## Local Computation with Valuations from a Commutative Semigroup (1996)

Venue: | Annals of Mathematics and Artificial Intelligence |

Citations: | 28 - 8 self |

### BibTeX

@ARTICLE{Lauritzen96localcomputation,

author = {S. L. Lauritzen and F. V. Jensen},

title = {Local Computation with Valuations from a Commutative Semigroup},

journal = {Annals of Mathematics and Artificial Intelligence},

year = {1996},

volume = {21},

pages = {51--69}

}

### OpenURL

### Abstract

This paper studies a variant of axioms originally developed by Shafer and Shenoy (1988). It is investigated which extra assumptions are needed to perform the local computations in a HUGIN-like architecture (Jensen et al. 1990) or in the architecture of Lauritzen and Spiegelhalter (1988). In particular it is shown that propagation of belief functions can be performed in these architectures. Keywords: articial intelligence, belief function, constraint propagation, expert system, probability propagation, valuation-based system. 1 Introduction An important development in articial intelligence is associated with an abstract theory of local computation known as the Shafer{Shenoy axioms (Shafer and Shenoy 1988; Shenoy and Shafer 1990). These describe in a very general setting how computations can be performed eciently and locally in a variety of problems, just if a few simple conditions are satised. Even though the axioms were developed to formalize computation with belief functions (Shaf...

### Citations

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(Show Context)
Citation Context ...ons can be performed eciently and locally in a variety of problems, just if a few simple conditions are satised. Even though the axioms were developed to formalize computation with belief functions (S=-=hafer 1976-=-), the general approach was valuable for a wide range of problems, including dynamic programming, constraint satisfaction, probabilistic inference, Spohn's belief revision, and others. Further, it enh... |

1324 | Local Computations with Probabilities on Graphical Structures and Their Application to Expert Systems - Lauritzen, Spiegelhalter - 1988 |

955 | An Introduction to Bayesian Networks - Jensen - 1996 |

290 |
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(Show Context)
Citation Context ...per studies a variant of axioms originally developed by Shafer and Shenoy (1988). It is investigated which extra assumptions are needed to perform the local computations in a HUGIN-like architecture (=-=Jensen et al. 199-=-0) or in the architecture of Lauritzen and Spiegelhalter (1988). In particular it is shown that propagation of belief functions can be performed in these architectures. Keywords: articial intelligence... |

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(Show Context)
Citation Context ...e local computation algorithms. Lauritzen and Spiegelhalter (1988) developed a related algorithm for probability calculations and this algorithm was improved in the implementation of the HUGIN shell (=-=Andersen et al. 1989-=-; Jensen et al. 1990). No abstract framework has been made that places the latter algorithms in a generality similar to that of Shafer and Shenoy although Dawid (1992) and Cowell and Dawid (1992) show... |

150 | Probabilistic Reasoning in Expert Systems - Neapolitan - 1990 |

143 | Propagation of probabilities, means, and variances in mixed graphical association models
- Lauritzen
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(Show Context)
Citation Context ...sen (1996) are recommended for further reading. The algorithms have been modied and generalized in a variety of ways, to include continuous variables based on the conditional Gaussian distributions (L=-=auritzen 1992-=-), to optimize decisions in in uence diagrams (Shenoy 1992; Jensen et al. 1994), etc. These algorithms fall outside the framework considered in the present paper as the valuations do not form semigrou... |

138 | Axioms for probability and belief-function propagation
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- 1990
(Show Context)
Citation Context ...n-based system. 1 Introduction An important development in articial intelligence is associated with an abstract theory of local computation known as the Shafer{Shenoy axioms (Shafer and Shenoy 1988; S=-=henoy and Shafer 199-=-0). These describe in a very general setting how computations can be performed eciently and locally in a variety of problems, just if a few simple conditions are satised. Even though the axioms were d... |

115 |
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(Show Context)
Citation Context ...s have been modied and generalized in a variety of ways, to include continuous variables based on the conditional Gaussian distributions (Lauritzen 1992), to optimize decisions in in uence diagrams (S=-=henoy 199-=-2; Jensen et al. 1994), etc. These algorithms fall outside the framework considered in the present paper as the valuations do not form semigroups or the marginalization operations do not full the assu... |

89 |
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- 1992
(Show Context)
Citation Context ...xA ; y V nA ): But another marginal of interest is the max-marginal dened as f ^A (x) = max y V nA 2X V nA f(xA ; y V nA ): Also, associated with retraction of evidence in expert systems (Cowell and D=-=awid 1-=-992) we have the out-marginal dened as f A (x) = P y V nA 2X V nA n f(xA ; y V nA ) Q v2V nA h v (y v ) o P y V nA 2X V nA n Q v2V nA h v (y v ) o ; where h v are functions on X v with values in f0; ... |

61 | Probabilistic Expert Systems - Shafer - 1996 |

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19 | An axiomatic study of computation in hypertrees - Shafer - 1991 |

11 |
Fast retraction of evidence in a probabilistic expert system
- Cowell, Dawid
- 1992
(Show Context)
Citation Context ... 2X V nA f(xA ; y V nA ): But another marginal of interest is the max-marginal dened as f ^A (x) = max y V nA 2X V nA f(xA ; y V nA ): Also, associated with retraction of evidence in expert systems (C=-=owell and Dawid 1-=-992) we have the out-marginal dened as f A (x) = P y V nA 2X V nA n f(xA ; y V nA ) Q v2V nA h v (y v ) o P y V nA 2X V nA n Q v2V nA h v (y v ) o ; where h v are functions on X v with values in f0; ... |

11 |
From In Diagrams to Junction Trees
- Jensen, Jensen, et al.
- 1994
(Show Context)
Citation Context ...odied and generalized in a variety of ways, to include continuous variables based on the conditional Gaussian distributions (Lauritzen 1992), to optimize decisions in in uence diagrams (Shenoy 1992; J=-=ensen et al. 199-=-4), etc. These algorithms fall outside the framework considered in the present paper as the valuations do not form semigroups or the marginalization operations do not full the assumptions. A number of... |

11 | Local computation in hypertrees
- Shafer, Shenoy
- 1988
(Show Context)
Citation Context ...ty propagation, valuation-based system. 1 Introduction An important development in articial intelligence is associated with an abstract theory of local computation known as the Shafer{Shenoy axioms (S=-=hafer and Shenoy 198-=-8; Shenoy and Shafer 1990). These describe in a very general setting how computations can be performed eciently and locally in a variety of problems, just if a few simple conditions are satised. Even ... |

1 | The l 1 -algebra of a commutative semigroup - Hewitt, Zuckermann - 1956 |