Deep Bayesian networks

Michael A. Kouritzin; Michael A. Kouritzin

doi:10.3934/math.2026011

AIMS Mathematics

2026, Volume 11, Issue 1: 272-290. doi: 10.3934/math.2026011

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Research article Special Issues

Deep Bayesian networks

Michael A. Kouritzin ^,

Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada

Received: 26 September 2025 Revised: 03 December 2025 Accepted: 23 December 2025 Published: 05 January 2026
MSC : 62M05, 60J22, 68T10

Deep Bayesian networks (DBNs) having deep recurrent neural network (DRNN) topography, effective forward-backward learning and innate model selection capabilities are introduced. DBNs provided randomness, Bayes' factor methods and efficient gradient-free expectation-maximization-based (EM) learning to the DRNN layout. DBN's learning, simulation and Bayes' factor capabilities provided an effective generative adversarial network (GAN) in the sequential (RNN) setting. Consequently, deep fakes with real probabilistic models could be created, based upon training data. Alternatively, DBNs could be thought of as some super generalization of hidden Markov models (HMMs), which have inputs and multiple hidden layers. The proofs establishing the above claims were based upon the novel idea to transform the whole network to a completely independent network where the analysis is trivial using a Girsanov like theorem.
- recurrent neural network,
- hidden Markov model,
- model selection,
- big data,
- Girsanov's theorem,
- machine learning
Citation: Michael A. Kouritzin. Deep Bayesian networks[J]. AIMS Mathematics, 2026, 11(1): 272-290. doi: 10.3934/math.2026011

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Abstract

Deep Bayesian networks (DBNs) having deep recurrent neural network (DRNN) topography, effective forward-backward learning and innate model selection capabilities are introduced. DBNs provided randomness, Bayes' factor methods and efficient gradient-free expectation-maximization-based (EM) learning to the DRNN layout. DBN's learning, simulation and Bayes' factor capabilities provided an effective generative adversarial network (GAN) in the sequential (RNN) setting. Consequently, deep fakes with real probabilistic models could be created, based upon training data. Alternatively, DBNs could be thought of as some super generalization of hidden Markov models (HMMs), which have inputs and multiple hidden layers. The proofs establishing the above claims were based upon the novel idea to transform the whole network to a completely independent network where the analysis is trivial using a Girsanov like theorem.

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