Abstract
Symbolic inference algorithms in Bayesian networks have now been applied in a variety of domains. These often require the computation of the derivatives of polynomials representing probabilities in such graphical models. In this paper we formalise a symbolic approach for staged trees, a model class making it possible to visualise asymmetric model constraints. We are able to show that the probability parametrisation associated to trees has several advantages over the one associated to Bayesian networks. We then continue to compute certain derivatives of staged trees’ polynomials and show their probabilistic interpretation. We are able to determine that these polynomials can be straightforwardly deduced by compiling a tree into an arithmetic circuit.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Antonucci, A., de Campos, C.P., Huber, D., Zaffalon, M.: Approximating Credal Network Inferences by Linear Programming. In: van der Gaag, L.C. (ed.) ECSQARU 2013. LNCS, vol. 7958, pp. 13–24. Springer, Heidelberg (2013)
Barclay, L.M., Hutton, J.L., Smith, J.Q.: Refining a Bayesian network using a Chain Event Graph. Int. J. Approx. Reason. 54, 1300–1309 (2013)
Barclay, L.M., Hutton, J.L., Smith, J.Q.: Chain event graphs for informed missingness. Bayesian Anal. 9(1), 53–76 (2014)
Brandherm, B., Jameson, A.: An extension of the differential approach for Bayesian network inference to dynamic Bayesian networks. Int. J. Intell. Syst. 19(8), 727–748 (2004)
Castillo, E., Gutiérrez, J.M., Hadi, A.S.: A new method for efficient symbolic propagation in discrete Bayesian Networks. Networks 28(1), 31–43 (1996)
Cowell, R.G., Smith, J.Q.: Causal discovery through MAP selection of stratified Chain Event Graphs. Electron. J. Stat. 8, 965–997 (2014)
Darwiche, A.: A differential approach to inference in Bayesian networks. J. ACM 50(3), 280–305 (2003)
Dawid, A.P.: Conditional independence in statistical theory. J. Roy. Stat. Soc. B 41(1), 1–31 (1979)
Görgen, C., Smith, J.Q.: Equivalence Classes of Chain Event Graph Models. In preparation
Jordan, M.I.: Graphical models. Stat. Sci. 19(1), 140–155 (2004)
Lauritzen, S.L., Spiegelhalter, D.J.: Local computations with probabilities on graphical structures and their application to expert systems. J. Roy. Stat. Soc. B 50, 157–224 (1988)
Leonelli, M., Smith, J.Q., Riccomagno, E.: Using computer algebra for the symbolic evaluation of discrete influence diagrams. Technical report, CRISM (2015)
Pearl, J.: Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Francisco (1988)
Pearl, J.: Causality: Models, Reasoning and Inference. Cambridge University Press, Cambidge (2000)
Pistone, G., Riccomagno, E., Wynn, E.P.: Gröbner bases and factorisation in discrete probability and Bayes. Stat. Comput. 11, 37–46 (2001)
Riccomagno, E.: A short history of algebraic statistics. Metrika 69(2–3), 397–418 (2009)
Shafer, G.: The Art of causal Conjecture. MIT Press, Cambridge (1996)
Smith, J.Q.: Bayesian Decision Analysis: Principles and Practice. Cambridge University Press, Cambridge (2010)
Smith, J.Q., Anderson, P.E.: Conditional independence and Chain Event Graphs. Artif. Intell. 172, 42–68 (2008)
Thwaites, P.A., Smith, J.Q.: Separation theorems for Chain Event Graphs. CRiSM 11–09 (2011)
Thwaites, P.A., Smith, J.Q., Riccomagno, E.: Causal analysis with Chain Event Graphs. Artif. Intell. 174, 889–909 (2010)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Görgen, C., Leonelli, M., Smith, J.Q. (2015). A Differential Approach for Staged Trees. In: Destercke, S., Denoeux, T. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2015. Lecture Notes in Computer Science(), vol 9161. Springer, Cham. https://doi.org/10.1007/978-3-319-20807-7_31
Download citation
DOI: https://doi.org/10.1007/978-3-319-20807-7_31
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-20806-0
Online ISBN: 978-3-319-20807-7
eBook Packages: Computer ScienceComputer Science (R0)