bayesian networks in reliability - ntnu · the good, the bad, and the ugly helge langseth ... an...

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

1 Introduction

2 The Good: Why Bayesian Nets are popularMathematical propertiesMaking decisionsApplications

3 The Bad: Building complex quantitative modelsThe model building processThe quantitative partUtility theory

4 The Ugly: Continuous variablesIntroductionApproximations

5 Concluding remarks


Introduction

A simple example: “Explosion”

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

E: Environment

L: Leak G: GD failed

X: Explosion

C: Casualties


Introduction


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

E: Environment


X: Explosion

C: Casualties


pa (X) = {L,G}X: Explosion


Introduction


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

E: Environment


X: Explosion

C: Casualties



E: Environment


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


Introduction


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

E: Environment


X: Explosion

C: Casualties



E: Environment


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



X⊥⊥E | {L,G}

Other d-sep. rules: Jensen&Nielsen (07)


Introduction


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

E: Environment


X: Explosion

C: Casualties



E: Environment


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



X⊥⊥E | {L,G}

Other d-sep. rules: Jensen&Nielsen (07)

X: Explosion

E: Environment


X: Explosion

C: Casualties

G E =hostile E =normal

yes λH · τ/2 λN · τ/2no 1− λH · τ/2 1− λN · τ/2

P (G |pa (G))

(

Hence, P (X |E,L,G) = P (X |L,G))

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


The Good: Why Bayesian Nets are popular

Outline

1 Introduction






The Good: Why Bayesian Nets are popular Mathematical properties

What the mathematical foundation has to offer

Intuitive representation: Almost defined as “box-diagram withformal meaning”. Causal interpretation natural inmany cases.

Efficient representation: The number of required parameters arereduced. If all variables are binary, the examplerequires 11 “local” parameters, compared to the 31“global” parameters of the full joint.

Efficient calculations: Efficient calculations of any jointdistribution P (xi, xj) or conditional distributionP (xk |xℓ, xm).

Model estimation: Estimating parameters (fixed structure) viaEM, estimating structure by discrete optimizationtechniques.


The Good: Why Bayesian Nets are popular Making decisions

Influence diagrams: The “Explosion” example revisited

E: Environment


X: Explosion

C: Casualties

SSM

Cost1

Cost2


The Good: Why Bayesian Nets are popular Making decisions

Influence diagrams: The “Explosion” example revisited

E: Environment


X: Explosion

C: Casualties

SSM

Cost1

Cost2

F : Effectiveness, SSM

Test interval

Cost3


The Good: Why Bayesian Nets are popular Applications

An application: Troubleshooting



Underlying model

TOP

C1 C2 C3 C4

X1 X2 X3 X4 X5 system-layer



Underlying model

TOP

C1 C2 C3 C4


E



Underlying model

TOP

C1 C2 C3 C4


A1 A2 A3 A4 A5

X1 X2 X3 X4 X5

action-layer



Underlying model

TOP

C1 C2 C3 C4


A1 A2 A3 A4 A5

X1 X2 X3 X4 X5

action-layerA1 A2 A3 A4 A5

C1 C2 C3 C4

R1 R2 R3 R4 R5 result-layer



Underlying model

TOP

C1 C2 C3 C4


A1 A2 A3 A4 A5

X1 X2 X3 X4 X5

action-layerA1 A2 A3 A4 A5

C1 C2 C3 C4

R1 R2 R3 R4 R5 result-layer

C1 C2

X3 X5

QSquestion-layer



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 latertoday


The Bad: Building complex quantitative models

Outline

1 Introduction






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 whatto include in the model.

Step 1 – Defining variables: Select the important variables inthe domain.

Step 2 – The qualitative part: Define the graphical structurethat connects the variables.

Step 3 – The quantitative part: Fix parameters to specify eachP (xi |pa (xi)). This is the ‘bad’ part.

Step 4 – Verification: Verification of the model.


The Bad: Building complex quantitative models The quantitative part

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

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

Y

Z1 Z2. . . Zm

Naïve approach: 2m conditional probabilities:All parameters are required if no other assumptions can bemade.





Y

Z1 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}.





Y

Z1 Z2. . . Zm


Deterministic relations: Parameter free

Noisy OR relation: m + 1 conditional probabilities:

Y

Q1 Q2 . . . Qm

Z1 Z2. . . Zm

Independent inhibitors Q1, . . . , Qm; Assume{Q1 = fail}∨. . .∨{Qm = fail} =⇒ {Y = fail}.

For each Qi we have

P (Qi = fail|Zi = fail) = qi,

P (Qi = fail|Zi = ¬fail) = 0.

“Leak probability”:P (Y = fail|Q1 = . . . = Qm = ¬fail) = q0.





Y

Z1 Z2. . . Zm



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”:

log

(

pz1,...,zm

1− pz1,...,zm

)

= η0 +∑

j

ηjzj +∑

i

∑

j

ηijzi · zj + . . .





Y

Z1 Z2. . . Zm




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 to fit CPR.– p′k(z, y)← pk−1(z, y) · p(z)/

∑

y pk−1(z, y)– pk(z, y)← p′k(z, y) · p(y)/

∑

z p′k(z, y)





Y

Z1 Z2. . . Zm





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, butP (Y = fail|z1, . . . , zm) = p2 for all other configurations z”.





Y

Z1 Z2. . . Zm






Special structures: From 2 to 2m conditional probabilities

Qualitative BNs: No quantitative parameters:Only qualitative effects modelled (and later calculated). Fromm to 2m qualitative effects (‘+’, ‘0’ or ‘−’).





Y

Z1 Z2. . . Zm






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.


The Bad: Building complex quantitative models Utility theory

Utility Theory


The Bad: Building complex quantitative models Utility theory

Utility Theory

Utility - 1

Utility - 2

Pareto boundary


The Ugly: Continuous variables

Outline

1 Introduction






The Ugly: Continuous variables Introduction

The Ugly: Continuous variables

The original calculation procedure only supports a restrictedset of distributional families:

Continuous variables must have Gaussian distributions.Discrete variables should only have discrete parents.Gaussian parents of Gaussians are partial regression coefficientsof their children.

X1

Disc.

Y1

Disc.

X2

Disc.

Y2

N (µx, σ2x)

X3

N (µ, σ2)

Y3

Disc.

X4

Disc.

Y 4

N (µx,Σx)

X5

Beta(α, β)

Y5

Bern(x)

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



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 Always known

T1 T2 T3 T4

w11 w24

Logistic regression

This is simple. The probability of a subject failing to performtask Ti is:

P (Ti = ti|z) =(

1 + exp(

−w′

iz))

−1



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 Z3 N (µ, σ2)

T1 T2 T3 T4 Logistic regression

We can also have latent traits, which are unknown and varybetween subjects (like “Omitting a step in a procedure”).

In this case, the model is a“ latent trait model ” (similar tobinary factory analyzer).

In the following we will focus on a situation with two latent“traits”, and one “task”.



Why is this difficult

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

Z1 Z2

T

The likelihood is given by

P (T = 1) =1

2πσ1σ2

∫

R2

exp(

−[

(z1−µ1)2

2σ2

1

+ (z2−µ2)2

2σ2

2

])

1 + exp(−z1 − z2)dz,

which has no known analytic representation in general. Hence, wecannot do the required calculations in this model.

Note! This is true even if we choose not to use Bayesian networksas our modelling language.


The Ugly: Continuous variables Approximations

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

Numerical approximation:

−3 −2 −1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

−3 −2 −1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

−6 −4 −2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Z1 Z2

T (Here: P (T = 1|z1 + z2))

P (T = 1) = 0.49945CPU = 600 msec.

f(z1, z2 |T = 1)

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

1000 × 1000 grid




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 toset split-points, including dynamic discretization.




Discretization:

−3 −2 −1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

−3 −2 −1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

−6 −4 −2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Z1 Z2

T (Here: P (T = 1|z1 + z2))

P (T = 1) = 0.49761CPU = 2 msec.

f(z1, z2 |T = 1)

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

5× 5 discretization grid




Mixtures of Truncated Exponentials:In standard discretization, the continuous variable isapproximated by a step-function.

Calculations are also possible if each ‘step’ is replaced by atruncated exponential.

A single variable density is split into n intervals Ik,k = 1, . . . , n, each approximated by

f∗(z) = a(k)0 +

m∑

i=1

a(k)i exp

(

b(k)i z

)

for z ∈ Ik

We typically see 1 ≤ n ≤ 4 and 0 ≤ m ≤ 2.

Clever parameter choices are tabulated for many standarddistributions.




Mixtures of Truncated Exponentials:

−3 −2 −1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

−3 −2 −1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

−6 −4 −2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Z1 Z2

T (Here: P (T = 1|z1 + z2))

P (T = 1) = 0.49914CPU = 4 msec.

f(z1, z2 |T = 1)

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

S. Acid et al.: ELVIRA




Markov Chain Monte Carlo:Works well with Bayesian Networks, as independencestatements can be exploited for fast simulation:

Metropolis-Hastings works directly out-of-the-box.Gibbs sampling might sometimes require clever adaption.




Markov Chain Monte Carlo:

−3 −2 −1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

−3 −2 −1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

−6 −4 −2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Z1 Z2

T (Here: P (T = 1|z1 + z2))

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

f(z1, z2 |T = 1)

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

W. Gilks et al.: BUGS




Variational Approximations:

-3 -2 -1 0 1 2 3

P (T = 1|x)

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

0 1 2 3 4 5 6 7 8 9

− log(exp(v/2) + exp(−v/2))

v2

Replace a tricky function h(v) with family of simple functionsindexed by ξ, h̃(v, ξ), such that h(v) = supξ h̃(v, ξ).

Note: log (P (T = 1|v)) = v/2− log(exp(v/2) + exp(−v/2)),where the last term is convex in v2.




Variational Approximations:Define

A(z1, z2) = z1 + z2

λ(ξ) = exp(−ξ)−14ξ(1+exp(−ξ))

Variational approximation

The variational approximation of P (T = 1 |z) is

P̃ (T = 1 |z, ξ) =exp

[

(A(z)− ξ)/2 + λi(ξ) · (A(z)2 − ξ2)]

1 + exp(−ξ).

Can be shown that the best choice is ξ ←

√

E

[

(Z1 + Z2)2∣

∣

∣T = 1

]

=⇒ We need to iterate




Variational Approximations:

−3 −2 −1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

−3 −2 −1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

−6 −4 −2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Z1 Z2

T (Here: P (T = 1|z1 + z2))

P (T = 1) = 0.49828CPU = 17 msec.

f(z1, z2 |T = 1)

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

J. M. Winn: VIBES




Other approaches:A number of other approaches are also being examined

Laplace-approximation

Transformation into Mixture-of-Gaussian models

Other frameworks, like Vines

etc.


Concluding remarks

Outline

1 Introduction






Concluding remarks

Summary

Bayesian Networks’ popularity is increasing, also in thereliability 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 conditionalindependence 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 findgood approximations to handle continuous variables.


Concluding remarks

Colleagues

A number of people have helped or worked with me on the topicscovered 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


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 Bayesiannetworks with mixtures of truncated exponentials.

Statistics and Computing, 46(3):293–308, 2006.


bayesian networks in reliability - ntnu · the good, the bad, and the ugly helge langseth ... an...

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