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Applying Bayesian networks. Risk forecasting & examples. The Key Problems. Rule Based decision-support systems cannot handle uncertainty Regression-based prediction systems cannot handle complex cause-effect relationships How to combine different types of evidence
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Applying Bayesian networks Risk forecasting & examples
The Key Problems Rule Based decision-support systems cannot handle uncertainty Regression-based prediction systems cannot handle complex cause-effect relationships How to combine different types of evidence How to combine both qualitative and quantitative information to arrive at a quantitative risk assessment How to make visible and auditable the assumptions of the assessor How to achieve more confidence in quantitative arguments
Bayesian Belief Nets (BBNs) Powerful graphical framework in which to reason about uncertainty using diverse forms of evidence Nodes of graph represent uncertain variables Arcs of graph represent casual or influential relationships between the variables Associated with each node is a probability table (CPT)
A very simple BBN Smoker? Has Lung Cancer Has Bronchitis
Bayes’ Theorem A: ‘Person has cancer’ p(A) = 0.1 (prior) B: ‘Person is smoker’ p(B) = 0.5 What is p(A\B)? (posterior) p(B\A) = 0.8 (likelihood) P(A\B).P(B) = P(B\A).P(A) P(A\B) = P(B\A).P(A)/P(B) So p(A\B) = 0.16
Bayesian Propagation Applying Bayes theorem to update all the probabilities when new evidence is entered Intractable even for small BBNs Breakthrough in late 1980s - fast algorithm Tools like Hugin implement efficient propagation Propagation is multi-directional Make predictions even with missing/incomplete data
BBN applications - external • Microsoft automated decision support: • Office 95 (and later) help wizards • Customer support/diagnostics • Hewlett Packard - fault diagnosis • NASA space shuttle VISTA system (display relevant telemetry data) • MUNIN system for medical diagnosis
A BBN safety argument System safety Faults in test/review Operational usage Intrinsic complexity Accuracy of testing Correctness of solution Complexity of solution System criticality Quality of supplier Quality of test team
Generic problems of building a BBN • Defining the BBN topology • What is the ‘right’ collection of nodes and arcs? • Use ‘idioms’ and join operations • Defining the Node Probability Table (CPTs) • Benefit of BBNs is that we can use empirical AND subjective data, but how to deal with combinatorial explosion and continuous variables? • Elicitation process that extrapolates a complete NPT based on a small number of inputs • Incorporating probabilistic and deterministic functions • Building BBN from database
BBNs and data-mining BBNs offer classic solution to data-mining problem Tools for constructing ‘optimal BBN’ from large databases Improved predictions over classical regression-based approaches
Summary of BBN Benefits Best Method for reasoning under uncertainty Computational tractability issues have largely been solved (unlike, e.g. neural nets) so BBNs can be used NOW on real, large-scale problems Can combine diverse data, including subjective beliefs and empirical data Can enter incomplete evidence and still obtain prediction Perform powerful ‘what-if’ analysis to test sensitivity of conclusions Visual reasoning tool and a major documentation aid
Advantages Over Classical statistics More adaptable to changes in risk characteristics Dynamic, as new risks are rated the Network learns Broad range of risk characteristics taken into account Allows investigations of risk characteristics to ultimate premium rates Avoids predetermined relationships as these are determined directly from experience Adds value to the Risk Management process
Never trust your guesses • The birthday puzzle What is the chance that in a group of 36 randomly selected people, two or more will be found to share the same birthday? The neatest way to work out the exact solution is to calculate 1 minus the probability that all 36 people will have different birthdays: 1-[(364x363x…x330)/36535]
Never trust your guesses • The birthday puzzle • The false-positive puzzle You are given the following:a) in random testing, you test positive for a disease,b) in 5% of cases, this test shows positive even when the subject does not have the disease,c) in the population at large, one person in 1000 has the disease. What is the probability that you have the disease?
Never trust your guesses • The birthday puzzle • The false-positive puzzle • The Monty Hall puzzle This is based on an old American game show in which contestants were offered a choice of three boxes. Open the correct one and you won a car; open either of the others and you won a goat. There was a twist, after the contestant had chosen, but before the box was opened, the host opened one of the other boxes to reveal a goat. Then he asked of the contestant wanted to stick with his first choice, or change his mind and open the third box instead. Question: is it a good idea to change your mind, a bad idea, or does it make no difference?
Getting the Goat The door containing the prize is known to Monty and thus “Prize” has an impact on “Monty Opens”. Monty will never choose to open the door of your first selection so also “First Selection “has impact on “Monty Opens”. This give us the BBN shown in the figure opposite.
