This document discusses the use of Bayesian networks for risk analysis. It begins by explaining the key elements of Bayesian networks, including that they are graphical models that represent probabilistic relationships between variables. It then discusses how Bayesian networks can be used to structure complex systems by modeling the dependencies and conditional independencies between variables. Finally, it provides examples of how Bayesian networks can be built and used for tasks like probabilistic inference.

1. Peter Friis-Hansen
12 January 2010
Bayesian Network and its use in risk analysis
Transportforum, 13-14 januari, 2010, Linköbing, Sweden

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Bayesian Network and its use in risk analysis
12 January 2010
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 Structure and materials
 Propulsion
 Compartmentation
 Manoeuvring characteristics
 Bridge layout
 Quality of crew
 +++
Frequency
Consequence
Risk based procedures requires insight deeply into very complex matters
Accidents:
?

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Bayesian Network and its use in risk analysis
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Structuring complex systems
REQUIREMENTS
 Transparency
 Uniformity in modelling complexity
 Verifiability of probabilistic modelling
 Bayesian Networks bridges the gab between
model formulation and analysis

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Bayesian Network and its use in risk analysis
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Content
 Why Bayesian Networks?
 Elements of Bayesian Network
 Building Bayesian Networks
 Modelling decisions

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Bayesian Network and its use in risk analysis
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Introducing Bayesian Networks
 A Bayesian Network
- is a graphical representation of uncertain quantities
- reveals explicitly the probabilistic dependence between the set variables
- is designed as a knowledge representation of the considered problem
 A BN is a network with directed arcs and no cycles
 The nodes represents random variables and/or decisions
 Arcs into random variables indicate probabilistic dependence
 Causal modelling most effectively does the model building

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Bayesian Network and its use in risk analysis
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What do Bayesian methods offer ?
1. Allows one to learn about causal relationships
- this knowledge allow to make predictions in the presence of interventions / observations
2. BN in conjunction with Bayesian statistical techniques facilitate the combination of
domain knowledge and data
- prior or domain knowledge
3. BN can readily handle incomplete data
- missing data
4. Bayesian methods in conjunction with BN and other methods offers efficient
methods to avoid over fitting of data

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Bayesian Network and its use in risk analysis
12 January 2010
BN for a set of variables
Battery
Gauge
Fuel
Turnover
Start
p(B)
p(T|B)
p(G|B, F)
p(F)
p(S| F, T)
Directed Acyclic Graph
low,
normal,
high
none,
click,
normal
low,
normal,
high
empty,
medium,
full
yes,
no

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Bayesian Network and its use in risk analysis
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BN - elements
BN for a set of variables consists of:
1. A network structure S that encodes a set of conditional independence assertions about the
variables in X
2. A set P of local, conditional probability distributions associated with each variable in X
 1. & 2. defines the joint probability distribution for X.
 S is a Directed Acyclic Graph (DAG)
 Nodes are in one-to-one correspondence with the variables in X
 denotes both the stochastic variable and the associated node
 denotes the parents to in S
 Lack of possible arcs in S encode conditional independence
X ={ , , }X Xn1 
Xi
pai Xi

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Bayesian Network and its use in risk analysis
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Description (nodes)
Probability node (discrete)
Decision node
Utility node
Link / arc
[ ]iixP pa| Local probability distribution (conditional)

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Bayesian Network and its use in risk analysis
12 January 2010
Bayesian Network
T
Turnover
- none
- click
- normal
S
Start
- yes
- no
P(T)
T = none 0.003
T = click 0.001
T = normal 0.996
P(S | T) T = none T = click T = normal
S = Yes 0.0 0.02 0.97
S = No 1.0 0.98 0.03
∑ =====
T
tTptTsSpsSp )()|()(

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Bayesian Network and its use in risk analysis
12 January 2010
Missing arcs encode conditional independence
Turnover
T
Gauge
G
P(G)
G = not empty 0.995
G = empty 0.005
P(T)
T = none 0.003
T = click 0.001
T = normal 0.996

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Bayesian Network and its use in risk analysis
12 January 2010
Bayesian Network Structure: Definition
1. Find the variables of the model
2. Build a DAG that encodes assertions of conditional independence
- Given an ordering of the variables
( ,..., )X Xn1
∏∏ ==
−
−
==
⇓
=
n
i
ii
n
i
iin
iiii
xpxxxpxxp
xpxxxp
11
111
11
)|(),....,|(),...,(
)|(),....,|(
pa
pa

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Bayesian Network and its use in risk analysis
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Example
Fuel Battery Turnover Gauge Start
p F( ) p B F p B( | ) ( )=

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Bayesian Network and its use in risk analysis
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Example
Fuel Battery Turnover Gauge Start
p F( ) p B F p B( | ) ( )=
p T B F p T B( | , ) ( | )=
p G F B T p G F B( | , , ) ( | , )=

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Bayesian Network and its use in risk analysis
12 January 2010
Example
Fuel Battery Turnover Gauge Start
p F( ) p B F p B( | ) ( )=
p T B F p T B( | , ) ( | )=
p G F B T p G F B( | , , ) ( | , )=
p S F B T G p S F T( | , , , ) ( | , )=
p F B T G S p F p B F p T B F p G F B T p S F B T G
p F p B p T B p G F B p S F T
( , , , , ) ( ) ( | ) ( | , ) ( | , , ) ( | , , , )
( ) ( ) ( | ) ( | , ) ( | , )
=
=

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Bayesian Network and its use in risk analysis
12 January 2010
Variable order is important!
Start Gauge Turnover Battery Fuel

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Bayesian Network and its use in risk analysis
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Causal knowledge simplifies the construction
Battery
Gauge
Fuel
Turnover
Start

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Bayesian Network and its use in risk analysis
12 January 2010
Conditional independence simplifies Probabilistic Inference
Battery
Gauge
Fuel
Turnover
Start
g
s
f
p F f S s G g
p f b t g s
p f b t g s
b t
b f t
( | , )
( , , , , )
( , , , , )
,
, ,
= = = =
∑
∑

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Bayesian Network and its use in risk analysis
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“Explaining Away”
Turnover
Start
Fuel
If the car does not start, hearing the engine turn over
makes no fuel more likely.

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Bayesian Network and its use in risk analysis
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Did Marco Pantani use EPO?
 Just before the last lap in the Giro d’Italia in 1999, the Italian Marco Pantani was
excluded from the race because of a positive EPO doping test. Marco Pantani was
leading the race when he was excluded.
 Question: does the bare fact of a positive EPO test reveal his quilt?
Assumptions:
 The EPO test is able to detect the use of EPO with a probability of 95%
 False positive test: Let us assume that this probability is 15%.
 Probability of riders are using EPO: say, 10% are using EPO.
HUGIN

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Bayesian Network and its use in risk analysis
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Max propagation – what is it ?
 That configuration in the joint probability distribution that has the largest value
 Identical to the ”FORM design point” in x-space
 Identical to finding the dominant cut set for fault trees

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Bayesian Network and its use in risk analysis
12 January 2010
Maximise expected utility
Party Location
- outdoor
- indoor
Utility
Weather
indoors
outdoors
.7
.3
.7
.3
dry
rain
dry
rain
50
60
100
0
EU[indoors] = 0.7 (50) + 0.3 (60) = 53
EU[outdoors] = 0.7 (100) + 0.3 (0) = 70 select “outdoor”

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Bayesian Network and its use in risk analysis
12 January 2010
Time critical decisions
System State
H, to
E1 E2 En
Action A, t
Duration of
Process
Utility
EU A t p H E u A H ti j i
j
n
j[ , ] ( | , ) ( , , )=
=
∑ ξ
1
probability of hypothesises for the different system states
given observations E and background information ξ
time dependent utility as a function of
action A and system state H

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Bayesian Network and its use in risk analysis
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OOW acts
OOW radar
OOW visual Radar freq
Looking freq
Alarm transfer
OOW Task
Bridge
Stress level
OOW trainingOther alarms
Time for radarTime for visual Maneuv. time
Traffic intensitVessel speed
Radar time
Visual time
Obj. rel. speed
Radar dist.
Visual dist.
Object type
Visibility
Day light
Radar statusWeather
Speed reducti
Basic network for
navigator reacting in time
Navigational route
Vessel Object
Visual distance
Time for detection
Minimum distance to avoid
critical situation
v1
v2

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Bayesian Network and its use in risk analysis
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Including the time aspect - DBN
a1(5)a1(4)a1(3)a1(2)a1(1)
SIF1(5)SIF1(4)SIF1 (3)SIF1 (2)SIF1(1)
Seastate5Seastate4Seastate3Seastate2Seastate1
Initial a_1
Model unc.
Fatigue modelling

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Bayesian Network and its use in risk analysis
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Fatigue inspection planning
Model uncer
a(0)
Seastate(2)
Seastate(4)
a(02) a(04)
CI(4)
CR(4)
CI(2)
CR(2)
Inspect(04)Inspect(02)
InspRes(04InspRes(02
CF(2)
a_rep(04)a_rep(02)
PF(02) PF(04)
CF(4) CF(6)
PF(06)
a_rep(06)
InspRes(06
Inspect(06)
CR(6)
CI(6)
a(06)
Seastate(6)
dPF(0-2) dPF(2-4) dPF(4-6)
PF(0)
CF(0)

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Bayesian Network and its use in risk analysis
12 January 2010
Where to get more information ?
 HUGIN expert AS
www.hugin.com
 Association for Uncertainty in Artificial Intelligence
www.auai.org
 Microsoft Decision Group
www.research.Microsoft.com/research/dtg
 Bibliography
www-users.cs.york.ac.uk/~sara/reference/biblios/

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Bayesian Network and its use in risk analysis
12 January 2010
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Two line Transformer station subjected
to earth quake

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Bayesian Network and its use in risk analysis
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Modelling the disconnect switch
ZiVarYVar
YVar
DSjDSi
+
=),(ρ

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Bayesian Network and its use in risk analysis
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Bulk carrier safety: MSC74/INF.15, 2001
?

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Bayesian Network and its use in risk analysis
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Safeguarding life, property
and the environment
www.dnv.com

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Bayesian Network and its use in risk analysis
12 January 2010
What is a complex system ?
 Complex: A whole made up of
dissimilar parts or parts of intricate
relationship
 Consisting of interconnected or
interwoven parts; composite
 Intricate: having a complicated
organisation, with many parts or
aspects difficult to follow or grasp

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Bayesian Network and its use in risk analysis
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Propagation in Bayesian Network
 U grows exponentially with number of variables and states – for binary O(2N
)
 Calls for efficient algorithm
 JUNCTION TREE
- The nodes of the junction tree are sets of variables called cliques
- Links are separators, which is the intersection of the adjacent cliques
∏
∏=
Separators
Cliques
UP ][

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Bayesian Network and its use in risk analysis
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Triangulated graph and junction tree
1
2
3
4
5
6 145
456
345
235
45
45
35
1
2
3
4
5
6

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Bayesian Network and its use in risk analysis
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Learning
 Learning probability distributions
- Uses EM algorithm
- Log likelihood optimisation reformulated to nested optimisation
- Assures better and faster convergence
- Beta distribution
- Dirichlet distribution
 Learning the structure – more ambitious
- Priors for all structures

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Bayesian Network and its use in risk analysis
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System knowledge and data
X1 X2
X4X3
Prior Network
α
Sample size
Data
X1 X2 X3 X4
x1 : T F T T
x2 : F T T F
……
xn : T T F F
X1 X2
X4X3
Priors for all structures
Learned structure
http://b-course.cs.helsinki.fi/

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Bayesian Network and its use in risk analysis
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Interpretation of critical situation
Navigational route
Vessel Object
Visual distance
Time for detection
Minimum distance to avoid
critical situation
Legend:
v1
v2
Considerations:
•Visual detection
•Radar detection
•Dependency of weather
•Correlation among variables
•Perception and assessment
of situation

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Bayesian Network and its use in risk analysis
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Description of the critical situation
“During the watch the considered vessel is on collision
course with an object. Moreover, machinery and steering
gear are functioning.”
“Does the Officer On the Watch react in time so that the
collision is avoided?”

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Bayesian Network and its use in risk analysis
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OOW acts
OOW radar
OOW visual Radar freq
Looking freq
Alarm transfer
OOW Task
Bridge
Stress level
OOW trainingOther alarms
Time for radarTime for visual Maneuv. time
Traffic intensitVessel speed
Radar time
Visual time
Obj. rel. speed
Radar dist.
Visual dist.
Object type
Visibility
Day light
Radar statusWeather
Speed reducti
Basic network

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Bayesian Network and its use in risk analysis
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Concept of the conventional bridge
 Conventional bridge is a modern bridge
 A rating lookout will (in principle) be present on the bridge from sunset to sunrise
 Calling of a rating lookout during daytime if conditions causes solo watch keeping
being unsafe
- conditions of weather,
- visibility,
- proximity of dangers to navigation,
- traffic situation

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Speed reducti
Weather Radar status
Day light
Visibility
Object type
Visual dist.
Radar dist.
Obj. rel. speed
Visual time
Radar time
Vessel speed Traffic intensit
Maneuv. timeTime for visual Time for radar
Other alarms OOW training
Stress level
Bridge
OOW Task
Alarm transfer
Looking freq
Radar freqOOW visual
OOW radar
OOW acts
Rating freq
Rating visual
Rating task
Rating pres
Rating inform
Rating Call
Conventional bridge

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Results - conventional bridge
Case P[OOW not acting]
Day and night 0.00270
Daylight 0.00330
Darkness 0.00209
Object P[OOW not acting]
(Day and night
P[OOW not acting]
(Daylight)
P[OOW not acting]
(Darkness)
Large vessel 0.00193 0.00264 0.00122
Small vessel 0.00191 0.00267 0.00116
Floating object 0.773 0.649 0.898

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Bayesian Network and its use in risk analysis
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Comparison with observations
log-log plot of probability of no action
y = -1.0792x - 1.7219
R2
= 0.9743-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
-0.5 0 0.5 1 1.5
log(Visibility)
log(P)
Log P
Linear (Log P)
Japanese observations in the period from 1966 to 1971 reveals a
proportionality between risk for collision and visibility r :
6.1
Risk −
∝ r
Causes ?
• Improved radar technology
• Difference in causes for
low visibility in DK and
Japan
Obtained factor is -1.1
Speed reducti
Weather Radar status
Day light
Visibility
Object type
Visual dist.
Radar dist.
Obj. rel. speed
Visual time
Radar time
Vessel speed Traffic intensit
Maneuv. timeTime for visual Time for radar
Other alarms OOW training
Stress level
Bridge
OOW Task
Alarm transfer
Looking freq
Radar freqOOW visual
OOW radar
OOW acts
Rating freq
Rating visual
Rating task
Rating pres
Rating inform
Rating Call