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

Conjunctive Bayesian networks

Niko Beerenwinkel, Nicholas Eriksson, and Bernd Sturmfels

Full-text: Open access


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

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

First available in Project Euclid: 9 November 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

Export citation


  • Andersson, S. and Perlman, M. (1993). Lattice models for conditional independence in a multivariate normal distribution., Ann. Statist. 21 1318--1358.
  • Andersson, S., Madigan, D., Perlman, M. and Triggs, C. (1995). On the relation between conditional independence models determined by finite distributive lattices and by directed acyclic graphs., J. Statist. Plann. Inference 48 25--46.
  • Beerenwinkel, N. and Drton, M. (2005). Mutagenetic tree models. In, Algebraic Statistics for Computational Biology Chapter 14 (L. Pachter and B. Sturmfels, eds.) 278--290. New York: Cambridge Univ. Press.
  • Beerenwinkel, N. and Drton, M. (2007). A mutagenetic tree hidden Markov model for longitudinal clonal HIV sequence data., Biostatistics 8 53--71. Available at
  • Beerenwinkel, N., Däumer, M., Sing, T., Rahnenführer, J., Lengauer, T., Selbig, J., Hoffmann, D. and Kaiser, R. (2005) Estimating HIV evolutionary pathways and the genetic barrier to drug resistance., J. Infect. Dis. 191 1953--1960. Available at
  • Beerenwinkel, N., Rahnenführer, J., Däumer, M., Hoffmann, D., Kaiser, R., Selbig, J. and Lengauer, T. (2005). Learning multiple evolutionary pathways from cross-sectional data., J. Comput. Biol. 12 584--598. Available at RECOMB 2004.
  • Beerenwinkel, N., Eriksson, N. and Sturmfels, B. (2006). Evolution on distributive lattices., J. Theor. Biol. 242 409--420. Available at
  • Catanese, F., Hoşten, S., Khetan, A. and Sturmfels, B. (2006). The maximum likelihood degree., Am. J. Math. 128 671--697.
  • Condra, J.H., Holder, D.J., Schleif, W.A., Blahy, O.M., Danovich, R.M., Gabryelski, L.J., Graham, D.J., Laird, D., Quintero, J.C., Rhodes, A., Robbins, H.L., Roth, E., Shivaprakash, M., Yang, T., Chodakewitz, J.A., Deutsch, P.J., Leavitt, R.Y., Massari, F.E., Mellors, J.W., Squires, K.E., Steigbigel, R.T., Teppler, H. and Emini, E.A. (1996). Genetic correlates of in vivo viral resistance to indinavir, a human immunodeficiency virus type 1 protease inhibitor., J. Virol. 70 8270--8276.
  • Desper, R., Jiang, F., Kallioniemi, O.P., Moch, H., Papadimitriou, C.H. and Schäffer, A.A. (1999). Inferring tree models for oncogenesis from comparative genome hybridization data., J. Comput. Biol. 6 37--51.
  • Drton, M. and Richardson, T.S. (2007). Binary models for marginal independence., J. Roy. Statist. Soc. Ser. B. To appear. Available at
  • Geiger, D., Meek, C. and Sturmfels, B. (2006). On the toric algebra of graphical models., Ann. Statist. 34 1463--1492.
  • Greuel, G.-M. and Pfister, G. (2002)., A Singular Introduction to Commutative Algebra. Berlin: Springer.
  • Hibi, T. (1987). Distributive lattices, affine semigroup rings and algebras with straightening laws. In, Commutative Algebra and Combinatorics (Kyoto, 1985). Adv. Stud. Pure. Math. 11 93--109. North-Holland, Amsterdam.
  • Iwasa, Y., Michor, F. and Nowak, M.A. (2004). Evolutionary dynamics of invasion and escape., J. Theoret. Biol. 226 205--214.
  • Lauritzen, S. (1996)., Graphical Models. Clarendon Press.
  • Meek, C. and Heckerman, D. (1997). Structure and parameter learning for causal independence and causal interaction models. In, Proceedings of the Thirteenth Conference on Uncertainty in Artificial Intelligence 366--375.
  • Molla, A., Korneyeva, M., Gao, Q., Vasavanonda, S., Schipper, P.J., Mo, H.M., Markowitz, M., Chernyavskiy, T., Niu, P., Lyons, N., Hsu, A., Granneman, G.R., Ho, D.D., Boucher, C.A., Leonard, J.M., Norbeck, D.W. and Kempf, D.J. (1996). Ordered accumulation of mutations in HIV protease confers resistance to ritonavir., Nat. Med. 2 760--766.
  • Pachter, L. and Sturmfels, B., eds. (2005)., Algebraic Statistics for Computational Biology. New York: Cambridge Univ. Press.
  • Pearl, J. (1986). Fusion, propagation, and structuring in belief networks., Artif. Intell. 29 241--288.
  • Pearl, J. (1988)., Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. San Mateo, CA: Morgan Kauffman.
  • Radmacher, M.D., Simon, R., Desper, R., Taetle, R., Schäffer, A.A. and Nelson, M.A. (2001). Graph models of oncogenesis with an application to melanoma., J. Theor. Biol. 212 535--548. Available at
  • Rahnenführer, J., Beerenwinkel, N., Schulz, W.A., Hartmann, C., von Deimling, A., Wullich, B. and Lengauer, T. (2005). Estimating cancer survival and clinical outcome based on genetic tumor progression scores., Bioinformatics 21 2438--2446. Available at
  • Rhee, S.-Y., Gonzales, M.J., Kantor, R., Betts, B.J., Ravela, J. and Shafer, R.W. (2003). Human immunodeficiency virus reverse transcriptase and protease sequence database., Nucleic Acids Res. 31 298--303.
  • Sturmfels, B. (1996)., Gröbner Bases and Convex Polytopes. Providence, RI: Amer. Math. Soc.
  • Sturmfels, B. and Sullivant, S. (2005). Toric ideals of phylogenetic invariants., J. Comput. Biol. 12 457--481. Available at
  • Szabo, A. and Boucher, K. (2002). Estimating an oncogenetic tree when false negatives and positives are present., Math. Biosci. 176 219--236.