Abstract
Influence diagrams provide a compact graphical representation of decision problems. Several algorithms for the quick computation of their associated expected utilities are available in the literature. However, often they rely on a full quantification of both probabilistic uncertainties and utility values. For problems where all random variables and decision spaces are finite and discrete, here we develop a symbolic way to calculate the expected utilities of influence diagrams that does not require a full numerical representation. Within this approach expected utilities correspond to families of polynomials. After characterizing their polynomial structure, we develop an efficient symbolic algorithm for the propagation of expected utilities through the diagram and provide an implementation of this algorithm using a computer algebra system. We then characterize many of the standard manipulations of influence diagrams as transformations of polynomials. We also generalize the decision analytic framework of these diagrams by defining asymmetries as operations over the expected utility polynomials.
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References
Bhattacharjya, D., Shachter, R. D.: Sensitivity Analysis in Decision Circuits Proceedings of the 24th Conf. Uncertainty in Artif. Intel, pp 34–42 (2008)
Bhattacharjya, D., Shachter, R. D.: Three New Sensitivity Analysis Methods for Influence Diagrams Proc. 26Th Conf. Uncertainty in Artif. Intel., pp 56–64 (2010)
Bhattacharjya, D., Shachter, R. D.: Formulating asymmetric decision problems as decision circuits. Decis. Anal. 9, 138–145 (2012)
Bielza, C., Gómez, M., Shenoy, P. P.: A review of representation issues and modeling challenges with influence diagrams. Omega 39, 227–241 (2011)
Blekherman, G., Parrilo, P. A., Thomas, R. R.: Semidefinite optimization and convex algebraic geometry. Siam philadelphia (2013)
Borgonovo, E., Tonoli, F.: Decision-network polynomials and the sensitivity of decision-support models. Eur. J. Oper. Res. 239, 490–503 (2014)
de Campos, C. P., Cozman, F. G.: Inference in Credal Networks Using Multilinear Programming Proceedings of the 2Nd Starting AI Researcher Symp., pp 50–61 (2004)
de Campos, C. P., Ji, Q.: Strategy Selection in Influence Diagrams Using Imprecise Probabilities Proceedings of the 24Th Conf. Uncertainty in Artif. Intel., pp 121–128 (2008)
Castillo, E., Gutierrez, J. M., Hadi, A. S.: Parametric Structure of Probabilities in Bayesian Networks ECSQARU 1995, pp 89–98. Springer (1995)
Castillo, E., Gutierrez, J. M., Hadi, A. S.: Sensitivity analysis in discrete Bayesian networks. IEEE T. Syst. Man. Cy. A 27, 412–423 (1997)
Castillo, E., Gutierrez, J. M., Hadi, A. S.: Expert Systems and Probabilistic Network Models. Springer, New York (2012)
Castillo, E., Kjærulff, U.: Sensitivity analysis in Gaussian Bayesian networks using a symbolic-numerical technique. Reliab. Eng. Syst. Safe. 79, 139–148 (2003)
Chan, H., Darwiche, A.: When Do Numbers Really Matter? Proceedings of the 17th Conf. Uncertainty in Artif. Intel., pp 65–74 (2001)
Chan, H., Darwiche, A.: Sensitivity Analysis in Bayesian Networks: from Single to Multiple Parameters Proceedings of the 20Th Conf. Uncertainty in Artif. Intel., pp 67–75 (2004)
Coupé, V. M. H., van der Gaag, L. C.: Properties of sensitivity analysis of Bayesian belief networks. Ann. Math. Artif. Intel. 36, 323–356 (2002)
Cox, D. A., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. Springer, New York (2007)
Cozman, F. G.: Credal networks. Artif. Intel. 120, 199–233 (2000)
Darwiche, A.: A differential approach to inference in Bayesian networks. J. ACM 50, 280–305 (2003)
Dawid, A. P., Constantinou, P.: A formal treatment of sequential ignorability. Stat. Biosci. 6, 166–188 (2014)
Demirer, R., Shenoy, P. P.: Sequential valuation networks for asymmetric decision problems. Eur. J. Oper. Res. 169, 286–309 (2006)
Felli, J. C., Hazen, G. B.: Javelin diagrams: a graphical tool for probabilistic sensitivity analysis. Decis. Anal. (2) (2004)
French, S.: Readings in Decision Analysis. CRC Press, Boca Raton (1989)
van der Gaag, L. C., Renooij, S., Coupé, V. M. H.: Sensitivity Analysis of Probabilistic Networks Advances in Probabilistic Graphical Models, pp 103–124. Springer (2007)
Görgen, C., Leonelli, M., Smith, J. Q.: A Differential Approach for Staged Trees ECSQARU 2015, pp 346–355. Springer (2015)
Howard, R.: The foundations of decision analysis. IEEE T. Syst. Man. Cyb. 4, 211–219 (1968)
Howard, R. A., Matheson, J. E.: Influence diagrams. Decis. Anal. 2, 127–143 (2005)
Jensen, F., Jensen, F. V., Dittmer, S. L.: From Influence Diagrams to Junction Trees Proceedings 10th Conf. Uncertainty in Artif. Intel., pp 367–373 (1994)
Jensen, F. V., Nielsen, T. D., Shenoy, P. P.: Sequential influence diagrams: a unified asymmetry framework. Int. J. Approx. Reason. 42, 101–118 (2006)
Keeney, R. L.: Multiplicative utility functions. Oper. Res. 22, 22–34 (1974)
Keeney, R. L., Raiffa, H.: Decision with Multiple Objectives. Cambridge University Press, Cambridge (1976)
Kikuti, D., Cozman, F. G., Shirota Filho, R.: Sequential decision making with partially ordered preferences. Artif. Intel. 175, 1346–1365 (2011)
Koller, D., Friedman, N.: Probabilistic Graphical Models. MIT press, Cambridge (2009)
Leonelli, M., Görgen, C., Smith, J. Q.: Sensitivity analysis, multilinearity and beyond. Tech. rep., arXiv:1512.02266 (2015)
Leonelli, M., Smith, J. Q.: Bayesian decision support for complex systems with many distributed experts. Ann. Oper. Res. 235, 517–542 (2015)
Nielsen, T. D., Jensen, F. V.: Sensitivity analysis in influence diagrams. IEEE T. Syst. Man. Cy. A 33, 223–234 (2003)
Nielsen, T. D., Jensen, F. V.: Bayesian Networks and Decision Graphs. Springer, New York (2009)
Pearl, J.: Probabilistic Reasoning in Intelligent Systems. Morgan Kaufmann, San Francisco (1988)
Sanner, S., Kersting, K.: Symbolic Dynamic Programming for First-Order POMDPs Proceedings 24Th AAAI Conf. Artif. Intel., pp 1140–1146 (2010)
Shachter, R. D.: Evaluating influence diagrams. Oper. Res. (6) (1986)
Shachter, R. D.: An ordered examination of influence diagrams. Networks (20) (1990)
Smith, J. E., Holtzman, S., Matheson, J. E.: Structuring conditional relationships in influence diagrams. Oper. Res. 41, 280–297 (1993)
Smith, J. Q.: Influence diagrams for Bayesian decision analysis. Eur. J. Oper. Res. 40, 363–376 (1989)
Smith, J. Q.: Influence diagrams for statistical modelling. Ann. Stat. 17, 654–672 (1989)
Smith, J. Q.: Bayesian Decision Analysis: Principles and Practice. Cambridge University Press, Cambridge (2010)
Smith, J. Q., Barons, M. J., Leonelli, M.: Coherent frameworks for statistical inference serving integrating decision support systems. Tech. rep., CRISM15-10, Warwick University (2015)
Tatman, J. A., Shachter, R. D.: Dynamic programming and influence diagrams. IEEE Trans. Systems Man Cybernet. 20, 365–379 (1990)
Zamani, Z., Sanner, S., Fang, C.: Symbolic Dynamic Programming for Continuous State and Action MDPs Proceedings 26th AAAI Conf. Artif. Intel., pp 1839–1845 (2012)
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Manuele Leonelli was funded by Capes, whilst J.Q. Smith was partly supported by EPSRC grant EP/K039628/1 and The Alan Turing Institute under EPSRC grant EP/N510129/1.
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Leonelli, M., Riccomagno, E. & Smith, J.Q. A symbolic algebra for the computation of expected utilities in multiplicative influence diagrams. Ann Math Artif Intell 81, 273–313 (2017). https://doi.org/10.1007/s10472-017-9553-y
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DOI: https://doi.org/10.1007/s10472-017-9553-y