishart distribution: let $W$ be an $n \\times n$, $n \\ge 3$, positive-definite symmetric matrix of random variables and $f(W)$ be a pdf of $W$. Then, f$(W)$ is a Wishart distribution if and only if $W_{11} - W_{12} W_{22}^{-1} W'_{12}$ is independent of $\\{W_{12},W_{22}\\}$ for every block partitioning $W_{11},W_{12}, W'_{12}, W_{22}$ of $W$. Similar characterizations of the normal and normal-Wishart distributions are provided as well.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-05T18:01:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "I.2",
      "G.3"
    ],
    "keywords": [
      "parameter prior",
      "directed acyclic graphical models",
      "probability distributions",
      "characterization",
      "complete gaussian dag models"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Institute of Mathematical Statistics",
      "journal": "Ann. Stat."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}