1. Probabilistic Reasoning (2)
Daehwan Kim,
Ravshan Khamidov,
Sehyong Kim

2. Contents Basics of Bayesian Networks (BNs) Construction Inference
Single and Multi connected BNs
Inference in Multi-connected BNs
Clustering algorithms
Cutset conditioning
Approximate Inference In Bayesian Network
Direct Sampling
Markov Chain Monte Carlo
Example applications of BNs

3. Representing Knowledge in an Uncertainty
Joint probability distribution
Answer any questions about the domain
Become intractably large as the number of variables grows
Specifying probabilities for atomic events is unnatural and difficult

4. Representing Knowledge in an Uncertainty
A Bayesian network
Provide concise way to present conditional independence relationships in the domain
Specifies full joint distribution but often exponentially smaller than the full joint distribution

5. Basics of Bayesian Network Definition
Topology of the network + CPT
A set of random variables makes up the nodes of the network
A set of directed links or arrows connects pairs of nodes. Example: X->Y  means X has a direct influence on Y.
Each node has a conditional probability table (CPT) that quantifies the effects that the parents have on the node. The parents of a node are all those nodes that have arrows pointing to it.
The graph has no directed cycles (hence is a directed, acyclic graph, or DAG).

6. Basics of Bayesian Network Construction
General procedure for incremental network construction:
choose the set of relevant variables Xi that describe the domain
choose the ordering for the variables
while there are variables left:
pick a variable Xi and add a node to the network for it
set Parent(Xi) by testing its conditional independence in the net
define the conditional probability table for Xi

7. Basics of Bayesian Network Topology of the network
Suppose we choose the ordering B,E,A,J,M
P(B|E)= P(E)? Yes
P(A|B)= P(A)? P(A|E)= P(A)? No
P(J|A,B,E)= P(J|A)? Yes
P(J |A)= P(J)? No
P(M|A)= P(M)? No
P(M|A,J)= P(M|A)? Yes
Burglary
Earthquake
Alarm
MaryCalls
JohnCalls

8. Basics of Bayesian Network Conditional Probability Table (CPT)
Once we get the topology of the network, conditional probability table (CPT) must be specified for each node.
Example of CPT for the variable WetGrass: C
S R
P(W=F) P(W=T)
F F
T F
F T
T T S R W

9. Basics of Bayesian Network Conditional Probability Table (CPT)
Each row in the table contains the conditional probability of each node value for a conditioning case. Each row must sum to 1, because the entries represent an exhaustive set of cases for the variable.
A conditioning case is a possible combination of values for the parent nodes.

10. Basics of Bayesian Network Example
Representing Knowledge Example
P(C=F) P(C=T)
Cloudy C
P(S=F) P(S=T) F T C
P(R=F) P(R=T) F T
Spinkler
Rain
WetGrass
S R
P(W=F) P(W=T)
F F
T F
F T
T T

11. Basics of Bayesian Network Example

12. Basics of Bayesian Network Example
In the example, notice that the two causes "compete" to "explain" the observed data. Hence S and R become conditionally dependent given that their common child, W, is observed, even though they are marginally independent.
For example, suppose the grass is wet, but that we also know that it is raining. Then the posterior probability that the sprinkler is on goes down:
Pr(S=1|W=1,R=1) =

13. Basics of Bayesian Network Inference
Inference in Bayesian network means computing the probability distribution of a set of query variables, given a set of evidence variables

14. Basics of Bayesian Network Exact inference
Inference by enumeration (with alarm example) B E A J M

15. Basics of Bayesian Network Exact inference
Variable elimination by distributive law
In general B E A J M Q H E

16. Basics of Bayesian Network Exact inference
Another example of variable elimination C W R S

17. Basics of Bayesian Network Complexity of exact inference
O(n) for polytree(singly connected network) - there exist at most one undirected path between any two nodes in the networks (e.g. alarm example)
Multiply connected network: exponential time (e.g. wet grass example)

18. Inference in Multi-connected BNs Clustering algorithm (aka Join Tree algorithm)
Basic idea
Transform network into probabilistically equivalent single connected BN (aka polytree) by merging (clustering) offending nodes
Most effective approach for exact evaluation of multiple connected BNs
The “new” node has only one parent
The time is reduced and is O(n)

19. Inference in Multi-connected BNs Clustering algorithm (aka Join Tree algorithm)
P(C)=.5
P(C)=.5
C P(S)
________
t .10
f .50
Cloudy
Cloudy
C P(R)
________
t .80
f .20
C P(S+R=x)
t t t f f t f f
_________________________ t f
Spr+Rain
Rain
Sprinkler
WetGrass
S R P(W)
________________
t t .99
t f .90
f t .90
f f .00
S R P(W)
______________
t t .99
t f .90
f t .90
f f .00
WetGrass

20. Inference in Multi-connected BNs Clustering algorithm (aka Join Tree algorithm)
Jensen Join-tree (Jensen, 1996) version the current most efficient algorithm in this class (e.g. was used in Hugin, Netica)
Network evaluation is done in two stages
Compile into join-tree
May be slow
May require too much memory if original network is highly connected
Do belief updating in join-tree (usually fast)
Note: clustered nodes have increased complexity; updates may be computationally complex

21. Inference in Multi-connected BNs Cutset conditioning
The Basic idea
find a minimal set of nodes whose instantiation will make the remainder of the network single connected and therefore safe for propagation
Historical note: This technique to deal with the problem of propagation was suggested by Pearl.

22. Inference in Multi-connected BNs Cutset conditioning methods
Once a variable is instantiated it can be duplicated and thus “break” a cycle
A cutset is a set of variables whose instantiation makes the graph a polytree
Each polytree’s likelihood is used as a weight when combining the results
Evaluating the most likely polytrees first is called bounded cutset conditioning

23. Inference in Multi-connected BNs Cutset conditioning - Examples
Eliminate Cloudy from the BN; Sum(Cloudy+,Cloudy-)
Cloudy+
Cloudy+
Cloudy-
Cloudy-
P(R)=0.8
P(R)=0.2
Sprinkler
Rain
Sprinkler
Rain
Wet Grass
Wet Grass
P(S)=0.1
P(S)=0.5
C P(S) C P(R)
T 0.10 T 0.80
F 0.50 F 0.20

24. Approximate Inference In Bayesian Network
Solution to intractably large, multiply connected networks
Monte Carlo algorithm
Widely used to estimate quantities that are difficult to calculate exactly
Randomized sampling algorithm
Accuracy depends on the number of samples
Two families
Direct sampling
Markov chaining sampling

25. Direct Sampling Method
Procedure
Sampling from known probability distribution
Estimate value as
(# of matched samples) / (# of total samples)
Sampling order
Sample each variable in turn, in topological order

26. Example in simple case Sampling Estimating Cloudy
P(C)=.5
Sampling
Cloudy
[Cloudy, Sprinkler, Rain, WetGrass]
C P(R)
________
t .80
f .20
C P(S)
________
t .10
f .50
[true, , , ]
[true, false, , ]
Rain
Sprinkler
[true, false, true, ]
[true, false, true, true]
WetGrass
S R P(W)
______________
t t .99
t f .90
f t .90
f f .00
N = 1000
N(Rain=true) = N([ _ , _ , true, _ ]) = 511
P(Rain=true) = 0.511
Estimating

27. Rejection Sampling Used in compute conditional probabilities Procedure
Generating sample from prior distribution specified by the Bayesian Network
Rejecting all that do not match the evidence
Estimating probability

28. Rejection Sampling Example
Let us assume we want to estimate P(Rain|Sprinkler = true) with 100 samples
100 samples
73 samples => Sprinkler = false
27 samples => Sprinkler = true
8 samples => Rain = true
19 samples => Rain = false
P(Rain|Sprinkler = true) = NORMALIZE({8,19}) = {0.296,0.704}
Problem
It rejects too many samples

29. Likelihood Weighting Advantage Idea
Avoiding inefficiency of rejection sampling
Idea
Generating only events consistent with evidence
Each event is weighted by likelihood that the event accords to the evidence

30. Likelihood Weighting Example
P(Rain|Sprinkler=true, WetGrass = true)?
Sampling
The weight is set to 1.0
1. Sample from P(Cloudy) = {0.5,0.5} => true
2. Sprinkler is an evidence variable with value true
w  w * P(Sprinkler=true | Cloudy = true) = 0.1
3. Sample from P(Rain|Cloudy=true)={0.8,0.2} => true
4. WetGrass is an evidence variable with value true
w w * P(WetGrass=true |Sprinkler=true, Rain = true) = 0.099
[true, true, true, true] with weight 0.099
Estimating
Accumulating weights to either Rain=true or Rain=false
Normalize

31. Markov Chain Monte Carlo
Let’s think of the network as being in a particular current state specifying a value for every variable
MCMC generates each event by making a random change to the preceding event
The next state is generated by randomly sampling a value for one of the nonevidence variables Xi, conditioned on the current values of the variables in the MarkovBlanket of Xi

32. Markov Blanket Markov blanket: Parents + children + children’s parents
Node is conditionally independent of all other nodes in network, given its Markov Blanket

33. Markov Chain Monte Carlo Example
Query P(Rain|Sprinkler = true, WetGrass = true)
Initial state is [true, true, false, true]
The following steps are executed repeatedly:
Cloudy is sampled, given the current values of its MarkovBlanket variables
So, we sample from P(Cloudy|Sprinkler = true, Rain=false)
Suppose the result is Cloudy = false.
Then current state is [false, true, false, true]
Rain is sampled, given the current values of its MarkovBlanket variables
So, we sample from P(Rain|Cloudy=false,Sprinkler = true, Rain=false)
Suppose the result is Rain = true.
Then current state is [false, true, true, true]
After all the iterations, let’s say the process visited 20 states where rain is true and 60 states where rain is false then the answer of the query is NORMALIZE({20,60})={0.25,0.75}

34. Example applications of BNs
Microsoft Belief Networks
Advantages
Easy to learn how to use
We can specify full and casually independent probability distributions
And finally it is free
Netica – from Norsys Software Corp/
Disadvatages
Not free, commercial product

35. Example applications of BNs Microsoft Belief Networks

36. Example applications of BNs Netica

37. Thank you!