Bayesian Networks in Reliability The Good, the Bad, and the Ugly Helge Langseth Department of Computer and Information Science Norwegian University of Science and Technology MMR’07 1 Helge Langseth Bayesian Networks in Reliability
Outline Introduction 1 The Good: Why Bayesian Nets are popular 2 Mathematical properties Making decisions Applications The Bad: Building complex quantitative models 3 The model building process The quantitative part Utility theory The Ugly: Continuous variables 4 Introduction Approximations Concluding remarks 5 2 Helge Langseth Bayesian Networks in Reliability
Introduction A simple example: “Explosion” E : Environment L : Leak G : GD failed X : Explosion C : Casualties P ( E, L, G, X, C ) 3 Helge Langseth Bayesian Networks in Reliability
Introduction A simple example: “Explosion” E : Environment L : Leak L : Leak G : GD failed G : GD failed X : Explosion X : Explosion pa ( X ) = { L, G } C : Casualties P ( E, L, G, X, C ) 3 Helge Langseth Bayesian Networks in Reliability
Introduction A simple example: “Explosion” E : Environment E : Environment L : Leak L : Leak L : Leak G : GD failed G : GD failed G : GD failed X : Explosion X : Explosion pa ( X ) = { L, G } nd( X ) = { E, L, G } C : Casualties P ( E, L, G, X, C ) 3 Helge Langseth Bayesian Networks in Reliability
Introduction A simple example: “Explosion” E : Environment E : Environment L : Leak L : Leak L : Leak L : Leak G : GD failed G : GD failed G : GD failed G : GD failed X : Explosion X : Explosion X : Explosion pa ( X ) = { L, G } pa ( X ) = { L, G } nd( X ) = { E, L, G } C : Casualties X ⊥ ⊥ E | { L, G } Other d-sep. rules: Jensen&Nielsen (07) P ( E, L, G, X, C ) 3 Helge Langseth Bayesian Networks in Reliability
Introduction A simple example: “Explosion” G E = hostile E = normal E : Environment E : Environment E : Environment λ H · τ/ 2 λ N · τ/ 2 yes 1 − λ H · τ/ 2 1 − λ N · τ/ 2 no L : Leak L : Leak L : Leak L : Leak L : Leak G : GD failed G : GD failed G : GD failed G : GD failed G : GD failed P ( G | pa ( G )) X : Explosion X : Explosion X : Explosion X : Explosion X : Explosion pa ( X ) = { L, G } pa ( X ) = { L, G } nd( X ) = { E, L, G } C : Casualties C : Casualties � � Hence, P ( X | E, L, G ) = P ( X | L, G ) X ⊥ ⊥ E | { L, G } Other d-sep. rules: Jensen&Nielsen (07) P ( E, L, G, X, C ) P ( E, L, G, X, C ) = P ( E ) · P ( L | E ) · P ( G | E, L ) · P ( X | E, L, G ) · P ( C | E, L, G, X ) = P ( E ) · P ( L | E ) · P ( G | E ) · P ( X | L, G ) · P ( C | X ) Markov properties ⇔ Factorization property 3 Helge Langseth Bayesian Networks in Reliability
The Good: Why Bayesian Nets are popular Outline Introduction 1 The Good: Why Bayesian Nets are popular 2 Mathematical properties Making decisions Applications The Bad: Building complex quantitative models 3 The model building process The quantitative part Utility theory The Ugly: Continuous variables 4 Introduction Approximations Concluding remarks 5 4 Helge Langseth Bayesian Networks in Reliability
The Good: Why Bayesian Nets are popular Mathematical properties What the mathematical foundation has to offer Intuitive representation: Almost defined as “box-diagram with formal meaning”. Causal interpretation natural in many cases. Efficient representation: The number of required parameters are reduced. If all variables are binary, the example requires 11 “local” parameters, compared to the 31 “global” parameters of the full joint. Efficient calculations: Efficient calculations of any joint distribution P ( x i , x j ) or conditional distribution P ( x k | x ℓ , x m ) . Model estimation: Estimating parameters (fixed structure) via EM, estimating structure by discrete optimization techniques. 5 Helge Langseth Bayesian Networks in Reliability
The Good: Why Bayesian Nets are popular Making decisions Influence diagrams: The “Explosion” example revisited Cost 1 E : Environment L : Leak G : GD failed SSM X : Explosion C : Casualties Cost 2 6 Helge Langseth Bayesian Networks in Reliability
The Good: Why Bayesian Nets are popular Making decisions Influence diagrams: The “Explosion” example revisited Cost 1 E : Environment L : Leak G : GD failed SSM X : Explosion F : Effectiveness, SSM C : Casualties Test interval Cost 2 Cost 3 6 Helge Langseth Bayesian Networks in Reliability
The Good: Why Bayesian Nets are popular Applications An application: Troubleshooting 7 Helge Langseth Bayesian Networks in Reliability
The Good: Why Bayesian Nets are popular Applications Underlying model TOP C 1 C 2 C 3 C 4 X 1 X 2 X 3 X 4 X 5 system-layer 8 Helge Langseth Bayesian Networks in Reliability
The Good: Why Bayesian Nets are popular Applications Underlying model TOP C 1 C 2 C 3 C 4 X 1 X 2 X 3 X 4 X 5 system-layer E 8 Helge Langseth Bayesian Networks in Reliability
The Good: Why Bayesian Nets are popular Applications Underlying model TOP C 1 C 2 C 3 C 4 X 1 X 1 X 2 X 2 X 3 X 3 X 4 X 4 X 5 X 5 system-layer A 1 A 2 A 3 A 4 A 5 action-layer 8 Helge Langseth Bayesian Networks in Reliability
The Good: Why Bayesian Nets are popular Applications Underlying model TOP C 1 C 1 C 2 C 2 C 3 C 3 C 4 C 4 X 1 X 1 X 2 X 2 X 3 X 3 X 4 X 4 X 5 X 5 system-layer A 1 A 1 A 2 A 2 A 3 A 3 A 4 A 4 A 5 A 5 action-layer R 1 R 2 R 3 R 4 R 5 result-layer 8 Helge Langseth Bayesian Networks in Reliability
The Good: Why Bayesian Nets are popular Applications Underlying model Q S question-layer TOP C 1 C 1 C 1 C 2 C 2 C 2 C 3 C 3 C 4 C 4 X 1 X 1 X 2 X 2 X 3 X 3 X 3 X 4 X 4 X 5 X 5 X 5 system-layer A 1 A 1 A 2 A 2 A 3 A 3 A 4 A 4 A 5 A 5 action-layer R 1 R 2 R 3 R 4 R 5 result-layer 8 Helge Langseth Bayesian Networks in Reliability
The Good: Why Bayesian Nets are popular Applications Other applications Software reliability Modelling Organizational factors (e.g., the SAM-Framework) Explicit models of dynamics (e.g., repairable systems, phase-mission-systems, monitoring systems) Some of these can be seen at the Bayes net sessions later today 9 Helge Langseth Bayesian Networks in Reliability
The Bad: Building complex quantitative models Outline Introduction 1 The Good: Why Bayesian Nets are popular 2 Mathematical properties Making decisions Applications The Bad: Building complex quantitative models 3 The model building process The quantitative part Utility theory The Ugly: Continuous variables 4 Introduction Approximations Concluding remarks 5 10 Helge Langseth Bayesian Networks in Reliability
The Bad: Building complex quantitative models The model building process Phases of the model building process Step 0 – Decide what to model: Select the boundary for what to include in the model. Step 1 – Defining variables: Select the important variables in the domain. Step 2 – The qualitative part: Define the graphical structure that connects the variables. Step 3 – The quantitative part: Fix parameters to specify each P ( x i | pa ( x i )) . This is the ‘bad’ part. Step 4 – Verification: Verification of the model. 11 Helge Langseth Bayesian Networks in Reliability
The Bad: Building complex quantitative models The quantitative part The quantitative part: Defining P ( y | pa ( y )) . . . Z 1 Z 2 Z m Consider a binary node with m binary parents. The CPT P ( y | z 1 , . . . , z m ) contains 2 m parameters. Y Naïve approach: 2 m conditional probabilities: All parameters are required if no other assumptions can be made. 12 Helge Langseth Bayesian Networks in Reliability
The Bad: Building complex quantitative models The quantitative part The quantitative part: Defining P ( y | pa ( y )) . . . Z 1 Z 2 Z m Consider a binary node with m binary parents. The CPT P ( y | z 1 , . . . , z m ) contains 2 m parameters. Y Naïve approach: 2 m conditional probabilities Deterministic relations: Parameter free: Y considered a function of its parents, e.g., { Y = fail } ⇐ ⇒ { Z 1 = fail }∨{ Z 2 = fail }∨ . . . ∨{ Z m = fail } . 12 Helge Langseth Bayesian Networks in Reliability
The Bad: Building complex quantitative models The quantitative part The quantitative part: Defining P ( y | pa ( y )) . . . Z 1 Z 2 Z m Consider a binary node with m binary parents. The CPT P ( y | z 1 , . . . , z m ) contains 2 m parameters. Y Naïve approach: 2 m conditional probabilities Deterministic relations: Parameter free Noisy OR relation: m + 1 conditional probabilities: Independent inhibitors Q 1 , . . . , Q m ; Assume . . . Z 1 Z 2 Z m { Q 1 = fail }∨ . . . ∨{ Q m = fail } = ⇒ { Y = fail } . For each Q i we have . . . Q 1 Q 2 Q m P ( Q i = fail | Z i = fail ) = q i , P ( Q i = fail | Z i = ¬ fail ) = 0 . Y “Leak probability”: P ( Y = fail | Q 1 = . . . = Q m = ¬ fail ) = q 0 . 12 Helge Langseth Bayesian Networks in Reliability
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