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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
2 Helge Langseth Bayesian Networks in Reliability
Introduction
A simple example: “Explosion”
P (E,L,G,X,C)
E: Environment
L: Leak G: GD failed
X: Explosion
C: Casualties
3 Helge Langseth Bayesian Networks in Reliability
Introduction
A simple example: “Explosion”
P (E,L,G,X,C)
E: Environment
L: Leak G: GD failed
X: Explosion
C: Casualties
L: Leak G: GD failed
pa (X) = {L,G}X: Explosion
3 Helge Langseth Bayesian Networks in Reliability
Introduction
A simple example: “Explosion”
P (E,L,G,X,C)
E: Environment
L: Leak G: GD failed
X: Explosion
C: Casualties
L: Leak G: GD failed
pa (X) = {L,G}X: Explosion
E: Environment
L: Leak G: GD failed
nd(X) = {E,L,G}
3 Helge Langseth Bayesian Networks in Reliability
Introduction
A simple example: “Explosion”
P (E,L,G,X,C)
E: Environment
L: Leak G: GD failed
X: Explosion
C: Casualties
L: Leak G: GD failed
pa (X) = {L,G}X: Explosion
E: Environment
L: Leak G: GD failed
nd(X) = {E,L,G}
L: Leak G: GD failed
pa (X) = {L,G}X: Explosion
X⊥⊥E | {L,G}
Other d-sep. rules: Jensen&Nielsen (07)
3 Helge Langseth Bayesian Networks in Reliability
Introduction
A simple example: “Explosion”
P (E,L,G,X,C)
E: Environment
L: Leak G: GD failed
X: Explosion
C: Casualties
L: Leak G: GD failed
pa (X) = {L,G}X: Explosion
E: Environment
L: Leak G: GD failed
nd(X) = {E,L,G}
L: Leak G: GD failed
pa (X) = {L,G}X: Explosion
X⊥⊥E | {L,G}
Other d-sep. rules: Jensen&Nielsen (07)
X: Explosion
E: Environment
L: Leak G: GD failed
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
3 Helge Langseth Bayesian Networks in Reliability
The Good: Why Bayesian Nets are popular
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
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 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.
5 Helge Langseth Bayesian Networks in Reliability
The Good: Why Bayesian Nets are popular Making decisions
Influence diagrams: The “Explosion” example revisited
E: Environment
L: Leak G: GD failed
X: Explosion
C: Casualties
SSM
Cost1
Cost2
6 Helge Langseth Bayesian Networks in Reliability
The Good: Why Bayesian Nets are popular Making decisions
Influence diagrams: The “Explosion” example revisited
E: Environment
L: Leak G: GD failed
X: Explosion
C: Casualties
SSM
Cost1
Cost2
F : Effectiveness, SSM
Test interval
Cost3
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
C1 C2 C3 C4
X1 X2 X3 X4 X5 system-layer
8 Helge Langseth Bayesian Networks in Reliability
The Good: Why Bayesian Nets are popular Applications
Underlying model
TOP
C1 C2 C3 C4
X1 X2 X3 X4 X5 system-layer
E
8 Helge Langseth Bayesian Networks in Reliability
The Good: Why Bayesian Nets are popular Applications
Underlying model
TOP
C1 C2 C3 C4
X1 X2 X3 X4 X5 system-layer
A1 A2 A3 A4 A5
X1 X2 X3 X4 X5
action-layer
8 Helge Langseth Bayesian Networks in Reliability
The Good: Why Bayesian Nets are popular Applications
Underlying model
TOP
C1 C2 C3 C4
X1 X2 X3 X4 X5 system-layer
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
8 Helge Langseth Bayesian Networks in Reliability
The Good: Why Bayesian Nets are popular Applications
Underlying model
TOP
C1 C2 C3 C4
X1 X2 X3 X4 X5 system-layer
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
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 latertoday
9 Helge Langseth Bayesian Networks in Reliability
The Bad: Building complex quantitative models
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
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 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.
11 Helge Langseth Bayesian Networks in Reliability
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.
12 Helge Langseth Bayesian Networks in Reliability
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
Deterministic relations: Parameter free:Y considered a function of its parents, e.g.,
{Y = fail} ⇐⇒ {Z1 = fail}∨{Z2 = fail}∨. . .∨{Zm = 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))
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
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.
12 Helge Langseth Bayesian Networks in Reliability
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
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”:
log
(
pz1,...,zm
1− pz1,...,zm
)
= η0 +∑
j
ηjzj +∑
i
∑
j
ηijzi · zj + . . .
12 Helge Langseth Bayesian Networks in Reliability
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
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 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)
12 Helge Langseth Bayesian Networks in Reliability
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
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, butP (Y = fail|z1, . . . , zm) = p2 for all other configurations z”.
12 Helge Langseth Bayesian Networks in Reliability
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
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). Fromm to 2m qualitative effects (‘+’, ‘0’ or ‘−’).
12 Helge Langseth Bayesian Networks in Reliability
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
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 Bayesian Networks in Reliability
The Bad: Building complex quantitative models Utility theory
Utility Theory
13 Helge Langseth Bayesian Networks in Reliability
The Bad: Building complex quantitative models Utility theory
Utility Theory
Utility - 1
Utility - 2
Pareto boundary
13 Helge Langseth Bayesian Networks in Reliability
The Ugly: Continuous variables
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
14 Helge Langseth Bayesian Networks in Reliability
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.
15 Helge Langseth Bayesian Networks in Reliability
The Ugly: Continuous variables 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 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
16 Helge Langseth Bayesian Networks in Reliability
The Ugly: Continuous variables 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 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”.
16 Helge Langseth Bayesian Networks in Reliability
The Ugly: Continuous variables Introduction
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.
17 Helge Langseth Bayesian Networks in Reliability
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
18 Helge Langseth Bayesian Networks in Reliability
The Ugly: Continuous variables 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 toset split-points, including dynamic discretization.
18 Helge Langseth Bayesian Networks in Reliability
The Ugly: Continuous variables Approximations
Attempts to find f(z1, z2 |T = 1) and P (T = 1)
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
18 Helge Langseth Bayesian Networks in Reliability
The Ugly: Continuous variables Approximations
Attempts to find f(z1, z2 |T = 1) and P (T = 1)
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.
18 Helge Langseth Bayesian Networks in Reliability
The Ugly: Continuous variables Approximations
Attempts to find f(z1, z2 |T = 1) and P (T = 1)
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
18 Helge Langseth Bayesian Networks in Reliability
The Ugly: Continuous variables Approximations
Attempts to find f(z1, z2 |T = 1) and P (T = 1)
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.
18 Helge Langseth Bayesian Networks in Reliability
The Ugly: Continuous variables Approximations
Attempts to find f(z1, z2 |T = 1) and P (T = 1)
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
18 Helge Langseth Bayesian Networks in Reliability
The Ugly: Continuous variables Approximations
Attempts to find f(z1, z2 |T = 1) and P (T = 1)
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.
18 Helge Langseth Bayesian Networks in Reliability
The Ugly: Continuous variables 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)−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
18 Helge Langseth Bayesian Networks in Reliability
The Ugly: Continuous variables Approximations
Attempts to find f(z1, z2 |T = 1) and P (T = 1)
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
18 Helge Langseth Bayesian Networks in Reliability
The Ugly: Continuous variables 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 Bayesian Networks in Reliability
Concluding remarks
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
19 Helge Langseth Bayesian Networks in Reliability
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.
20 Helge Langseth Bayesian Networks in Reliability
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
21 Helge Langseth Bayesian Networks in Reliability
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.
22 Helge Langseth Bayesian Networks in Reliability