## On triangulating dynamic graphical models (2003)

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Venue: | In Proceed. 19th Conf. on Uncertainty in Artificial Intelligence |

Citations: | 25 - 15 self |

### BibTeX

@INPROCEEDINGS{Bartels03ontriangulating,

author = {Jeff Bilmes Chris Bartels and Jeff Bilmes Chris Bartels},

title = {On triangulating dynamic graphical models},

booktitle = {In Proceed. 19th Conf. on Uncertainty in Artificial Intelligence},

year = {2003}

}

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### Abstract

This paper introduces improved methodology to triangulate dynamic graphical models and dynamic Bayesian networks (DBNs). In this approach, a standard DBN template can be modified so the repeating and unrolled graph section may dissect the original DBN time slice and may also span (and partially intersect) many such slices. We introduce the notion of a “boundary ” which divides a graph into multi-slice partitions each of which has an interface, and define the “boundary algorithm”, a method to find the best boundary (and corresponding interface) between partitions in such models. We prove that, after using this algorithm, the sizes of the best forward- and backward- interface (and also the corresponding fill-ins) are identical. The boundary algorithm allows for constrained elimination orders (and therefore graph triangulations) that are impossible using standard slice-by-slice constrained elimination. We describe the above using the Graphical Model ToolKit (GMTK)’s notion of dynamic graphical model, slightly generalizing standard DBN templates. We report triangulation results on hand-concocted graphs, novel speech recognition DBNs, and random graphs, and find that the boundary algorithm can significantly improve both tree width and graph weight. 1

### Citations

1131 | Graphical Models
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(Show Context)
Citation Context ...equal in quality. Section 5 describes the new GMTK triangulation engine. Section 6 describes our results, and Section 7 concludes. Throughout this paper, we assume basic knowledge of graphical models =-=[11]-=- and their set-theoretic description. 2 Technical Background A DBN [5] of length T is a directed acyclic graph G = (V, E) = ( �T t=1 Vt, ET ∪ �T −1 t=1 Et∪E → t ) with node set V and edge set E compri... |

588 | Dynamic Bayesian Networks: Representation, Inference and Learning
- Murphy
- 2002
(Show Context)
Citation Context ...oes the number of possible triangulations thereby making it more difficult (and less likely) to find high quality triangulations. Of course, one can resort to approximate inference techniques in DBNs =-=[6, 12]-=- but with a good triangulation, some even quite complex networks can be utilized exactly. The most promising work on DBN triangulation and exact inference uses a constrained elimination scheme [8, 10,... |

297 |
Algorithmic aspects of vertex elimination on graphs
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Citation Context ...es of adjacent slices. This template is then unrolled to any desired length T to yield the DBN G. The following theorem is relied upon by most work on DBN triangulation: Theorem 2.1. Rose (Lemma 4 in =-=[14]-=-). Let G = (V, E) be an undirected graph with a given elimination ordering that maps G to G ′ = (V, E ′ ) where E ′ = E ∪ F , and where F are the fill-in edges added during elimination. Then uv ∈ E ′ ... |

110 | Probabilistic temporal reasoning - Dean, Kanazawa - 1988 |

109 | The Graphical Models Toolkit: An open source software system for speech and time-series processing
- Bilmes, Zweig
- 2002
(Show Context)
Citation Context ...e our novel triangulation procedures. Note, however, that the triangulation procedures described in this paper are entirely applicable to standard DBN templates. The graphical modeling toolkit (GMTK) =-=[2]-=- is a general purpose software system for developing graphical-model based speech and language systems. While being graphically oriented, GMTK also has features that are contained in common speech/lan... |

102 |
A graph-theoretic study of the numerical solution of sparse positive-definite systems of linear equations
- Rose
- 1972
(Show Context)
Citation Context ... effective on such graphs. Standard triangulation heuristics include greedy schemes where elimination orders are produced by choosing next nodes according to their current fill-ins, sizes, or weights =-=[15]-=-. These schemes, however, can easily start eliminating nodes with neighbors that span many time slices and thereby produce correspondingly large cliques. Second, evidence will typically come in at dif... |

31 | Probabilistic modeling with Bayesian Networks for automatic speech recognition
- Zweig, Russell
- 1998
(Show Context)
Citation Context ...|CL| was identical for the two strategies. Table 3 shows weights for the speech research systems 3 . The first column shows our baseline results using the triangulation method (the Frontier algorithm =-=[18]-=-) used in [2]. The second column is the best weight from partitions created from the standard forward/backward interface with minimum size. The third column is the best weight from a variety of bounda... |

30 | Dynamic Bayesian networks
- Murphy
- 2002
(Show Context)
Citation Context ...currently being used for connected-word continuous speech recognition. The bottom left (C) is a graph used to illustrate a property of the boundary algorithm below. The middle right (D) shows a graph =-=[13]-=- and its 2×-unrolled version where standard sliceby-slice elimination fails to achieve the obvious size-2 maxclique. The bottom right (E) shows a “snake-like” graph, one where no constrained eliminati... |

24 | A computational scheme for reasoning in dynamic probabilistic networks
- Kjærulff
- 1992
(Show Context)
Citation Context ...[6, 12] but with a good triangulation, some even quite complex networks can be utilized exactly. The most promising work on DBN triangulation and exact inference uses a constrained elimination scheme =-=[8, 10, 16, 4, 12, 13]-=-. In this case, rather than considering all possible elimination orders in an unrolled graph, one places a priori constraints on the elimination order that severely restrict the number of elimination ... |

14 | dHugin: A Computational System for Dynamic Time-Sliced Bayesian Networks
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- 1995
(Show Context)
Citation Context ...[6, 12] but with a good triangulation, some even quite complex networks can be utilized exactly. The most promising work on DBN triangulation and exact inference uses a constrained elimination scheme =-=[8, 10, 16, 4, 12, 13]-=-. In this case, rather than considering all possible elimination orders in an unrolled graph, one places a priori constraints on the elimination order that severely restrict the number of elimination ... |

13 | DBN based multi-stream models for speech
- Zhang, Diao, et al.
- 2003
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Citation Context ...aseline min(|I ← t |, |I→ Figure 3-A 6.40814 t |) 5.8020 Boundary 5.6603 Figure 3-B 14.2418 11.7260 11.7260 Livescu Decode A 11.2024 10.9910 10.9907 Livescu Decode B 7.03116 6.7382 6.7382 Muli-Stream =-=[17]-=- 8.36556 7.4553 7.3595 7 Discussion In this paper, we introduced the boundary algorithm, a new method for facilitating the triangulation of dynamic graphical models. We plan in future work to define a... |

9 | Optimal decomposition of probabilistic networks by simulated annealing
- Kjærulff
- 1992
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Citation Context ...r by the user. The highest-priority heuristic is used to determine an elimination order, with lower priority heuristics used only to break ties when they occur. GMTK also supports simulated annealing =-=[9]-=- and maximum cardinality search. If chunks are small enough, it is possible even to exhaustively search all elimination orders. More interestingly, it is possible to produce an exhaustive search over ... |

7 | Temporally invariant junction tree for inference in dynamic Bayesian network
- Xiang
- 1998
(Show Context)
Citation Context ...[6, 12] but with a good triangulation, some even quite complex networks can be utilized exactly. The most promising work on DBN triangulation and exact inference uses a constrained elimination scheme =-=[8, 10, 16, 4, 12, 13]-=-. In this case, rather than considering all possible elimination orders in an unrolled graph, one places a priori constraints on the elimination order that severely restrict the number of elimination ... |

6 |
Lecture Notes in Artificial Intelligence, chapter Learning Dynamic Bayesian Networks
- Ghahramani
- 1998
(Show Context)
Citation Context ...oes the number of possible triangulations thereby making it more difficult (and less likely) to find high quality triangulations. Of course, one can resort to approximate inference techniques in DBNs =-=[6, 12]-=- but with a good triangulation, some even quite complex networks can be utilized exactly. The most promising work on DBN triangulation and exact inference uses a constrained elimination scheme [8, 10,... |

5 |
Generating random bayesiannetworks
- Ide, Cozman
- 2002
(Show Context)
Citation Context ... the left interface size |CL| is reported. As can be seen, both the interface and the clique size can improve dramatically. Table 2 shows results for randomly generated graphs (using methods based on =-=[7]-=-). The first five graphs contain forward only temporal edges and the second five contain both forward and backward. Each network contains 5, 10, 15, or 20 nodes per frame with random variable cardinal... |

5 |
The GMTK Documentation
- Bilmes
(Show Context)
Citation Context ...tures that are contained in common speech/language toolkits (e.g., pruning, scaling factors, etc.). In this section, we describe only its extended DBN representation — other features are described in =-=[2, 1]-=-. As mentioned above, a typical DBN template is described using slice nodes and their intra- and inter- set of edges. A GMTK template extends a standard DBN template in three ways: first, it allows fo... |

4 |
Constant Space Reasoning in Dynamic Bayesian Networks. Intl
- Darwiche
(Show Context)
Citation Context |

2 | The 2001 GMTK-based SPINE ASR system
- Çetin, Nock, et al.
- 2002
(Show Context)
Citation Context ... are given (e.g., certain dependencies might be deterministic). The graphs are displayed in Figure 3. 1 The top left (A) shows a standard GMTK template used for a number of speech recognition systems =-=[2, 3]-=-. The top right (B) shows a template currently being used for connected-word continuous speech recognition. The bottom left (C) is a graph used to illustrate a property of the boundary algorithm below... |

1 |
The GMTK documentation, 2002. http: //ssli.ee.washington.edu/˜bilmes/gmtk
- Bilmes
(Show Context)
Citation Context ...tures that are contained in common speech/language toolkits (e.g., pruning, scaling factors, etc.). In this section, we describe only its extended DBN representation — other features are described in =-=[2, 1]-=-. As mentioned above, a typical DBN template is deS T-1 O T-1 E T-1sscribed using slice nodes and their intra- and interset of edges. A GMTK template extends a standard DBN template in three ways: fir... |