NTNU Bayesian Networks in Reliability: A primer Helge Langseth helgel@math.ntnu.no Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway MMR 2004 – p.1/13
Outline NTNU Basics of Bayesian networks -framework Definition Representation Calculation scheme Apparent shortcomings: Sparser representations Knowledge acquisition Continuous variables Some cute features Causal models Model estimation Reliability applications MMR 2004 – p.2/13
A simple example: “Car start” NTNU B : Battery F : Fuel level H : Head-lights G : Fuel gauge E : Engine turns C : Car starts P ( B, F, H, G, E, C ) MMR 2004 – p.3/13
A simple example: “Car start” NTNU B : Battery B : Battery F : Fuel level H : Head-lights G : Fuel gauge E : Engine turns E : Engine turns pa( E ) = { B } C : Car starts P ( B, F, H, G, E, C ) MMR 2004 – p.3/13
A simple example: “Car start” NTNU B : Battery B : Battery B : Battery F : Fuel level F : Fuel level H : Head-lights H : Head-lights G : Fuel gauge G : Fuel gauge E : Engine turns E : Engine turns pa( E ) = { B } nd( E ) = { H, G, F, B } C : Car starts P ( B, F, H, G, E, C ) MMR 2004 – p.3/13
A simple example: “Car start” NTNU B : Battery B : Battery B : Battery B : Battery F : Fuel level F : Fuel level H : Head-lights H : Head-lights G : Fuel gauge G : Fuel gauge E : Engine turns E : Engine turns pa( E ) = { B } nd( E ) = { H, G, F, B } C : Car starts X ⊥ ⊥ nd( X ) \ pa( X ) | pa( X ) E ⊥ ⊥{ H, G, F } | B Other d-sep. rules: Pearl(88) P ( B, F, H, G, E, C ) MMR 2004 – p.3/13
A simple example: “Car start” NTNU B = empty B � = empty E . 01 . 97 yes B : Battery B : Battery B : Battery B : Battery B : Battery F : Fuel level F : Fuel level F : Fuel level . 99 . 03 no H : Head-lights H : Head-lights H : Head-lights G : Fuel gauge G : Fuel gauge G : Fuel gauge E : Engine turns E : Engine turns E : Engine turns E : Engine turns E : Engine turns P ( E | pa ( E )) pa( E ) = { B } nd( E ) = { H, G, F, B } C : Car starts C : Car starts X ⊥ ⊥ nd( X ) \ pa( X ) | pa( X ) E ⊥ ⊥{ H, G, F } | B Other d-sep. rules: Pearl(88) = P ( B ) P ( F ) P ( H | B ) P ( G | F ) P ( B, F, H, G, E, C ) · P ( E | B ) P ( C | E, F ) Markov properties ⇔ Factorization property MMR 2004 – p.3/13
They crop up everywhere NTNU M latent, multinomial M Y p ( y | M = m ) Mixture models MMR 2004 – p.4/13
They crop up everywhere NTNU X latent, N ( 0 , I ) X 1 X 2 Linear regression Y 1 Y 2 Y 3 Y | x ∼ N ( µ + Lx , Ψ ) , Ψ diagonal Factor analyzers MMR 2004 – p.4/13
They crop up everywhere NTNU X latent, N ( 0 , I ) M latent, multinomial X 1 X 2 M Linear regression given M = m Y 1 Y 2 Y 3 Y | { x , M = m } ∼ N ( µ m + L m x , Ψ m ) , Ψ m diagonal Mixture of Factor analyzers MMR 2004 – p.4/13
Calculation scheme NTNU X 1 X 2 X 3 X 4 X 5 MMR 2004 – p.5/13
Calculation scheme NTNU X 1 X 2 X 1 X 2 X 3 X 4 X 3 X 4 X 5 X 5 MMR 2004 – p.5/13
Calculation scheme NTNU X 1 X 2 X 1 X 2 X 1 X 2 X 3 X 4 X 3 X 4 X 3 X 4 X 5 X 5 X 5 MMR 2004 – p.5/13
Calculation scheme NTNU X 1 X 2 X 1 X 2 X 1 X 2 X 1 X 2 X 3 X 4 X 3 X 4 X 3 X 4 X 3 X 4 X 5 X 5 X 5 X 5 MMR 2004 – p.5/13
Calculation scheme NTNU X 1 X 2 X 1 X 2 X 1 X 2 X 1 X 2 X 3 X 4 X 3 X 4 X 3 X 4 X 3 X 4 X 5 X 5 X 5 X 5 ψ 1 ( x 3 , x 4 ) ψ 2 ( x 1 , x 4 ) φ 1 ( x 3 , x 4 , x 5 ) φ 2 ( x 1 , x 3 , x 4 ) φ 3 ( x 1 , x 2 , x 4 ) X 3 , X 4 , X 5 X 1 , X 3 , X 4 X 1 , X 2 , X 4 X 3 , X 4 X 1 , X 4 ‘Divide-and-Conquer’ strategy: We can look at 3 variables at a time instead of 5 . Important, as the cost is exponential in # variables. MMR 2004 – p.5/13
Cheaper representations NTNU Consider a binary node with m Y binary parents. The CPT requires 2 m parameters. This . . . must be bad, right? Z 1 Z m MMR 2004 – p.6/13
Cheaper representations NTNU Consider a binary node with m Y binary parents. The CPT requires 2 m parameters. This must be bad, right? Wrong! . . . Z 1 Z m All parameters are required if we do not make additional assumptions! But: p ( y | z 1 , . . . , z m ) Functional relations z i = 0 z i = 1 ( if-then-else , AND -gates, ...). p 1 Sparser representations z j = 1 z j = 0 than CPTs, e.g., probability trees. p 2 . . . MMR 2004 – p.6/13
Continuous variables NTNU Not all families of distributions can be handled by the calculation scheme. Works for: Multinomial variables Multivariate Gaussian distributions Conditional Gaussian distributions What can we do when these models are unrealistic? MMR 2004 – p.7/13
Continuous variables NTNU Not all families of distributions can be handled by the calculation scheme. Works for: Multinomial variables Multivariate Gaussian distributions Conditional Gaussian distributions What can we do when these models are unrealistic? Discretization: Difficult tradeoff between precision and model complexity Mixtures of Truncated exponentials: A new family of distributions that can cope with the calculation scheme. Can approximate any distribution arbitrarily well MMR 2004 – p.7/13
KA: When p ( x i | pa ( x i )) is not available NTNU All elements of the set { p ( x i | pa ( x i )) } n i =1 are required to fully specify a BN. Experts sometimes prefer to give { p ( x i ) } n i =1 and correlations (e.g., in the form of cross-product ratios) instead. Alternative frameworks: Vines, Chain graphs, ... Iterative proportional fitting procedure: 1. p 0 ( x i , x j ) initialized to obtain correct correlation. 2. for k = 1 , 2 , . . . : � j p k − 1 ( x i ,x j ) ( i ) p ′ k ( x i , x j ) ← p k − 1 ( x i , x j ) · p ( x i ) � i p ′ k ( x i ,x j ) ( ii ) p k ( x i , x j ) ← p ′ k ( x i , x j ) · p ( x j ) MMR 2004 – p.8/13
KA: When p ( x i | pa ( x i )) is not available NTNU All elements of the set { p ( x i | pa ( x i )) } n i =1 are required to fully specify a BN. Experts sometimes prefer to give { p ( x i ) } n i =1 and correlations (e.g., in the form of cross-product ratios) instead. Alternative frameworks: Vines, Chain graphs, ... Iterative proportional fitting procedure: In BNs: Work with the cliques! Iterate over cliques l : p k ( x ) ← p k − 1 ( x ) p k − 1 ( x l ) p ( x l ) Gives minimum info model It also works for inconsistent input MMR 2004 – p.8/13
Calculating causal (!) effects NTNU Cause Effect Can we estimate causal strength? The key is that p ( x | observe ( Y = y )) = p ( x | do ( Y ← y )) not holds in general! MMR 2004 – p.9/13
Calculating causal (!) effects NTNU Management TTF Planned PM Can we estimate causal strength? NO! Destroyed by confounder The key is that p ( x | observe ( Y = y )) = p ( x | do ( Y ← y )) not holds in general! MMR 2004 – p.9/13
Calculating causal (!) effects NTNU Management TTF Planned PM Actual PM Can we estimate causal strength? Yes! Intermediate (observable) effect saves the day! The key is that p ( x | observe ( Y = y )) = p ( x | do ( Y ← y )) not holds in general! MMR 2004 – p.9/13
Estimating models from data NTNU Estimating parameters: No missing values: Counting Missing values: EM-algorithm Estimating structure: Only discrete (or discretized) variables: Constrain-based (hypothesis testing) Fully Bayesian approach Continuous/Mixed variables: Purely Gaussian and conditional Gaussian models: “Simple” General distributions: Difficult MMR 2004 – p.10/13
Troubleshooting NTNU Find a “useful” repair strategy; i.e. , a sequence of steps with a low expected cost of repair Example: The BATS system (developed by HP). Can be employed in many domains, initially intended for troubleshooting printers Bobbio et al. ’s FTA ⇒ BN algorithm gives troubleshooter systems new expressive power: Refined system models Common cause failures NOT - events Modeling user interaction Non-perfect repair actions Questions MMR 2004 – p.11/13
Safety-critical software NTNU Safety-assessment requires: Safety-critical software is special: Disparate sources of information; several types of evidence, many which are not quantitative in No-fault-criterion: Bugs nature. immediately removed The relation between evidence and safety Test in a traditional way assessment is not always direct or quantifiable is not sufficient (PIE algorithm; 50% of tested We need a framework to combine these locations hide their disparate sources of information in a faults). transparent way Qsystem Complexity Testing Experience Reliability Consequences Fault tol. System safety MMR 2004 – p.12/13
Safety-critical software NTNU Safety-assessment requires: Safety-critical software is special: Disparate sources of information; several types of evidence, many which are not quantitative in No-fault-criterion: Bugs nature. immediately removed The relation between evidence and safety Test in a traditional way assessment is not always direct or quantifiable is not sufficient (PIE algorithm; 50% of tested We need a framework to combine these locations hide their disparate sources of information in a faults). transparent way Each node in the top-level model is refined using a “sub-net” The safety standard for safety critical software in aviation (RTCA/DO-17B) was implemented in this way See Gran (2002) for details MMR 2004 – p.12/13
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