## A New Graphical Model for the Representation of Marginalized DAG-Representable Relations ∗

### BibTeX

@MISC{Paz_anew,

author = {Azaria Paz},

title = {A New Graphical Model for the Representation of Marginalized DAG-Representable Relations ∗},

year = {}

}

### OpenURL

### Abstract

A new model for representing PD-induced relations that are derived from DAG-representable relations through marginalization over a subset of their variables is introduced. The new model requires polynomial space and a polynomial algorithm is given for testing whether a given triplet is represented in the model. In addition a polynomial algorithm is derived for testing whether a marginalized DAG-representable relation is DAGrepresentable. 1

### Citations

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Probabilistic Reasoning in Intelligent Systems
- Pearl
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(Show Context)
Citation Context ...d the Annotated Graph (AG) model. The reader is supposed to be familiar with the DAG-model which was studied extensively in the literature. An ample introduction to the DAG model is included in Pearl =-=[7, 1988]-=-, Pearl [8, 2000], and Lauritzen [3, 1996]. ∗ The main part of this work was done while the author visited the Cognitive Systems Laboratory at UCLA and was supported in part by grants from Air Force, ... |

1117 |
Causality: Models, Reasoning, and Inference
- Pearl
- 2000
(Show Context)
Citation Context ...raph (AG) model. The reader is supposed to be familiar with the DAG-model which was studied extensively in the literature. An ample introduction to the DAG model is included in Pearl [7, 1988], Pearl =-=[8, 2000]-=-, and Lauritzen [3, 1996]. ∗ The main part of this work was done while the author visited the Cognitive Systems Laboratory at UCLA and was supported in part by grants from Air Force, NSF and ONR (MURI... |

217 |
Equivalence and synthesis of causal models
- Verma, Pearl
- 1990
(Show Context)
Citation Context ...nnecting vw,p ′ reset W = W ∪ q ′ . • Mark the edge (p ′ , q ′ ). 3.6 Go to 3.2 4. End of algorithm 3.2.2.2 An Example to v, else The graph D shown below is borrowed from the paper of Verma and Pearl =-=[11, 1991]-=- and we want to marginalize this graph over {a, b, c, e, f} = S. The annotated graph representation of D is shown in Fig. 4. Notice that all ranges in K(D) have nonempty intersection with S. The graph... |

140 |
Independence properties of directed markov fields
- Lauritzen, Dawid, et al.
- 1990
(Show Context)
Citation Context ... is interpreted as “X is independent of Y given Z” and t ∈ D means: t is represented in D. To check whether a given triplet t is represented in D we use the Algorithm L1 below due to Lauritzen et al. =-=[2, 1990]-=-. Algorithm L1: Input: D = (V, E) and t = (X; Y |Z). 1. Let V ′ be the set of ancestor of v(t) = X ∪ Y ∪ Z and let D ′ (t) be the subgraph of D over V ′ . 2. Moralize D ′ (t) (i.e., join all uncoupled... |

9 |
Equivalence of causal models with latent variables
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(Show Context)
Citation Context ...ivalent DAG can be found and is shown in Fig. 19. 4.5 Checking whether two DAG’s are equivalent when both are marginalized over the same subset of their common vertices This problem was considered in =-=[9, 1992]-=- where a polynomial algorithm is suggested for solving it. It is reasonable to assume that the annotated graph rep-A new graphical model 135 p q a b c d r s Figure 16: DAG representing relation p q a... |

4 | On testing whether an embedded Bayesian network represents a probability model
- Geiger, Paz, et al.
- 1994
(Show Context)
Citation Context ...ose presence dictates the disconnecting of u from v when the MDT procedure is applied to AS(D) for different triplets t. This feature is illustrated in the example below – introduced in Geiger et al. =-=[1, 1994]-=-. Consider the DAG D = (V, E) with V = {xi|i ∈ N} ∪ {yi|i ∈ N} ∪ {uij|i, j ∈ N, i ̸= j} ∪ {vij|i, j ∈ N, i ̸= j} ∪ {wii|i ∈ N} E = {xi ← uij → xj | i, j ∈ N, i ̸= j}∪ {yi ← vij → yj | i, j ∈ N, i ̸= j... |

4 | The annotated graph model for representing DAG-representable relations – algorithms approach. Research report - Paz - 2007 |

3 |
A sweep operator for triangular matrices and its statistical applications
- Wermuth, Cox
- 2000
(Show Context)
Citation Context ...ocedure is provided for checking whether a marginalized DAG-representable relation is itself DAG-representable. A different approach to marginalized relations as above can be found in Wermuth and Cox =-=[12, 2000]-=-. 2 Preliminaries 2.1 Definitions and notations UGs will denote undirected graphs G = (V, E) where V is a set of vertices and E is a set of undirected edges connecting between two vertices. Two vertic... |

2 |
Graphical Models. Claredon
- Lauritzen
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(Show Context)
Citation Context ...er is supposed to be familiar with the DAG-model which was studied extensively in the literature. An ample introduction to the DAG model is included in Pearl [7, 1988], Pearl [8, 2000], and Lauritzen =-=[3, 1996]-=-. ∗ The main part of this work was done while the author visited the Cognitive Systems Laboratory at UCLA and was supported in part by grants from Air Force, NSF and ONR (MURI).112 A. PAZ The AG-mode... |

2 | An alternative version of Lauritzen et al’s algorithm for checking representation of independencies - Paz |

1 | Representation of irrelevance relations by annotated graphs
- Paz, Geva, et al.
(Show Context)
Citation Context ... the Cognitive Systems Laboratory at UCLA and was supported in part by grants from Air Force, NSF and ONR (MURI).112 A. PAZ The AG-model in a general form was introduced by Paz, Geva, and Studeny in =-=[6, 2000]-=- and a restricted form of this model, which is all we need for this paper, was introduced by Paz [4, 2003a] and investigated further in Paz [5, 2003b]. For the sake of completeness, we shall reproduce... |

1 | Private communication - Studeny - 2000 |