Bayesian Networks in Reliability - Department of Computer and ...

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

2

1

Introduction

2

The Good: Why Bayesian Nets are popular Mathematical properties Making decisions Applications

3

The Bad: Building complex quantitative models The model building process The quantitative part Utility theory

4

The Ugly: Continuous variables Introduction Approximations

5

Concluding remarks Helge Langseth


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


Introduction


L: Leak

G: GD failed

X: Explosion

pa (X) = {L, G}

C: Casualties

P (E, L, G, X, C)

3

Helge Langseth


Introduction


L: Leak

G: GD failed

X: Explosion

pa (X) = {L, G} nd(X) = {E, L, G}

C: Casualties

P (E, L, G, X, C)

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Helge Langseth


Introduction


L: Leak

G: GD failed

X: Explosion

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)

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Helge Langseth


Introduction

A simple example: “Explosion” G yes no

E: Environment

L: Leak

G: GD failed

X: Explosion

E =hostile λH · τ /2 1 − λH · τ /2

E =normal λN · τ /2 1 − λN · τ /2

P (G | pa (G))

pa (X) = {L, G} nd(X) = {E, L, G}

C: Casualties

X⊥⊥E | {L, G} Hence, P (X | E, L, G) = P (X | L, G) Other d-sep. rules: Jensen&Nielsen (07)

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)

Markov properties ⇔ Factorization property 3

Helge Langseth


· P (C | X)

The Good: Why Bayesian Nets are popular

Outline

4

1

Introduction

2


3


4


5




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 (xi , xj ) or conditional distribution P (xk | xℓ , xm ). Model estimation: Estimating parameters (fixed structure) via EM, estimating structure by discrete optimization techniques.

5

Helge Langseth



Making decisions

Influence diagrams: The “Explosion” example revisited Cost1

E: Environment

L: Leak

G: GD failed

X: Explosion

C: Casualties

Cost2

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Helge Langseth


SSM


Making decisions

Influence diagrams: The “Explosion” example revisited Cost1

E: Environment

L: Leak

6

SSM

G: GD failed

X: Explosion

F : Effectiveness, SSM

C: Casualties

Test interval

Cost2

Cost3

Helge Langseth



Applications

An application: Troubleshooting

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Helge Langseth



Applications

Underlying model

TOP C1 X1

8

C2 X2

C3 X3

Helge Langseth

C4 X4

X5


system-layer


Applications

Underlying model

TOP C1 X1

C2 X2

C3 X3

C4 X4

X5

E

8

Helge Langseth


system-layer


Applications

Underlying model

TOP C1

8

C2

C3

C4

X1

X2

X3

X4

X5

A1

A2

A3

A4

A5

Helge Langseth


system-layer

action-layer


Applications

Underlying model

TOP C1

8

C2

C3

C4

X1

X2

X3

X4

X5

A1

A2

A3

A4

A5

R1

R2

R3

R4

R5

Helge Langseth


system-layer

action-layer result-layer


Applications

Underlying model QS

question-layer TOP

C1

8

C2

C3

C4

X1

X2

X3

X4

X5

A1

A2

A3

A4

A5

R1

R2

R3

R4

R5

Helge Langseth


system-layer

action-layer result-layer


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

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Helge Langseth


The Bad: Building complex quantitative models

Outline

10

1

Introduction

2


3


4


5




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 (xi | pa (xi )). This is the ‘bad’ part. Step 4 – Verification: Verification of the model.

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Helge Langseth



The quantitative part

The quantitative part: Defining P (y|pa (y)) Z1

...

Z2

Consider a binary node with m binary parents. The CPT P (y|z1 , . . . , zm ) contains 2m parameters. Y

2m

Naïve approach: conditional probabilities: All parameters are required if no other assumptions can be made.

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Helge Langseth


Zm




...

Z2

Zm


Naïve approach:

2m

conditional probabilities

Deterministic relations: Parameter free: Y considered a function of its parents, e.g., {Y = fail} ⇐⇒ {Z1 = fail}∨{Z2 = fail}∨. . .∨{Zm = fail}.

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Helge Langseth





...

Z2

Zm


Naïve approach:

2m


Deterministic relations: Parameter free Z1

Q1

Noisy OR relation: m + 1 conditional probabilities: Independent inhibitors Q1 , . . . , Qm ; Assume Z2 . . . Zm {Q1 = fail}∨. . .∨{Qm = fail} =⇒ {Y = fail}. For each Qi we have . . . Qm Q2 P (Qi = fail|Zi = fail) = qi , Y

12

P (Qi = fail|Zi = ¬fail) = 0. “Leak probability”: P (Y = fail|Q1 = . . . = Qm = ¬fail) = q0 . Helge Langseth





...

Z2


Naïve approach:

2m


Deterministic relations: Parameter free Noisy OR relation: m + 1 conditional probabilities Logistic regression: From m + 1 to 2m regression parameters: Y is dependent variable in logistic regression with Zi ’s as “covariates”: X XX pz1 ,...,zm log ηj zj + ηij zi · zj + . . . = η0 + 1 − pz1 ,...,zm j

12

Helge Langseth

i

j


Zm




...

Z2


Naïve approach:

2m


Deterministic relations: Parameter free Noisy OR relation: m + 1 conditional probabilities Logistic regression: From m + 1 to 2m regression parameters IPF procedure: m + 1 marginal distributions, m CPRs: Find a joint PT over Z1 , . . . , Zm , Y with given CPRs. Assume m = 1, p0 (z, y) initialized P to fit CPR. – p′k (z, y) ← pk−1 (z, y) · p(z)/ P y′ pk−1 (z, y) ′ – pk (z, y) ← pk (z, y) · p(y)/ z pk (z, y) 12

Helge Langseth


Zm




...

Z2


Naïve approach:

2m


Deterministic relations: Parameter free Noisy OR relation: m + 1 conditional probabilities Logistic regression: From m + 1 to 2m regression parameters IPF procedure: m + 1 marginal distributions, m CPRs Special structures: From 2 to 2m conditional probabilities: Y defined, e.g., by rules such as “P (Y = fail|Z1 = fail, . . . , Zm = fail) = p1 , but P (Y = fail|z1 , . . . , zm ) = p2 for all other configurations z”.

12

Helge Langseth


Zm




...

Z2


Naïve approach:

2m


Deterministic relations: Parameter free Noisy OR relation: m + 1 conditional probabilities Logistic regression: From m + 1 to 2m regression parameters IPF procedure: m + 1 marginal distributions, m CPRs Special structures: From 2 to 2m conditional probabilities Qualitative BNs: No quantitative parameters: Only qualitative effects modelled (and later calculated). From m to 2m qualitative effects (‘+’, ‘0’ or ‘−’).

12

Helge Langseth


Zm




...

Z2


Naïve approach:

2m


Deterministic relations: Parameter free Noisy OR relation: m + 1 conditional probabilities Logistic regression: From m + 1 to 2m regression parameters IPF procedure: m + 1 marginal distributions, m CPRs Special structures: From 2 to 2m conditional probabilities Qualitative BNs: No quantitative parameters Alternative solutions: No conditional probabilities: Other frameworks (like vines), or parameter estimation. 12

Helge Langseth


Zm


Utility theory

Utility Theory

13

Helge Langseth



Utility theory

Utility Theory Utility - 2

Pareto boundary

Utility - 1

13

Helge Langseth


The Ugly: Continuous variables

Outline

14

1

Introduction

2


3


4


5




Introduction

The Ugly: Continuous variables The original calculation procedure only supports a restricted set of distributional families: Continuous variables must have Gaussian distributions. Discrete variables should only have discrete parents. Gaussian parents of Gaussians are partial regression coefficients of their children. Disc.

Disc.

N (µ, σ 2 )

Disc.

Beta(α, β)

X1

X2

X3

X4

X5

Y1

Y2

Y3

Y4

Y5

Disc.

N (µx , σx2 )

Disc.

N (µx , Σx )

Bern(x)

These classes of distributions are not sufficient for reliability analysis. This is the ‘ugly’ part. 15

Helge Langseth



Introduction

An example model: The THERP methodology Used to model human ability to perform in certain settings (measured as binary variables) Known environment variables, like “Level of feedback” Z1

Z2

w11 T1

Always known w24

T2

T3

T4 Logistic regression

This is simple. The probability of a subject failing to perform task Ti is: −1 P (Ti = ti |z) = 1 + exp −w′i z

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Helge Langseth



Introduction

An example model: The THERP methodology Used to model human ability to perform in certain settings (measured as binary variables) Known environment variables, like “Level of feedback”

T1

N (µ, σ 2 )

Z1

Z2

Z3

T2

T3

T4 Logistic regression

We can also have latent traits, which are unknown and vary between subjects (like “Omitting a step in a procedure”). In this case, the model is a“latent trait model ” (similar to binary factory analyzer). In the following we will focus on a situation with two latent “traits”, and one “task”. 16

Helge Langseth



Introduction

Why is this difficult Z1

Z2

Assume we have one observation D1 = {1}, and parameters w1 = [1 1]T . T

The likelihood is given by P (T = 1) =

1 2πσ1 σ2

Z

h −µ1 )2 exp − (z12σ + 2

R2

1

(z2 −µ2 )2 2σ22

1 + exp(−z1 − z2 )

i

dz,

which has no known analytic representation in general. Hence, we cannot do the required calculations in this model. Note! This is true even if we choose not to use Bayesian networks as our modelling language.

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Helge Langseth



Approximations

Attempts to find f (z1, z2 | T = 1) and P (T = 1) Numerical approximation: 0.4

0.4

0.35

0.35

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05 0 −3

P (T = 1) = 0.49945 CPU = 600 msec. f (z1 , z2 | T = 1)

0.05

−2

−1

0

1

2

0 −3

3

−2

−1

Z1

0

Z2

1

2

3

3

2

1

0

T (Here: P (T = 1|z1 + z2 )) −1

1 0.9 0.8

−2

0.7 0.6 0.5

−3 −3

0.4

−2

−1

0

1

0.3 0.2 0.1 0 −6

18

−4

−2

0

2

4

6

Helge Langseth

1000 × 1000 grid Bayesian Networks in Reliability

2

3


Approximations

Attempts to find f (z1, z2 | T = 1) and P (T = 1) Discretization: Every continuous variable is “translated” into a discrete one. The more discrete states used the higher . . . - approximation quality. - computational complexity.

“Tricks” are available to find number of states and where to set split-points, including dynamic discretization.

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Helge Langseth



Approximations

Attempts to find f (z1, z2 | T = 1) and P (T = 1) Discretization: 0.4

0.4

0.35

0.35

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0 −3

−2

−1

0

1

2

0 −3

3

P (T = 1) = 0.49761 CPU = 2 msec. −2

−1

Z1

0

Z2

1

2

f (z1 , z2 | T = 1)

3

3

2

1

0

T (Here: P (T = 1|z1 + z2 )) −1

1 0.9 0.8

−2

0.7 0.6 0.5

−3 −3

0.4

−2

−1

0

1

2

0.3 0.2 0.1 0 −6

18

−4

−2

0

2

4

6

Helge Langseth

5 × 5 discretization grid Bayesian Networks in Reliability

3


Approximations

Attempts to find f (z1, z2 | T = 1) and P (T = 1) Mixtures of Truncated Exponentials: In standard discretization, the continuous variable is approximated by a step-function. Calculations are also possible if each ‘step’ is replaced by a truncated exponential. A single variable density is split into n intervals I k , k = 1, . . . , n, each approximated by ∗

f (z) =

(k) a0

+

m X i=1

(k) (k) ai exp bi z for z ∈ I k

We typically see 1 ≤ n ≤ 4 and 0 ≤ m ≤ 2. Clever parameter choices are tabulated for many standard distributions. 18

Helge Langseth



Approximations

Attempts to find f (z1, z2 | T = 1) and P (T = 1) Mixtures of Truncated Exponentials: 0.4

0.4

0.35

0.35

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05 0 −3

P (T = 1) = 0.49914 CPU = 4 msec. f (z1 , z2 | T = 1)

0.05

−2

−1

0

1

2

0 −3

3

−2

−1

Z1

0

Z2

1

2

3

3

2

1

0

T (Here: P (T = 1|z1 + z2 )) −1

1 0.9 0.8

−2

0.7 0.6 0.5

−3 −3

0.4

−2

−1

0

1

2

0.3 0.2 0.1 0 −6

18

−4

−2

0

2

4

6

Helge Langseth

S. Acid et al.: ELVIRA Bayesian Networks in Reliability

3


Approximations

Attempts to find f (z1, z2 | T = 1) and P (T = 1) Markov Chain Monte Carlo: Works well with Bayesian Networks, as independence statements can be exploited for fast simulation: Metropolis-Hastings works directly out-of-the-box. Gibbs sampling might sometimes require clever adaption.

18

Helge Langseth



Approximations

Attempts to find f (z1, z2 | T = 1) and P (T = 1) Markov Chain Monte Carlo: 0.4

0.4

0.35

0.35

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0 −3

−2

−1

0

1

2

0 −3

3

P (T = 1) = 0.49821 CPU = 32 · 103 msec. −2

−1

Z1

0

Z2

1

2

f (z1 , z2 | T = 1)

3

3

2

1

0

T (Here: P (T = 1|z1 + z2 )) −1

1 0.9 0.8

−2

0.7 0.6 0.5

−3 −3

0.4

−2

−1

0

1

2

0.3 0.2 0.1 0 −6

18

−4

−2

0

2

4

6

Helge Langseth

W. Gilks et al.: BUGS Bayesian Networks in Reliability

3


Approximations

Attempts to find f (z1, z2 | T = 1) and P (T = 1) Variational Approximations: − log(exp(v/2) + exp(−v/2))

P (T = 1|x)

-0.6 -0.8 -1 -1.2 -1.4 -1.6 -1.8 -3

-2

-1

0

1

2

3

0

1

2

3

4

5

6

7

8

Replace a tricky function h(v) with family of simple functions ˜ ξ), such that h(v) = sup h(v, ˜ ξ). indexed by ξ, h(v, ξ Note: log (P (T = 1|v)) = v/2 − log(exp(v/2) + exp(−v/2)), where the last term is convex in v 2 . 18

Helge Langseth


9

v2


Approximations

Attempts to find f (z1, z2 | T = 1) and P (T = 1) Variational Approximations: Define A(z1 , z2 ) = z1 + z2 λ(ξ) =

exp(−ξ)−1 4ξ(1+exp(−ξ))

Variational approximation The variational approximation of P (T = 1 | z) is exp (A(z) − ξ)/2 + λi (ξ) · (A(z)2 − ξ 2 ) ˜ . P (T = 1 | z, ξ) = 1 + exp(−ξ) r h i Can be shown that the best choice is ξ ← E (Z1 + Z2 )2 T = 1 =⇒ We need to iterate

18

Helge Langseth



Approximations

Attempts to find f (z1, z2 | T = 1) and P (T = 1) Variational Approximations: 0.4

0.4

0.35

0.35

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05 0 −3

P (T = 1) = 0.49828 CPU = 17 msec. f (z1 , z2 | T = 1)

0.05

−2

−1

0

1

2

0 −3

3

−2

−1

Z1

0

Z2

1

2

3

3

2

1

0

T (Here: P (T = 1|z1 + z2 )) −1

1 0.9 0.8

−2

0.7 0.6 0.5

−3 −3

0.4

−2

−1

0

1

0.3 0.2 0.1 0 −6

18

−4

−2

0

2

4

6

Helge Langseth

J. M. Winn: VIBES Bayesian Networks in Reliability

2

3


Approximations

Attempts to find f (z1, z2 | T = 1) and P (T = 1) Other approaches: A number of other approaches are also being examined Laplace-approximation Transformation into Mixture-of-Gaussian models Other frameworks, like Vines etc.

18

Helge Langseth


Concluding remarks

Outline

19

1

Introduction

2


3


4


5



Concluding remarks

Summary Bayesian Networks’ popularity is increasing, also in the reliability community. The main features (as seen from our community) are: Constitute an intuitive modelling ‘language’. High level of modelling flexibility. Efficient calculations based on utilization of the conditional independence structures encoded in the graph. Cost efficient representation.

Building models can still be time consuming. Problem owners lack training in using BNs: Users more confident when using traditional frameworks, like, e.g., FT modelling. The calculations may be too complex to understand.

Most important research focus (for this community) is to find good approximations to handle continuous variables. 20

Helge Langseth


Concluding remarks

Colleagues A number of people have helped or worked with me on the topics covered in this presentation: Bayesian Network Models: Thomas D. Nielsen, Finn V. Jensen, Jiří Vomlel Bayesian Networks in Reliability: Luigi Portinale, Claus Skaanning Continuous Variables: Antonio Salmerón, Rafael Rumí Reliability Models: Bo Lindqvist, Tim Bedford, Roger M. Cooke, Jørn Vatn

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Helge Langseth


Concluding remarks

Further reading Helge Langseth and Finn V. Jensen. Bayesian networks and decision graphs in reliability. In Encyclopedia of Statistics in Quality and Reliability. John Wiley & Sons, In press. Helge Langseth and Luigi Portinale. Bayesian networks in reliability. Reliability Engineering and System Safety, 92(1):92–108, 2007. Finn V. Jensen and Thomas D. Nielsen. Bayesian Networks and Decision Graphs. Springer-Verlag, Berlin, Germany, 2007. Barry R. Cobb, Prakash P. Shenoy, and Rafael Rumí. Approximating probability density functions in hybrid Bayesian networks with mixtures of truncated exponentials. Statistics and Computing, 46(3):293–308, 2006. 22

Helge Langseth


Bayesian Networks in Reliability - Department of Computer and ...

Bayesian Networks in Reliability - Department of Computer and ...

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