An Implementation of Multiply Sectioned Bayesian Networks

An Implementation of Multiply An Implementation of Multiply Sectioned Bayesian NetworksSectioned Bayesian Networks

Metron, Inc.

Chris Boner

Thor Whalen

OutlineOutline

• Multiply sectioned Bayes nets (MSBN)

• Problem formulation and elements of a solution

• Using a junction tree to construct an MSBN

• Matlab tool

Multiply Sectioned Bayes NetMultiply Sectioned Bayes Net

• What is a Bayes Net?• What is a Multiply Sectioned Bayes Net (MSBN)?• Motivation

What is a Bayes Net?What is a Bayes Net?• A Bayes Net is a representation of

a probability distribution P(V) on a set V=X1, ..., Xn of variables

• A BN consists of – A Directed Acyclic Graph (DAG)

• Nodes: Variables of V

• Edges: Causal relations

A DAG is a directed graph with no directed cycles

The above directed graph is a DAG

Now this graph IS NOT a DAG because it has a directed cycle

Directed cycle

X1 X2 X3

X9 X10

X11 X12 X13

• Nodes: Variables of V

– A list of conditional probability distributions (CPDs); one for every node of the DAG

X1 X2 X3

X9 X10

X11 X12 X13

Etc...

- i.e. P(A , B | C) = P(A | C) P(B | C)

• Nodes: variables of V

• The DAG characterizes the (in)dependence structure of P(V)

X1 X2 X3

X9 X10

X11 X12 X13

BA and are independent given

- i.e. P(A | B, C) = P(A | C)

We will say that C separates A and B

• Nodes: variables of V

• The DAG characterizes the (in)dependency structure of P(V)

• The CPDs characterize the probabilistic and/or deterministic relations between parent states and children states

X1 X2 X3

X9 X10

X11 X12 X13

X3X1 X2

X11 X13

X12Evidence

What is a Bayes Net?What is a Bayes Net?

• The prior distributions on the variables of parentless nodes, along with the CPDs of the BN, induce prior distribution—called “beliefs” in the literature—on all the variables

• If the system receives evidence on a variable: – this evidence impacts its belief,– along with the beliefs of all other

variables

X9 X10

Parentless nodes

• Subnets of the BN are maintained independently

• Each subnet locally integrates evidence it receives

• When inter-subnets communication is possible, messages are passed that enable fusion of evidence received by other subnets

Evidence

What is a Multiply Sectioned Bayes Net?What is a Multiply Sectioned Bayes Net?

• Multi-agent systems where:– each agent only has partial knowledge of the

domain– communication among agents is limited

• some agent-agent connections may be impossible, sporadic and/or low bandwidth

– decisions must be made by agents based on local observations and limited information from other agents

MotivationMotivation

Motivation (2)Motivation (2)

• Distributed computing for reusable systems where:– probabilistic knowledge can be captured once

and used for multiple cases– queries and evidence will be localized, that is,

there are phases when• new evidence and queries are repeatedly directed

to small parts of network• only a small part of the network is needed for

decision-making

• Problem specification

• A naïve solution

• A less naïve solution

• Sufficient information

• Communication graph considerations

Problem Formulation Problem Formulation and elements of a solutionand elements of a solution

Problem SpecificationProblem SpecificationGiven: • A BN on V={X1, ..., Xn}

• A number of agents, each having:– Qi: a set of query variables

– Ei: a set of evidence variables

Problem SpecificationProblem SpecificationGiven: • A BN on V={X1, ..., Xn}

• A number of agents, each having:– Qi: a set of query variables

– Ei: a set of evidence variables

Determine: • An agent communication graph

• A subset Si of V for each agent

• An inference protocol that specifies – How to fuse evidence and messages

received from other agents– The content of messages between

agents

Every agent has a copy of the entire Bayes net

Evidence

Agents communicate evidence (findings or likelihood functions) that are re-propagated through each copy of the BN

A Naïve SolutionA Naïve Solution

Pros• Each agent’s queries

are as informed as possible once all the evidence it has received is propagated

• Inter-agent communications require relatively low bandwidth

Cons• Could be a colossal

waste of memory and processing time

• Each agent may be able to achieve fully informed queries by representing a much smaller section of BN

A Naïve SolutionA Naïve Solution

• The previous solution allows each agent to compute the posterior prob. of all the variables

• But all the agent is interested in is the posterior of its query variables

• Hence it is sufficient for every agent to only represent

- its query variables, - its evidence variables, - the evidence variables

of the other agents• Contra: Could be a

colossal waste of memory and processing time

A Less Naïve SolutionA Less Naïve Solution

variables

Evidence

variables

Evidence

variables

Evidence

variables

Evidence

variables

Sufficient informationSufficient information

Evidence variables

Query variables

Agent 1

Agent 2

Agent 3

Agent 4

Specifications

• A Bayes net

• A number of agents, each having

- query variables

- evidence variables

Agent 1

E F G K L MH I J

Specifications

• A Bayes net

- query variables

Agent 2

Agent 3

Agent 4

The naïve solution

• Agents contain their own query and evidence variables

• In order to receive evidence from the other agents, agent 1 must represent variables E, F, G, H, I, J, K, L, and M

Agent 1

Agent 2 Agent 3 Agent 4

E F G K L MH I J

Agent 1

E F G K L MH I J

Specifications

• A Bayes net

- query variables

Agent 2

Agent 3

Agent 4

The naïve solution

separatesZ and X Y

Note that

whether Y is equal to:

• {K,L,M},

• {H,J,I}, or

• {E,F,G}.YY

Agent 1 must

represent many

variables!

How else could

the other agents

communicate their

evidence?

P(X,Z|eY)

separatesZ and X Y

X = {A,B}

Z = {C,D}

Y = {E,G,F}

P(Y|Z) = P(Y|X,Z)

→→

Likelihood given X and Z

of evidence on Y

Likelihood given Z

of evidence on Y=

→ It is sufficient for agent 2 to send its posterior on Z

to agent 1 for the latter to compute its posterior on X

Agent 1

Agent 2

P(Y,Z|eY)

P(X,Z)

P(Z|eY)

P(Y,Z)

x P(X,Z)P(Z)-1

evidence eY

P(X,Z|eY)

separatesZ and X Y

X = {A,B}

Z = {C,D}

Y = {E,G,F}

P(Y|X,Z) = P(Y|Z)

→→

Likelihood given X and Z

of evidence on Y

Likelihood given Z

of evidence on Y=

→ It is sufficient for agent 2 to send its posterior on Z

to agent 1 for the latter to compute its posterior on X

Agent 1

Agent 2

P(Y,Z|eY)

P(Z|eY)

P(eY)-1

P(Z)-1P(X,Z) P(Z|eY)

P(Z)-1= P(X,Z) P(Z,eY)

P(eY)-1= P(X,Z) P(eY|Z)

P(eY)-1= P(X,Z) P(eY|X,Z)

P(X,Z|eY)P(eY)-1= P(X,Z,eY) =

Because:

P(eY|Z)P(eY|X,Z)

E F G K L MH I J

Specifications

• A Bayes net

- query variables

The naïve solution

E F G K L MH I J

C D C D C D

Using separation

• Agent 1 only needs to represent two extra variables

• Agent 1 may compute its posterior queries faster from CD than from EFGHIJK

• Communication lines need to transmit two variables instead of three

E F G K L MH I JE F G K L MH I J

C D C D C DC D C D C D

Communication Graph ConsiderationsCommunication Graph Considerations

Agent 6 receives info from agent 1 through both agent 4 and 5.

How should subnet 6 deal with possible redundancy?

A communication graphOne solution (often adopted) would be to impose a tree structure to the communication graph

• When choosing the communication graph, one should take into consideration

- The quality of the possible communication lines- The processing speed of the agents- The importance of given queries

If this is the key decision-making agent

...then this communication graph is more appropriate… than this one

• In a tree communication graph every edge is the only communication line

between two parts of the network• Hence it must deliver enough information so that the evidence received in

one part may convey its impact to the query variables of the other part• We restrict ourselves to the case where every node represented by an agent can be queried or receive evidence• In this case it is sufficient that the set of variables Z, that will be represented in any communication line, separates the set X of variables of one side of the network from the set Y of variables of the other side

Using a Junction Tree Using a Junction Tree to construct an MSBNto construct an MSBN

• The junction tree and its use• Building a Junction tree

– Moralization– Triangulation– The junction graph– From junction graph to junction tree

• Partitioning the junction tree• Adding and removing agents• A note on continuous variables

Secondary Structure/Junction Tree• multi-dim. random variables• joint probabilities (potentials)

Bayesian Network• one-dim. random variables• conditional probabilities

eghdef

ad ae ce

The junction tree and its useThe junction tree and its use

eghdef

• A junction tree is a graphical model of a probability space:

- Nodes of a JT are sets of variables

- Edges of a JT (called sepsets) are labeled by the intersection of the set of variables of the nodes they join

• The set of variables Z of any edge of a JT separates the set of variables of the sub-trees of both sides of this edge

e.g.{a,e} separates {a,b,d,e,f} and {a,c,e,g,h}

eghdef

So any partition a junction tree into sub-trees will allow for distributed inference

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Example of the Junction Tree ApproachExample of the Junction Tree Approach

Agent 1

Agent 2

Agent 4

query nodes

evidence nodes

query nodesevidence nodes

query nodes

evidence nodes

Building a Junction TreeBuilding a Junction Tree

Moral Graph

Triangulated Graph

Junction Tree

Identifying Cliques

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Building a Junction TreeBuilding a Junction Tree: MoralizationMoralization

1) Add an edge between every node having a common child.

2) Drop the directions of all other edges.

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Building a Junction TreeBuilding a Junction Tree: TriangulationTriangulation

Add edges to the graph to triangulate (induced) cycles of length greater than three.

This is the only

induced cycle

of length greater

than three

There are only two ways

to triangulate it...So we’ll choose this

way for our example

This way can be shown

to be problematic

Building a Junction TreeBuilding a Junction Tree: Junction GraphJunction Graph

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• A complete subgraph is one with edges between every vertex of the subgraph.

• A clique is a complete subgraph contained in no other complete subgraph.

This is a clique.This is a clique.

This is NOT a clique.Now it IS a clique.

Building a Junction TreeBuilding a Junction Tree: Junction GraphJunction Graph

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1) Identify Cliques.Every clique corresponds to a node in the JG.

3) Label edges of JG with intersection of cliques.

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2) Draw an edge between two nodes if they share variables.

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From Junction Graph to Junction TreeFrom Junction Graph to Junction Tree

1) Weight every edge with the number of variables it is labeled with

2) Find a maximal weight spanning (i.e. covering all JG nodes) tree

3) The corresponding subgraph of the JG is a Junction Tree (JT)

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The cliques are

the nodes of the JT

The edges of the JT

are called sepsets

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Partitioning the Junction TreePartitioning the Junction Tree

• Partition the nodes of the JT:

Sets of the partition → subnets

• Edges between two nodes of different subnets → communication lines

• Desirable for portion of JT inside a subnet to be connected

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Corresponding SubnetsCorresponding Subnets

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Evidence EvidenceEvidence

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Partitioning the Junction TreePartitioning the Junction Tree

Adding and removing AgentsAdding and removing Agents

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• Here we address the problem of adding and removing agents to the network

• Consider the BN given earlier

• Is it possible to add and remove agents (containing sensor variables) to the

network and perform inference without reconfiguring the network

Adding and removing Agents (2)Adding and removing Agents (2)

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• Adding the cliques containing the new variables does not change the structure of the Junction tree, so new agents containing these variables may easily be added and removed, along with a single communication line to the rest of the network.

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• Adding the cliques containing the new variables does not change the structure of the Junction tree, so new agents containing these variables may easily be added and removed, along with a single communication line to the rest of the network.

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• It is not desirable to have sensors chained as such since evidence received in one agent must pass through other agents to reach the central agent

• It would be preferable to have the sensor agents communicate with the central agent directly

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• By adding an extra variable the appropriate clique, we now have a different junction tree structure more fit for our application

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A note on continuous variablesA note on continuous variables

• Wish to extend JT inference to handle continuous variables.

• JT inference ↔ Potential manipulation• In general, a potential Φ on X=X1, ..., Xn is

a function Φ: X1 x ... x Xn → [0,+∞)• We need to define

– Multiplication (and division) of two potentials• Use function multiplication for this

– marginalization of a potential• use integration instead of summation

• Prior distributions and all possible evidence likelihood functions must be represented algebraically by a class of functions closed under multiplication and integration

• If the class of functions doesn’t encompass the prior distributions and evidence exactly, the question arises whether approximate inference or discretization might yield better results

• Algebraic manipulation is not a straightforward computational task

• Exact integration is not always possible

• Is numerical integration necessarily better than approximate inference or discretization?

MatLab ToolMatLab Tool

• Setup

• Functionality

• Adding and removing subnets

>> setBN('benzene.dne')

The global BN is now "benzene.dne".

The CPDs are:------------------ ------------------------ ------------------------| P(A) | | P(C|A) | | P(B|A) | |----------------| |----------------------| |----------------------| | A | Values | | A C | Values | | A B | Values | |----------------| |----------------------| |----------------------| | True | 0.3000 | | True True | 0.7500 | | True True | 1.0000 | | False | 0.7000 | | True False | 0.2500 | | True False | 0.0000 | ------------------ | False True | 0.1000 | | False True | 0.3000 |

| False False | 0.9000 | | False False | 0.7000 | ------------------------ ------------------------

------------------------ ------------------------ ------------------------------| P(E|C) | | P(D|B) | | P(F|D,E) | |----------------------| |----------------------| |----------------------------| | C E | Values | | B D | Values | | E D F | Values | |----------------------| |----------------------| |----------------------------| | True True | 0.2000 | | True True | 0.2500 | | True True True | 0.1000 | | True False | 0.8000 | | True False | 0.7500 | | True True False | 0.9000 | | False True | 0.6000 | | False True | 0.3000 | | True False True | 0.3000 | | False False | 0.4000 | | False False | 0.7000 | | True False False | 0.7000 | ------------------------ ------------------------ | False True True | 0.2000 |

• Bayes Net

– variables and states

– CPDs

– entered as simple text format or translated from a .dne Netica file

• Junction tree from Bayes Net

• Partition of the junction tree

Inputs

i + [Enter] to Initialize, v + [Enter] to view all variables (even those

containing no information),e + [Enter] to enter evidence, c + [Enter] to perform a inter-subnet

communication,p + [Enter] to go to the previous step, n + [Enter] to go to the next step,a + [Enter] to add a sensor, r + [Enter] to remove a sensor,t + [Enter] to turn true marginals view ON, m + [Enter] to turn discrepancy marking OFF,s + [Enter] to save to a file, q + [Enter] to quit.

Enter Command:

• Insert evidence into given subnets and propagate their impact inside the subnet

• Initiate communication between subnets, followed by the propagation of new information

• View the marginal distributions of the different subnets at every step

• Step forward and backward• Save eye-friendly logs to a

Main functionality

MatLab Tool: DisplayMatLab Tool: Display* Configuration 2: After evidence L(e|C) = (2,5) has been entered into subnet number 2

The TRUTH:------------------ ------------------ ------------------ ------------------ ------------------ ------------------| A | Values | | C | Values | | B | Values | | E | Values | | D | Values | | F | Values | |----------------| |----------------| |----------------| |----------------| |----------------| |----------------| | True | 0.2005 | | True | 0.1434 | | True | 0.4403 | | True | 0.5426 | | True | 0.2780 | | True | 0.2901 | | False | 0.7995 | | False | 0.8566 | | False | 0.5597 | | False | 0.4574 | | False | 0.7220 | | False | 0.7099 | ------------------ ------------------ ------------------ ------------------ ------------------ ------------------

SUBNET 1 (adjacent to subnets 2): Err(ACB) = 0.0527.~~~~ AD = 0.0704 / ~~~~ AD = 0.1072 / ~~~~ AD = 0.0493 / ------------------ ------------------ ------------------/ A / Values / / C / Values / / B / Values / | E | Values | | D | Values | | F | Values | /~~~~~~~~~~~~~~~~/ /~~~~~~~~~~~~~~~~/ /~~~~~~~~~~~~~~~~/ |----------------| |----------------| |----------------| / True / 0.3000 / / True / 0.2950 / / True / 0.5100 / | True | ###### | | True | ###### | | True | ###### | / False / 0.7000 / / False / 0.7050 / / False / 0.4900 / | False | ###### | | False | ###### | | False | ###### | ~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~ ------------------ ------------------ ------------------

SUBNET 2 (adjacent to subnets 1, 3):------------------ ------------------ ------------------ ------------------ ------------------ ------------------| A | Values | | C | Values | | B | Values | | E | Values | | D | Values | | F | Values | |----------------| |----------------| |----------------| |----------------| |----------------| |----------------| | True | ###### | | True | 0.1434 | | True | 0.4403 | | True | 0.5426 | | True | 0.2780 | | True | ###### | | False | ###### | | False | 0.8566 | | False | 0.5597 | | False | 0.4574 | | False | 0.7220 | | False | ###### | ------------------ ------------------ ------------------ ------------------ ------------------ ------------------

SUBNET 3 (adjacent to subnets 2): Err(EDF) = 0.0169.------------------ ------------------ ------------------ ~~~~ AD = 0.0429 / ~~~~ AD = 0.0025 / ~~~~ AD = 0.0048 / | A | Values | | C | Values | | B | Values | / E / Values / / D / Values / / F / Values / |----------------| |----------------| |----------------| /~~~~~~~~~~~~~~~~/ /~~~~~~~~~~~~~~~~/ /~~~~~~~~~~~~~~~~/ | True | ###### | | True | ###### | | True | ###### | / True / 0.4820 / / True / 0.2745 / / True / 0.2969 / | False | ###### | | False | ###### | | False | ###### | / False / 0.5180 / / False / 0.7255 / / False / 0.7031 / ------------------ ------------------ ------------------ ~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~

Enter a command (enter h + [Enter] for help):

Indicates step number and last action that was taken

Shows the marginal distributions that would have been obtained by infering on the entire Bayes Net

Shows the marginal distributions of the variables represented in each subnet

Prompts for new action

MatLab Tool: DisplayMatLab Tool: Display* Configuration 2: After evidence L(e|C) = (2,5) has been entered into subnet number 2

The TRUTH:------------------ ------------------ ------------------ ------------------ ------------------ ------------------| A | Values | | C | Values | | B | Values | | E | Values | | D | Values | | F | Values | |----------------| |----------------| |----------------| |----------------| |----------------| |----------------| | True | 0.2005 | | True | 0.1434 | | True | 0.4403 | | True | 0.5426 | | True | 0.2780 | | True | 0.2901 | | False | 0.7995 | | False | 0.8566 | | False | 0.5597 | | False | 0.4574 | | False | 0.7220 | | False | 0.7099 | ------------------ ------------------ ------------------ ------------------ ------------------ ------------------

SUBNET 1 (adjacent to subnets 2): Err(ACB) = 0.0527.~~~~ AD = 0.0704 / ~~~~ AD = 0.1072 / ~~~~ AD = 0.0493 / ------------------ ------------------ ------------------/ A / Values / / C / Values / / B / Values / | E | Values | | D | Values | | F | Values | /~~~~~~~~~~~~~~~~/ /~~~~~~~~~~~~~~~~/ /~~~~~~~~~~~~~~~~/ |----------------| |----------------| |----------------| / True / 0.3000 / / True / 0.2950 / / True / 0.5100 / | True | ###### | | True | ###### | | True | ###### | / False / 0.7000 / / False / 0.7050 / / False / 0.4900 / | False | ###### | | False | ###### | | False | ###### | ~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~ ------------------ ------------------ ------------------

SUBNET 2 (adjacent to subnets 1, 3):------------------ ------------------ ------------------ ------------------ ------------------ ------------------| A | Values | | C | Values | | B | Values | | E | Values | | D | Values | | F | Values | |----------------| |----------------| |----------------| |----------------| |----------------| |----------------| | True | ###### | | True | 0.1434 | | True | 0.4403 | | True | 0.5426 | | True | 0.2780 | | True | ###### | | False | ###### | | False | 0.8566 | | False | 0.5597 | | False | 0.4574 | | False | 0.7220 | | False | ###### | ------------------ ------------------ ------------------ ------------------ ------------------ ------------------

SUBNET 3 (adjacent to subnets 2): Err(EDF) = 0.0169.------------------ ------------------ ------------------ ~~~~ AD = 0.0429 / ~~~~ AD = 0.0025 / ~~~~ AD = 0.0048 / | A | Values | | C | Values | | B | Values | / E / Values / / D / Values / / F / Values / |----------------| |----------------| |----------------| /~~~~~~~~~~~~~~~~/ /~~~~~~~~~~~~~~~~/ /~~~~~~~~~~~~~~~~/ | True | ###### | | True | ###### | | True | ###### | / True / 0.4820 / / True / 0.2745 / / True / 0.2969 / | False | ###### | | False | ###### | | False | ###### | / False / 0.5180 / / False / 0.7255 / / False / 0.7031 / ------------------ ------------------ ------------------ ~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~

Enter a command (enter h + [Enter] for help): Indicates that subnet does not represent this variable

Indicates that marginal variable distribution matches “true marginal”

Indicates that marginal variable distribution differs from “true marginal”

Shows average discrapancy between subnet marginals and “true marginals”

Shows average discrepancy between subnet and true joint distributions of the variables of the subnet

• Bayes Net

– variables and states

– CPDs

• Junction tree from Bayes Net

• Partition of the junction tree

>> setBN('benzene.dne')

The global BN is now "benzene.dne".

The CPDs are: ------------------ ------------------------ ------------------------ | P(A) | | P(C|A) | | P(B|A) | |----------------| |----------------------| |----------------------| | A | Values | | A C | Values | | A B | Values | |----------------| |----------------------| |----------------------| | True | 0.3000 | | True True | 0.7500 | | True True | 1.0000 | | False | 0.7000 | | True False | 0.2500 | | True False | 0.0000 | ------------------ | False True | 0.1000 | | False True | 0.3000 | | False False | 0.9000 | | False False | 0.7000 | ------------------------ ------------------------

------------------------ ------------------------ ------------------------------ | P(E|C) | | P(D|B) | | P(F|D,E) | |----------------------| |----------------------| |----------------------------| | C E | Values | | B D | Values | | E D F | Values | |----------------------| |----------------------| |----------------------------| | True True | 0.2000 | | True True | 0.2500 | | True True True | 0.1000 | | True False | 0.8000 | | True False | 0.7500 | | True True False | 0.9000 | | False True | 0.6000 | | False True | 0.3000 | | True False True | 0.3000 | | False False | 0.4000 | | False False | 0.7000 | | True False False | 0.7000 | ------------------------ ------------------------ | False True True | 0.2000 | | False True False | 0.8000 | Description of cliques: | False False True | 0.4000 | Clique 1 has nodes A, C, B. | False False False | 0.6000 | Clique 2 has nodes C, B, E. ------------------------------ Clique 3 has nodes B, E, D.Clique 4 has nodes E, D, F.

Inputs

i + [Enter] to Initialize,

v + [Enter] to view all variables (even those containing no information),

e + [Enter] to enter evidence,

c + [Enter] to perform a inter-subnet communication,

p + [Enter] to go to the previous step,

n + [Enter] to go to the next step,a + [Enter] to add a sensor,

r + [Enter] to remove a sensor,t + [Enter] to turn true marginals view ON,

m + [Enter] to turn discrepancy marking

OFF,s + [Enter] to save to a file,

q + [Enter] to quit.

Enter Command:

• Insert evidence into given subnets and propagate their impact inside the subnet

• Initiate communication between subnets, followed by the propagation of new information

• View the marginal distributions of the different subnets at every step

• Step forward and backward• Save eye-friendly logs to a

Main functionality

MatLab Tool: DisplayMatLab Tool: Display * Configuration 2: After evidence L(e|C) = (2,5) has been entered into subnet number 2

The TRUTH:

------------------ ------------------ ------------------ ------------------ ------------------ ------------------

| A | Values | | C | Values | | B | Values | | E | Values | | D | Values | | F | Values |

|----------------| |----------------| |----------------| |----------------| |----------------| |----------------|

| True | 0.2005 | | True | 0.1434 | | True | 0.4403 | | True | 0.5426 | | True | 0.2780 | | True | 0.2901 |

| False | 0.7995 | | False | 0.8566 | | False | 0.5597 | | False | 0.4574 | | False | 0.7220 | | False | 0.7099 |

------------------ ------------------ ------------------ ------------------ ------------------ ------------------

SUBNET 1 (adjacent to subnets 2): Err(ACB) = 0.0527.

~~~~ AD = 0.0704 / ~~~~ AD = 0.1072 / ~~~~ AD = 0.0493 / ------------------ ------------------ ------------------

/ A / Values / / C / Values / / B / Values / | E | Values | | D | Values | | F | Values |

/~~~~~~~~~~~~~~~~/ /~~~~~~~~~~~~~~~~/ /~~~~~~~~~~~~~~~~/ |----------------| |----------------| |----------------|

/ True / 0.3000 / / True / 0.2950 / / True / 0.5100 / | True | ###### | | True | ###### | | True | ###### |

/ False / 0.7000 / / False / 0.7050 / / False / 0.4900 / | False | ###### | | False | ###### | | False | ###### |

~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~ ------------------ ------------------ ------------------

SUBNET 2 (adjacent to subnets 1, 3):

------------------ ------------------ ------------------ ------------------ ------------------ ------------------

|----------------| |----------------| |----------------| |----------------| |----------------| |----------------|

| True | ###### | | True | 0.1434 | | True | 0.4403 | | True | 0.5426 | | True | 0.2780 | | True | ###### |

| False | ###### | | False | 0.8566 | | False | 0.5597 | | False | 0.4574 | | False | 0.7220 | | False | ###### |

------------------ ------------------ ------------------ ------------------ ------------------ ------------------

SUBNET 3 (adjacent to subnets 2): Err(EDF) = 0.0169.

------------------ ------------------ ------------------ ~~~~ AD = 0.0429 / ~~~~ AD = 0.0025 / ~~~~ AD = 0.0048 /

| A | Values | | C | Values | | B | Values | / E / Values / / D / Values / / F / Values /

|----------------| |----------------| |----------------| /~~~~~~~~~~~~~~~~/ /~~~~~~~~~~~~~~~~/ /~~~~~~~~~~~~~~~~/

| True | ###### | | True | ###### | | True | ###### | / True / 0.4820 / / True / 0.2745 / / True / 0.2969 /

| False | ###### | | False | ###### | | False | ###### | / False / 0.5180 / / False / 0.7255 / / False / 0.7031 /

------------------ ------------------ ------------------ ~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~

Indicates step number and last action that was taken

Shows the marginal distributions that would have been obtained by infering on the entire Bayes Net

Shows the marginal distributions of the variables represented in each subnet

Prompts for new action

MatLab Tool: DisplayMatLab Tool: Display * Configuration 2: After evidence L(e|C) = (2,5) has been entered into subnet number 2

The TRUTH:

------------------ ------------------ ------------------ ------------------ ------------------ ------------------

|----------------| |----------------| |----------------| |----------------| |----------------| |----------------|

| True | 0.2005 | | True | 0.1434 | | True | 0.4403 | | True | 0.5426 | | True | 0.2780 | | True | 0.2901 |

| False | 0.7995 | | False | 0.8566 | | False | 0.5597 | | False | 0.4574 | | False | 0.7220 | | False | 0.7099 |

------------------ ------------------ ------------------ ------------------ ------------------ ------------------

SUBNET 1 (adjacent to subnets 2): Err(ACB) = 0.0527.

~~~~ AD = 0.0704 / ~~~~ AD = 0.1072 / ~~~~ AD = 0.0493 / ------------------ ------------------ ------------------

/ A / Values / / C / Values / / B / Values / | E | Values | | D | Values | | F | Values |

/~~~~~~~~~~~~~~~~/ /~~~~~~~~~~~~~~~~/ /~~~~~~~~~~~~~~~~/ |----------------| |----------------| |----------------|

/ True / 0.3000 / / True / 0.2950 / / True / 0.5100 / | True | ###### | | True | ###### | | True | ###### |

/ False / 0.7000 / / False / 0.7050 / / False / 0.4900 / | False | ###### | | False | ###### | | False | ###### |

~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~ ------------------ ------------------ ------------------

SUBNET 2 (adjacent to subnets 1, 3):

------------------ ------------------ ------------------ ------------------ ------------------ ------------------

|----------------| |----------------| |----------------| |----------------| |----------------| |----------------|

| True | ###### | | True | 0.1434 | | True | 0.4403 | | True | 0.5426 | | True | 0.2780 | | True | ###### |

| False | ###### | | False | 0.8566 | | False | 0.5597 | | False | 0.4574 | | False | 0.7220 | | False | ###### |

------------------ ------------------ ------------------ ------------------ ------------------ ------------------

SUBNET 3 (adjacent to subnets 2): Err(EDF) = 0.0169.

------------------ ------------------ ------------------ ~~~~ AD = 0.0429 / ~~~~ AD = 0.0025 / ~~~~ AD = 0.0048 /

| A | Values | | C | Values | | B | Values | / E / Values / / D / Values / / F / Values /

|----------------| |----------------| |----------------| /~~~~~~~~~~~~~~~~/ /~~~~~~~~~~~~~~~~/ /~~~~~~~~~~~~~~~~/

| True | ###### | | True | ###### | | True | ###### | / True / 0.4820 / / True / 0.2745 / / True / 0.2969 /

| False | ###### | | False | ###### | | False | ###### | / False / 0.5180 / / False / 0.7255 / / False / 0.7031 /

------------------ ------------------ ------------------ ~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~

Indicates that subnet does not represent this variable

Indicates that marginal variable distribution differs from “true marginal”

Shows average discrapancy between subnet marginals and “true marginals”

Shows average discrepancy between subnet and true joint distributions of the variables of the subnet

Indicates that marginal variable distribution matches “true marginal”

MatLab Tool: ControlMatLab Tool: ControlEnter a command (enter h + [Enter] for help): hType:

i + [Enter] to Initialize, v + [Enter] to view all variables (even those containing no information),e + [Enter] to enter evidence, c + [Enter] to perform a inter-subnet communication,

p + [Enter] to go to the previous step, n + [Enter] to go to the next step,

a + [Enter] to add a sensor, r + [Enter] to remove a sensor,t + [Enter] to turn true marginals view OFF, m + [Enter] to turn discrepancy marking OFF,s + [Enter] to save to a file, q + [Enter] to quit.

So, what do you want to do?

MatLab Tool: Initial stateMatLab Tool: Initial state * Configuration 1: Initial configuration:

SUBNET 1 (adjacent to subnets 2, 3, 4, 5): ------------------ ------------------ ------------------ ------------------ ------------------ ------------------ | CT | Values | | RF1 | Values | | RF2 | Values | | CF3 | Values | | CM7 | Values | | RF5 | Values | |----------------| |----------------| |----------------| |----------------| |----------------| |----------------| | True | 0.0069 | | True | 0.8064 | | True | 0.6287 | | True | 0.5420 | | True | 0.2257 | | True | 0.4738 | | False | 0.9931 | | False | 0.1936 | | False | 0.3713 | | False | 0.4580 | | False | 0.7743 | | False | 0.5262 | ------------------ ------------------ ------------------ ------------------ ------------------ ------------------

SUBNET 2 (adjacent to subnets 1): ------------------ ------------------ ------------------ | CM7 | Values | | RM8 | Values | | RF5 | Values | |----------------| |----------------| |----------------| | True | 0.2257 | | True | 0.8332 | | True | 0.4738 | | False | 0.7743 | | False | 0.1668 | | False | 0.5262 | ------------------ ------------------ ------------------

SUBNET 3 (adjacent to subnets 1): ------------------ ------------------ ------------------ ------------------ ------------------ | CF3 | Values | | RF5 | Values | | RF4 | Values | | RM6 | Values | | RM5 | Values | |----------------| |----------------| |----------------| |----------------| |----------------| | True | 0.5420 | | True | 0.4738 | | True | 0.4616 | | True | 0.5000 | | True | 0.2202 | | False | 0.4580 | | False | 0.5262 | | False | 0.5384 | | False | 0.5000 | | False | 0.7798 | ------------------ ------------------ ------------------ ------------------ ------------------

SUBNET 4 (adjacent to subnets 1): ------------------ ------------------ ------------------ ------------------ | RF1 | Values | | RF2 | Values | | RM1 | Values | | RM2 | Values | |----------------| |----------------| |----------------| |----------------| | True | 0.8064 | | True | 0.6287 | | True | 0.2472 | | True | 0.5120 | | False | 0.1936 | | False | 0.3713 | | False | 0.7528 | | False | 0.4880 | ------------------ ------------------ ------------------ ------------------

MatLab Tool: Initial stateMatLab Tool: Initial state * Configuration 1: Initial configuration:

RF2RF1

RF5RF2

RM1RM2

RM3RM4

CM7RF5CF3

RF5RF4RM5

MatLab Tool: Remove a sensorMatLab Tool: Remove a sensor * Configuration 2: The radar-subnet number 5 was removed.

SUBNET 1 (adjacent to subnets 2, 3, 4): ------------------ ------------------ ------------------ ------------------ ------------------ ------------------ | CT | Values | | RF1 | Values | | RF2 | Values | | CF3 | Values | | CM7 | Values | | RF5 | Values | |----------------| |----------------| |----------------| |----------------| |----------------| |----------------| | True | 0.0069 | | True | 0.8064 | | True | 0.6287 | | True | 0.5420 | | True | 0.2257 | | True | 0.4738 | | False | 0.9931 | | False | 0.1936 | | False | 0.3713 | | False | 0.4580 | | False | 0.7743 | | False | 0.5262 | ------------------ ------------------ ------------------ ------------------ ------------------ ------------------

RF2RF1

RF5RF2

RM1RM2

CM7RF5CF3

RF5RF4RM5

MatLab Tool: Receive evidenceMatLab Tool: Receive evidenceEnter a command (enter h + [Enter] for help): e

In which subnet (number) do you wish to insert evidence? 2

The variables handled by subnet number 2 of this cell are: 5: CM7 6: RM8 7: RF5 Which variable NUMBER does the evidence pertain to? 6

Enter likelihood for state "True": 0

Enter likelihood for state "False": 1

RF2RF1

RF5RF2

RM1RM2

CM7RF5CF3

RF5RF4RM5

Subnet 2 receives evidence on RM8.

MatLab Tool: Receive evidenceMatLab Tool: Receive evidence * Configuration 3: After evidence L(e|RM8) = (0,1) has been entered into subnet number 2

SUBNET 1 (adjacent to subnets 2, 3, 4): Err(CTCF3RF5) = 0.0368. Err(CTCM7RF5) = 0.0395. ------------------ ------------------ ------------------ ------------------ ------------------ ~~~~ AD = 0.0784 / | CT | Values | | RF1 | Values | | RF2 | Values | | CF3 | Values | | CM7 | Values | / RF5 / Values / |----------------| |----------------| |----------------| |----------------| |----------------| /~~~~~~~~~~~~~~~~/ | True | 0.0069 | | True | 0.8064 | | True | 0.6287 | | True | 0.5420 | | True | 0.2257 | / True / 0.4738 / | False | 0.9931 | | False | 0.1936 | | False | 0.3713 | | False | 0.4580 | | False | 0.7743 | / False / 0.5262 / ------------------ ------------------ ------------------ ------------------ ------------------ ~~~~~~~~~~~~~~~~~~

SUBNET 3 (adjacent to subnets 1): Err(CF3RM5RF5RF4) = 0.0169. Err(RF5RM6) = 0.0555. ------------------ ~~~~ AD = 0.0784 / ~~~~ AD = 0.0119 / ------------------ ------------------ | CF3 | Values | / RF5 / Values / / RF4 / Values / | RM6 | Values | | RM5 | Values | |----------------| /~~~~~~~~~~~~~~~~/ /~~~~~~~~~~~~~~~~/ |----------------| |----------------| | True | 0.5420 | / True / 0.4738 / / True / 0.4616 / | True | 0.5000 | | True | 0.2202 | | False | 0.4580 | / False / 0.5262 / / False / 0.5384 / | False | 0.5000 | | False | 0.7798 | ------------------ ~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~ ------------------ ------------------

RF2RF1

RF5RF2

RM1RM2

CM7RF5CF3

RF5RF4RM5

MatLab Tool: Receive evidenceMatLab Tool: Receive evidenceEnter a command (enter h + [Enter] for help): e

In which subnet (number) do you wish to insert evidence? 4

The variables handled by subnet number 4 of this cell are: 2: RF1 3: RF2 11: RM1 12: RM2 Which variable NUMBER does the evidence pertain to? 12

Enter likelihood for state "True": 3

Enter likelihood for state "False": 5

RF2RF1

RF5RF2

RM1RM2

CM7RF5CF3

RF5RF4RM5

Subnet 4 receives evidence on RM2.

MatLab Tool: Receive evidenceMatLab Tool: Receive evidence * Configuration 4: After evidence L(e|RM2) = (3,5) has been entered into subnet number 4

SUBNET 1 (adjacent to subnets 2, 3, 4): Err(CTRF1RF2) = 0.0059. Err(CTCF3RF5) = 0.0368. Err(CTCM7RF5) = 0.0395. ------------------ ~~~~ AD = 0.0012 / ~~~~ AD = 0.0063 / ------------------ ------------------ ~~~~ AD = 0.0784 / | CT | Values | / RF1 / Values / / RF2 / Values / | CF3 | Values | | CM7 | Values | / RF5 / Values / |----------------| /~~~~~~~~~~~~~~~~/ /~~~~~~~~~~~~~~~~/ |----------------| |----------------| /~~~~~~~~~~~~~~~~/ | True | 0.0069 | / True / 0.8064 / / True / 0.6287 / | True | 0.5420 | | True | 0.2257 | / True / 0.4738 / | False | 0.9931 | / False / 0.1936 / / False / 0.3713 / | False | 0.4580 | | False | 0.7743 | / False / 0.5262 / ------------------ ~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~ ------------------ ------------------ ~~~~~~~~~~~~~~~~~~

RF2RF1

RF5RF2

RM1RM2

CM7RF5CF3

RF5RF4RM5

* Configuration 5: After subnet 4 sent information to subnet 1

SUBNET 1 (adjacent to subnets 2, 3, 4): Err(CTCF3RF5) = 0.0368. Err(CTCM7RF5) = 0.0395. ------------------ ------------------ ------------------ ------------------ ------------------ ~~~~ AD = 0.0784 / | CT | Values | | RF1 | Values | | RF2 | Values | | CF3 | Values | | CM7 | Values | / RF5 / Values / |----------------| |----------------| |----------------| |----------------| |----------------| /~~~~~~~~~~~~~~~~/ | True | 0.0068 | | True | 0.8081 | | True | 0.6376 | | True | 0.5420 | | True | 0.2257 | / True / 0.4738 / | False | 0.9932 | | False | 0.1919 | | False | 0.3624 | | False | 0.4580 | | False | 0.7743 | / False / 0.5262 / ------------------ ------------------ ------------------ ------------------ ------------------ ~~~~~~~~~~~~~~~~~~

MatLab Tool: Subnet communicationMatLab Tool: Subnet communication

RF2RF1

RF5RF2

RM1RM2

CM7RF5CF3

RF5RF4RM5

234Enter a command (enter h + [Enter] for help): c Send information FROM subnet number: 4 Subnet 4 can communicate with the following subnets: 1 Enter the subnet number that subnet 4 should send information

RF2RF1

RF5RF2

RM1RM2

CM7RF5CF3

RF5RF4RM5

234Enter a command (enter h + [Enter] for help): c Send information FROM subnet number: 2 Subnet 4 can communicate with the following subnets: 1 Enter the subnet number that subnet 4 should send information

RF2RF1

RF5RF2

RM1RM2

CM7RF5CF3

RF5RF4RM5

234Enter a command (enter h + [Enter] for help): c Send information FROM subnet number: 1 Subnet 4 can communicate with the following subnets: 2, 3, 4 Enter the subnet number that subnet 4 should send information TO:

MatLab Tool: Add a subnetMatLab Tool: Add a subnet * Configuration 8: The radar-subnet number 5 was added.

SUBNET 1 (adjacent to subnets 2, 3, 4, 5): Err(CTRF1RF2) = 0.0227. Err(CTCF3RF5) = 0.0004. Err(CTCM7RF5) = 0.0004. ~~~~ AD = 0.0011 / ~~~~ AD = 0.0464 / ~~~~ AD = 0.0087 / ------------------ ~~~~ AD = 0.0002 / ------------------ / CT / Values / / RF1 / Values / / RF2 / Values / | CF3 | Values | / CM7 / Values / | RF5 | Values | /~~~~~~~~~~~~~~~~/ /~~~~~~~~~~~~~~~~/ /~~~~~~~~~~~~~~~~/ |----------------| /~~~~~~~~~~~~~~~~/ |----------------| / True / 0.0068 / / True / 0.8081 / / True / 0.6376 / | True | 0.5420 | / True / 0.2257 / | True | 0.3628 | / False / 0.9932 / / False / 0.1919 / / False / 0.3624 / | False | 0.4580 | / False / 0.7743 / | False | 0.6372 | ~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~ ------------------ ~~~~~~~~~~~~~~~~~~ ------------------

SUBNET 2 (adjacent to subnets 1): Err(CM7RM8RF5) = 0.0001. ~~~~ AD = 0.0002 / ------------------ ------------------ / CM7 / Values / | RM8 | Values | | RF5 | Values | /~~~~~~~~~~~~~~~~/ |----------------| |----------------| / True / 0.2257 / | True | 0.0000 | | True | 0.3629 | / False / 0.7743 / | False | 1.0000 | | False | 0.6371 | ~~~~~~~~~~~~~~~~~~ ------------------ ------------------

SUBNET 4 (adjacent to subnets 1): Err(RF1RF2RM1RM2) = 0.0099. ~~~~ AD = 0.0464 / ~~~~ AD = 0.0087 / ~~~~ AD = 0.0009 / ~~~~ AD = 0.0011 / / RF1 / Values / / RF2 / Values / / RM1 / Values / / RM2 / Values / /~~~~~~~~~~~~~~~~/ /~~~~~~~~~~~~~~~~/ /~~~~~~~~~~~~~~~~/ /~~~~~~~~~~~~~~~~/ / True / 0.8081 / / True / 0.6376 / / True / 0.2633 / / True / 0.3863 / / False / 0.1919 / / False / 0.3624 / / False / 0.7367 / / False / 0.6137 / ~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~

RF2RF1

RF5RF2

RM1RM2

RM3RM4

CM7RF5CF3

RF5RF4RM5

Enter a command (enter h + [Enter] for help): a

MatLab Tool: Subnet communicationMatLab Tool: Subnet communication * Configuration 9: After subnet 5 sent information to subnet 1

SUBNET 2 (adjacent to subnets 1): Err(CM7RM8RF5) = 0.0001. ~~~~ AD = 0.0002 / ------------------ ------------------ / CM7 / Values / | RM8 | Values | | RF5 | Values | /~~~~~~~~~~~~~~~~/ |----------------| |----------------| / True / 0.2257 / | True | 0.0000 | | True | 0.3629 | / False / 0.7743 / | False | 1.0000 | | False | 0.6371 | ~~~~~~~~~~~~~~~~~~ ------------------ ------------------

RF2RF1

RF5RF2

RM1RM2

RM3RM4

CM7RF5CF3

RF5RF4RM5

RF2RF1

RF5RF2

RM1RM2

RM3RM4

CM7RF5CF3

RF5RF4RM5

RF2RF1

RF5RF2

RM1RM2

RM3RM4

CM7RF5CF3

RF5RF4RM5

RF2RF1

RF5RF2

RM1RM2

RM3RM4

CM7RF5CF3

RF5RF4RM5

An Implementation of Multiply Sectioned Bayesian Networks

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Transcript of An Implementation of Multiply Sectioned Bayesian Networks

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