Improving Construction of Conditional Probability Tables for Ranked
Improving Construction of Conditional Probability Tables for Ranked Nodes in Bayesian Networks Pekka Laitila, Kai Virtanen pekka. laitila@aalto. fi, kai. virtanen@aalto. fi Systems Analysis Laboratory, Department of Mathematics and Systems Analysis
Bayesian networks (BNs) Ø Represent uncertain knowledge Ø Reasoning under uncertainty
Bayesian networks (BNs) Ø Represent uncertain knowledge Ø Reasoning under uncertainty
Bayesian networks (BNs) Ø Represent uncertain knowledge Ø Reasoning under uncertainty Conditional probability tables (CPTs) Ø Quantify dependence between linked nodes Work Efficiency Skill Level Very High … Very Low Activity Level Very High … Very Low Disturbance Level Very High … Very Low Very High 0. 00 … 0. 00 High 0. 27 … 0. 00 Medium 0. 72 … 0. 00 Low 0. 01 … 0. 49 Very Low 0. 00 … 0. 51
Bayesian networks (BNs) Ø Represent uncertain knowledge Ø Reasoning under uncertainty Conditional probability tables (CPTs) Ø Quantify dependence between linked nodes Work Efficiency Skill Level Very High … Very Low Activity Level Very High … Very Low Disturbance Level Very High … Very Low Very High 0. 00 … 0. 00 High 0. 27 … 0. 00 Medium 0. 72 … 0. 00 Low 0. 01 … 0. 49 Very Low 0. 00 … 0. 51 Challenge of expert elicitation Construction of CPTs based on expert elicitation is time consuming and prone to biases
Bayesian networks (BNs) Ø Represent uncertain knowledge Ø Reasoning under uncertainty Conditional probability tables (CPTs) Ø Quantify dependence between linked nodes Work Efficiency Skill Level Very High … Very Low Activity Level Very High … Very Low Disturbance Level Very High … Very Low Very High 0. 00 … 0. 00 High 0. 27 … 0. 00 Medium 0. 72 … 0. 00 Low 0. 01 … 0. 49 Very Low 0. 00 … 0. 51 Probability elicitation in DA • Direct estimation • Probability scale methods • Gamble-like methods • Probability wheel • etc.
Bayesian networks (BNs) Ø Represent uncertain knowledge Ø Reasoning under uncertainty Conditional probability tables (CPTs) Ø Quantify dependence between linked nodes Work Efficiency Skill Level Very High … Very Low Activity Level Very High … Very Low Disturbance Level Very High … Very Low Very High 0. 00 … 0. 00 High 0. 27 … 0. 00 Medium 0. 72 … 0. 00 Low 0. 01 … 0. 49 Very Low 0. 00 … 0. 51 Probability elicitation in DA • Direct estimation • Probability scale methods Inadequate! • Gamble-like methods • Probability wheel • etc.
Construction of CPTs with parametric methods • Idea: 1. 2. Probabilistic relationship between nodes fits a standard pattern Expert assigns parameters characterizing the pattern CPT • Benefit: – Number of parameters ˂< Number of CPT entries Expert saves time!
Construction of CPTs with parametric methods • Idea: 1. 2. Probabilistic relationship between nodes fits a standard pattern Expert assigns parameters characterizing the pattern CPT • Benefit: – Number of parameters ˂< Number of CPT entries Expert saves time! • Challenges we have recognized in ”Ranked Nodes Method” (RNM) – – Parameters lack clear interpretations Hampers assignment Application requires technical insight Use inefficient
Construction of CPTs with parametric methods • Idea: 1. 2. Probabilistic relationship between nodes fits a standard pattern Expert assigns parameters characterizing the pattern CPT • Benefit: – Number of parameters ˂< Number of CPT entries Expert saves time! • Challenges we have recognized in ”Ranked Nodes Method” (RNM) – – • Parameters lack clear interpretations Hampers assignment Application requires technical insight Use inefficient Our contribution for alleviating efforts of the expert – – Interpretations that facilitate determination of parameters Guidelines for efficient use of RNM
Ranked Nodes (Fenton, Neil, and Caballero, 2007) Represent by ordinal scales continuous quantities that lack a well-established interval scale
Ranked Nodes Method (RNM) (Fenton, Neil, and Caballero, 2007) w 1 w 2 w 3 § Uncertainty parameter σ X 3: Disturbance Level 1 Very Low Medium 0. 2 0. 4 0. 6 0. 8 High 0 Very High Low Medium 0. 6 0. 8 Very High § Weights of parent nodes 0 0. 2 0. 4 High F 1 Y: Work Efficiency Medium § Aggregation function 0. 2 0. 4 0. 6 0. 8 Low Parameters to be elicited 0 Very Low 1 Very High Medium Low 0. 2 0. 4 0. 6 0. 8 Very Low 0 X 2: Activity Level Very Low X 1: Skill Level 1
Ranked Nodes Method (RNM) (Fenton, Neil, and Caballero, 2007) Parameters to be elicited § Aggregation function F w 2 w 3 § Uncertainty parameter σ X 3: Disturbance Level Very Low Medium Very High 1 TNormal( µ , σ , 0, 1), µ µ 0. 2 0. 4 0. 6 0. 8 x 3 Very High 0 High 1 Y: Work Efficiency § Weights of parent nodes w 1 High 0. 2 0. 4 0. 6 0. 8 x 2 Medium 0 Low 1 Very High Medium Low 0. 2 0. 4 0. 6 0. 8 x 1 Very Low 0 X 2: Activity Level Very Low X 1: Skill Level = F ( x 1 , x 2 , x 3 , w 1 , w 2 , w 3 ) P(Y=Low | X 1=Low, X 2=Medium, X 3=Low)
Challenges recognized with RNM 1. Parameters lack interpretations Values determined by trial and error Slow and difficult!
Challenges recognized with RNM 1. Parameters lack interpretations Values determined by trial and error Slow and difficult! 2. Application to nodes with interval scales: Ignorant discretizations may prevent construction of sensible CPTs
RNM and nodes with interval scales: New approach (Laitila and Virtanen, 2016) Formation of suitable ordinal scales 1. Divide interval scales freely into equal number of subintervals 2. Assess the mode of child node in scenarios corresponding to equal ordinal states of parent nodes Update discretizations accordingly ”What is the most likely rent for a 45 m 2 apartment that is 2 km from the centre and has 5 years since overhaul? ” ”I’d say its 1000 €. ”
RNM and nodes with interval scales: New approach (Laitila and Virtanen, 2016) Determination of aggregation function F and weights w 1, . . . , wn Assess the mode of child node in scenarios corresponding to extreme states of parent nodes F and w 1, . . . wn determined based on interpretations derived for weights ”What is the most likely rent for a 20 m 2 apartment that is right in the centre and has just been renovated? ” ”I’d say its 600 €. ”
Conclusion • Parametric methods ease up construction of CPTs for BNs by expert elicitation • New approach facilitates use of RNM Further relief to expert elicitation – Real-life application in performance model of air surveillance network – Applicable to BNs and Influence Diagrams Supports decision making under uncertainty • Future research – Human experiment: new approach vs. direct parameter estimation – Generalisation of the approach to nodes without interval scales
References • • • P. Laitila and K. Virtanen, “Improving Construction of Conditional Probability Tables for Ranked Nodes in Bayesian Networks, ” IEEE Transactions on Knowledge and Data Engineering, vol. 28, no. 7, pp. 1691– 1705, 2016 N. Fenton, M. Neil, and J. Caballero, “Using Ranked Nodes to Model Qualitative Judgments in Bayesian Networks, ” IEEE Transactions on Knowledge and Data Engineering, vol. 19, no. 10, pp. 1420– 1432, 2007 J. Pearl, Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, 1988 F. Diez, “Parameter Adjustment in Bayes Networks. The Generalized Noisy Or. Gate, ” in Proceedings of the 9 th Conference on Uncertainty in Artificial Intelligence. Washington, D. C. , USA, July 9 -11, 1993, pp. 99– 105 S. Srinivas, “A Generalization of the Noisy-Or Model, ” in Proceedings of the 9 th Conference on Uncertainty in Artificial Intelligence. Washington, D. C. , USA, July 9– 11, 1993, pp. 208– 215 S. Russell and P. Norvig, Artificial Intelligence: A Modern Approach. Prentice Hall, 2003
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