Bernoulli

  • Bernoulli
  • Volume 13, Number 4 (2007), 893-909.

Conjunctive Bayesian networks

Niko Beerenwinkel, Nicholas Eriksson, and Bernd Sturmfels

Full-text: Open access

Abstract

Conjunctive Bayesian networks (CBNs) are graphical models that describe the accumulation of events which are constrained in the order of their occurrence. A CBN is given by a partial order on a (finite) set of events. CBNs generalize the oncogenetic tree models of Desper et al. by allowing the occurrence of an event to depend on more than one predecessor event. The present paper studies the statistical and algebraic properties of CBNs. We determine the maximum likelihood parameters and present a combinatorial solution to the model selection problem. Our method performs well on two datasets where the events are HIV mutations associated with drug resistance. Concluding with a study of the algebraic properties of CBNs, we show that CBNs are toric varieties after a coordinate transformation and that their ideals possess a quadratic Gröbner basis.

Article information

Source
Bernoulli Volume 13, Number 4 (2007), 893-909.

Dates
First available in Project Euclid: 9 November 2007

Permanent link to this document
https://projecteuclid.org/euclid.bj/1194625594

Digital Object Identifier
doi:10.3150/07-BEJ6133

Mathematical Reviews number (MathSciNet)
MR2364218

Zentralblatt MATH identifier
1129.62100

Keywords
Bayesian network distributive lattice Gröbner basis maximum likelihood estimation Möbius transform mutagenetic tree oncogenetic tree sagbi basis toric variety

Citation

Beerenwinkel, Niko; Eriksson, Nicholas; Sturmfels, Bernd. Conjunctive Bayesian networks. Bernoulli 13 (2007), no. 4, 893--909. doi:10.3150/07-BEJ6133. https://projecteuclid.org/euclid.bj/1194625594


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