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Structure of the high-order Boltzmann machine from independence maps
F X Albizuri1, A d'Anjou, M Grana
1Dept. of Comput. Sci., Univ. of the Basque Country.
IEEE Transactions on Neural Networks
|January 1, 1997
Summary
This study determines the structure of high-order Boltzmann machines (HOBMs) using conditional independences. A minimal hypergraph is constructed from Bayesian and Markov networks for efficient probability distribution approximation.
Area of Science:
- Artificial Intelligence
- Machine Learning
- Computational Statistics
Background:
- High-order Boltzmann machines (HOBMs) are stochastic recurrent networks used for approximating complex probability distributions.
- Determining the optimal structure of HOBMs is crucial for their effective application.
- Existing methods may not efficiently capture the underlying conditional independences required for network structure.
Purpose of the Study:
- To derive the structure of high-order Boltzmann machines (HOBMs) from conditional independences.
- To develop a method for constructing a minimal hypergraph representation of the HOBM.
- To investigate the relationship between Bayesian networks, Markov networks, and the resulting HOBM hypergraph.
Main Methods:
- Conditional independences of a probability distribution are used to infer network structures.
- Independence maps, Markov networks (undirected graphs), and Bayesian networks (directed acyclic graphs) are constructed.
- The intersection hypergraph of all possible Bayesian networks is analyzed and compared to the Markov network.
Main Results:
- A procedure is presented to construct the HOBM hypergraph from conditional independences.
- The intersection hypergraph of all Bayesian networks is proven to be contained within the Markov network hypergraph.
- The Markov network graph provides minimum connectivity for Bayesian network-derived hypergraphs.
Conclusions:
- The proposed method enables the determination of a minimal and efficient HOBM structure.
- Leveraging conditional independences and network graph theory leads to simplified network representations.
- This approach enhances the practical application of HOBMs in probability distribution modeling.
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