## Faithfulness in Chain Graphs: The Discrete Case

Citations: | 2 - 2 self |

### BibTeX

@MISC{Peña_faithfulnessin,

author = {Jose M. Peña},

title = {Faithfulness in Chain Graphs: The Discrete Case},

year = {}

}

### OpenURL

### Abstract

This paper deals with chain graphs under the classic Lauritzen-Wermuth-Frydenberg interpretation. We prove that the strictly positive discrete probability distributions with the prescribed sample space that factorize according to a chain graph G with dimension d have positive Lebesgue measure wrt R d, whereas those that factorize according to G but are not faithful to it have zero Lebesgue measure wrt R d. This means that, in the measuretheoretic sense described, almost all the strictly positive discrete probability distributions with the prescribed sample space that factorize according to G are faithful to it.

### Citations

1239 | Spatial Interaction and the Statistical Analysis of Lattice Systems (with Discussion - Besag - 1974 |

1155 | Graphical Models
- Lauritzen
- 1996
(Show Context)
Citation Context ... parameterization is inspired by Besag (1974, p. 197). We say that a strictly positive probability distribution p factorizes according to a CG G with n blocks if the following two conditions are met (=-=Lauritzen, 1996-=-, p. 53): 1. p(x) = ∏ n i=1 p(xBi |x P a(Bi)) where 2. p(xBi |x P a(Bi)) = ∏ C∈C((G Bi P a(B i )) m ) ψi C (xC) where ψ i C (xC) are positive real functions. Let 0U denote that every random variable i... |

113 |
The chain graph Markov property
- Frydenberg
- 1990
(Show Context)
Citation Context ...ovian wrt G and such that U ̸⊥ qV |W (Studen´y & Bouckaert, 1998, Consequence 5.2). Then, there is some state xUV W of UV W such that q(xUV W )q(xW ) − q(xUW )q(xV W ) ̸= 0. (6) Moreover, q ∈ D(G) + (=-=Frydenberg, 1990-=-, Theorem 4.1). Then, q corresponds to some fa parameter values by Lemma 2. We can expand q to a probability distribution p ∈ D(G) + over the original cardinalities of the random variables in X by ass... |

62 | Probabilistic conditional independence structures - Studen´y - 2005 |

46 |
Strong completeness and faithfulness in bayesian networks
- Meek
- 1995
(Show Context)
Citation Context ...for any undirected graph there exists a discrete probability distribution with the prescribed sample space that is faithful to it. This result 1Peña has also been proven for acyclic directed graphs (=-=Meek, 1995-=-, Theorem 7). The result in Meek (1995) is actually stronger, as it proves that, in a certain measure-theoretic sense, almost all the discrete probability distributions with the prescribed sample spac... |

20 | On chain graphs models for description of conditional independence structures - Studený, Bouckaert - 1998 |

15 |
Distinctness of the eigenvalues of a quadratic form in a multivariate sample
- Okamoto
- 1973
(Show Context)
Citation Context ...e polynomial for xUV W referred above. Then, sol(xUV W ) has zero Lebesgue measure wrt R d because it consists of the solutions to a non-trivial polynomial in real variables (i.e. the fa parameters) (=-=Okamoto, 1973-=-). Let sol = ⋃ ⋂ {U,V,W ⊆X disjoint : U̸⊥GV |W } xUV W sol(xUV W ). Then, sol has zero Lebesgue measure wrt R d , because the finite union and intersection of sets of zero Lebesgue measure has zero Le... |

5 | Bayesian networks from the point of view of chain graphs - Studen´y - 1998 |

3 | Approximate counting of Graphical Models via MCMC
- Peña
(Show Context)
Citation Context ...ed graphs and acyclic directed graphs. However, the vast majority of independence models that can be represented by chain graphs cannot be represented by undirected graphs or acyclic directed graphs (=-=Peña, 2007-=-). As Studen´y (2005, Section 1.1) points out, something that would help to judge whether this is an advantage of chain graphs would be proving that any independence model represented by a chain graph... |

2 |
Björkegren and Jesper Tegnér. An Algorithm for Reading Dependencies from the Minimal Undirected Independence Map of a Graphoid 13 that Satisfies Weak Transitivity
- Peña, Nilsson, et al.
- 2009
(Show Context)
Citation Context ...ability distribution that is faithful to it for some sample space, most likely different from the prescribed sample space (Studen´y & Bouckaert, 1998, Theorem 7.2). Another related result is that in (=-=Peña et al., 2009-=-, Theorem 3), where it is proven that for any undirected graph there exists a discrete probability distribution with the prescribed sample space that is faithful to it. This result 1Peña has also bee... |