## A characterization of the Dirichlet distribution through global and local parameter independence (1997)

Venue: | THE ANNALS OF STATISTICS |

Citations: | 29 - 5 self |

### BibTeX

@MISC{Geiger97acharacterization,

author = {Dan Geiger and David Heckerman},

title = {A characterization of the Dirichlet distribution through global and local parameter independence},

year = {1997}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

7436 | Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference - Pearl - 1988 |

1341 | Local computations with probabilities on graphical structures and their application to expert system (with discussion - LAURITZEN, SPIEGELHALTER - 1988 |

1132 | A Bayesian method for the induction of probabilistic networks from data - Cooper, Herskovits - 1992 |

948 | Learning bayesian networks: The combination of knowledge and statistical data
- Heckerman, Geiger, et al.
- 1995
(Show Context)
Citation Context ...cture of a graphical model. The latter problem has been addressed by several researchers who have investigated Bayesian methods for model averaging and selection when the models are Bayesian networks =-=[2, 4, 12, 30]-=-. Such a task is often referred to as learning. These approaches all have the same basic components: a scoring rule and a search procedure. The scoring rule takes data and a network structure and retu... |

805 | Optimal Statistical Decisions - DeGroot - 1970 |

405 | Lectures on Functional Equations and Their Applications - Aczél - 1966 |

395 | Fusion, propagation and structuring in belief networks
- Pearl
- 1986
(Show Context)
Citation Context ...ents a conditional independence assertion. Such models have been extensively studied in the fields of statistics (e.g.,[17, 34, 18, 15, 31, 6, 25]), artificial intelligence and computer science (e.g.,=-=[20, 21, 10, 23]-=-), operations research (e.g.,[28, 29]) and philosophy (e.g.,[32]). For an introduction to graphical models, see [22, 35] and references therein. Graphical models are based on directed acyclic graphs, ... |

325 | Bayesian networks - Heckerman, MP - 1995 |

206 |
1990. Sequential updating of conditional probabilities on directed graphical structures
- Spiegelhalter, Lauritzen
(Show Context)
Citation Context ...ph represents a random variable, and the lack of an edge between two nodes represents a conditional independence assertion. Such models have been extensively studied in the fields of statistics (e.g.,=-=[17, 34, 18, 15, 31, 6, 25]-=-), artificial intelligence and computer science (e.g.,[20, 21, 10, 23]), operations research (e.g.,[28, 29]) and philosophy (e.g.,[32]). For an introduction to graphical models, see [22, 35] and refer... |

197 | Theory Refinement of Bayesian Networks - Buntine - 1991 |

197 | Bayesian analysis in expert systems - Spiegelhalter, Dawid, et al. - 1993 |

161 | Mathematical Statistics - Wilks - 1962 |

144 |
Probabilistic Similarity Networks
- Heckerman
- 1990
(Show Context)
Citation Context ...of uncertainty. In recent years, researchers have realized that, although Bayesian networks can be constructed directly from expert knowledge (often using advanced computerized elicitation techniques =-=[11]-=-), it is advantageous to use data to update both the parameters and structure of a graphical model. The latter problem has been addressed by several researchers who have investigated Bayesian methods ... |

123 |
Hyper markov laws and statistical analysis of decomposable graphical models
- Dawid, Lauritzen
- 1993
(Show Context)
Citation Context ...ph represents a random variable, and the lack of an edge between two nodes represents a conditional independence assertion. Such models have been extensively studied in the fields of statistics (e.g.,=-=[17, 34, 18, 15, 31, 6, 25]-=-), artificial intelligence and computer science (e.g.,[20, 21, 10, 23]), operations research (e.g.,[28, 29]) and philosophy (e.g.,[32]). For an introduction to graphical models, see [22, 35] and refer... |

113 |
Identifying independence in Bayesian networks
- Geiger, Verma, et al.
- 1990
(Show Context)
Citation Context ...ents a conditional independence assertion. Such models have been extensively studied in the fields of statistics (e.g.,[17, 34, 18, 15, 31, 6, 25]), artificial intelligence and computer science (e.g.,=-=[20, 21, 10, 23]-=-), operations research (e.g.,[28, 29]) and philosophy (e.g.,[32]). For an introduction to graphical models, see [22, 35] and references therein. Graphical models are based on directed acyclic graphs, ... |

99 |
Evidential Reasoning Using Stochastic Simulation of Causal Models
- Pearl
- 1987
(Show Context)
Citation Context ...ents a conditional independence assertion. Such models have been extensively studied in the fields of statistics (e.g.,[17, 34, 18, 15, 31, 6, 25]), artificial intelligence and computer science (e.g.,=-=[20, 21, 10, 23]-=-), operations research (e.g.,[28, 29]) and philosophy (e.g.,[32]). For an introduction to graphical models, see [22, 35] and references therein. Graphical models are based on directed acyclic graphs, ... |

70 | A bayesian method for constructing bayesian belief networks from databases - Cooper, Herskovits |

23 | A characterization of the Dirichlet distribution with application to learning Bayesian networks
- Geiger, Heckerman
- 1995
(Show Context)
Citation Context ...ected edge is drawn into node i. These nodes are called the parents of i. A simple example of a Bayesian Received February 1995; revised April 1996. 1 A preliminary version of this article appears in =-=[9]-=-. 2 Part of this work was done while the author visited Microsoft Research Center. AMS 1991 subject classifications. Primary 62E10, 60E05; secondary 62A15, 62C10, 39B99. Key words and phrases. Bayesia... |

17 | Lectures on Contingency Tables - Lauritzen - 1982 |

9 |
Choquet-Deny type functional equations with applications to stochastic models
- Rao, Shanbhag
- 1994
(Show Context)
Citation Context ...cable as is. In this case, one may resort to the functional equations defined by the characteristic functions near the origin. A review of many characterization problems in statistics can be found in =-=[16, 26,27]-=-. These texts do not use the elementary solution method used herein. Another well-known characterization of the Dirichlet distribution which is described by several authors is based on W. E. Johnson's... |

4 | Two characterizations of the Dirichlet distribution - Fabius - 1973 |

3 | A new characterization of the Dirichlet distribution through neutrality. Ann. Statist. 8 183–189. MR0557563 Kshirsagar, A.M - James, Mosimann - 1980 |

2 |
A characterization of the Dirichlet distribution
- DARROCH, RATCLIFF
- 1971
(Show Context)
Citation Context ... they "upgrade" results proved for smooth pdfs to any pdfs. We shall now demonstrate their usefulness for another well-known characterization of the Dirichlet distribution due to Darroch and Ratcliff =-=[5]-=-. Their bivariate theorem states: Let X and Y be two continuous, positive random variables which satisfy the inequality X + Y < s. Assume the pdf of X and Y on (0, s) and X/s - Y and Y/s - X on (0, 1)... |

1 |
Lectures on Functional Equations and Their Applications
- Acz2L
- 1966
(Show Context)
Citation Context ...dependence assumptionstated in Theorem 2. Methods for solving functional equations such as 4, that is, finding all functions that satisfy them under different regularity assumptions, are discussed in =-=[1]-=-. We use the following technique. First, we argue that any positive solution to (4) must be differentiable in any order ([1], Section 4.2.2, "Deduction of differentiability from integrability"). Then ... |

1 |
Two characterizations of the Dirichlet distribution
- FABrUs
- 1973
(Show Context)
Citation Context ... consequently, the theorem by Darroch and1350 D. GEIGER AND D. HECKERMAN Ratcliff holds even without assuming continuous pdfs. Indeed, among other results, this was shown, using other techniques, by =-=[8, 14]-=-. Note that if a pdf is in fact a gpdf, that is, it contains a discrete element, then Lebesgue integrability is not satisfied and this technique is not applicable as is. In this case, one may resort t... |