Probabilistic Reasoning System 2: Bayesian network

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Probabilistic Reasoning System 2Bayesian network

• CS 570
• Team12
• ??? ??, ???, ???, ???

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Todays topics

• 1. Inference in Multiply Connected Network
• Clustering algorithms
• Cutset conditioning methods
• 2. Approximate inference (Monte Carlo Algorithms)
• Direct sampling methods
• Rejection sampling in Bayesian networks
• Likelihood weighting
• Markov Chain Monte Carlo algorithm
• 3. Implementation tools and applications of
  Bayesian network

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Singly connected Network/ Polytree
P(E) ____ .002
P(B) ____ .001
Burglary
Earthquake

•  A singly connected network is a network in
  which there is at most one undirected path
  between any nodes in it 
• Size is defined as the number of CPT entries.

B E P(A) _______________ t t .95 t f .94 f t .29 f
f .001
Alarm
MaryCalls
JohnCalls
A P(M) _________ t .70 f .01
A P(J) _______ t .90 f .05
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Multiply connected Network

•  Network in which two nodes are connected by
  more than one path 
• That means
• Two or more possible causes for some variable
• One variable can influence another variable
  through more than one causal mechanism

C P(R) ________ t .80 f .20
Cloudy
P(C).5
Rain
Sprinkler
S R P(W) ________________ t t .99 t f .90 f t .90
f f .00
C P(S) ________ t .10 f .50
WetGrass
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Clustering Algorithms (1)

• Combining individual nodes of the network to form
  cluster nodes (meganode) in such a way that the
  resulting network is a polytree (Singly connected
  network)
• The meganode has only one parent.
• Time can be reduced to O(n)

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Clustering Algorithms (2)
P(C).5
Cloudy
P(C).5
Cloudy
C P(R) ________ t .80 f .20
C P(S) ________ t .10 f .50
SprRain
C P(SRx) t t t f f t f f _____________________
_______________ t .08 .02 .72 .18 f .10 .40 .10 40
Rain
Sprinkler
WetGrass
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
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Cutset conditioning

• a cutset is a set of variables that can be
  instantiated to yield polytree
• each polytree has one or more variables
  instantiated to a definite value
• the number of resulting polytrees is exponential
  in the size of the cutset, so the smallest cutset
  is the best
• evaluating the most likely polytrees first is
  called bounded cutset conditioning

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Cutset conditioning example
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• 1. Inference in Multiply Connected Network
• Clustering algorithms
• Cutset conditioning methods
• 2. Approximate inference (Monte Carlo Algorithms)
• Direct sampling methods
• Rejection sampling in Bayesian networks
• Likelihood weighting
• Markov Chain Monte Carlo algorithm
• 3. Implementation tools and applications of
  Bayesian network

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Monte Carlo Algorithms

• Provide approximate answers whose accuracy
  depends on the number of samples generated
• Two families
• Direct sampling
• Markov Chain sampling

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Direct sampling methods (1)

• No evidence associated with the network
• The algorithm samples each variable in turn
• For nodes without parents, sample from their
  distribution for nodes with parents, sample from
  the conditional distribution.
• Example
• Sample from P(Cloudy) 0.5,0.5 gt true
• Sample from P(Sprinklercloudy True) 0.1,0.9
  gt false
• Sample from P(RainCloudy True) 0.8,0.2 gt
  true
• Sample from P(WetGrassSprinkler false,
  RainTrue)0.9,0.1 gt true

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P(b).05
FamilyOut
P(f).2
BowelProblems
LightsOn
DogOut
D P(hD) t .6 f .25
F B P(dF,B) t t .994 t f .88 f t .96 f f .2
HearDogBark
F P(lF) t .99 f .1
Query What is the probability of family at home,
dog has no bowel problems and isnt out, the
light is off, and dogs barking can be
heard? We generate 100,000 samples from the
network We obtain NS(f,b,l,d,h)
13740 13740/100000 0.1374 P(f,b,l,d,h)
0.80.950.90.80.25 0.1368
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Direct sampling methods(2)Prior-Sample algorithm

• Function PRIOR-SAMPLE(bn) returns an event
  sampled from the prior specified by bn
• inputs bn, a Bayesian network specifying joint
  distribution P(X1,,Xn)
• x lt- an event with n elements
• for i 1 to n do
• xi lt- a random sample from P(Xi parents(Xi))
• return x

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Direct sampling methods(3)

• Looking at the sampling process, we have
• Which remind
• This estimated probability becomes exact in the
  large-sample limit. We call it  consistent 

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Rejection sampling (1)

• Method to approximate conditional probabilities
  P(Xe) given evidence e
• P(Xe) Ns(X,e) / Ns(e)
• 1. First, generates samples from the prior
  distribution specified by the network.
• 2. Then, rejects all those that do not match the
  evidence
• 3. Finally, the estimate P(Xxe) is obtained by
  counting how often Xx occurs in the remaining
  samples.

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Rejection sampling (2) algorithm

• Function Rejection-Sampling(X,e,b,N) returns
  an estimate of P(Xe)
• Inputs X, the query variable
• e, evidence specified as an event
• bn, a Bayesian network
• N, the total number of samples to be
  generated
• Local variables N, a vector of counts over X,
  initially zero
• For j1 to N do
• x lt- PRIOR-SAMPLE(bn)
• if x is consistent with e then
• Nx lt- Nx 1 where x is the value of X in x
• return NORMALIZE(NX)

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Rejection sampling (3)Example

• Let us assume we want to estimate
  P(rainSprinkler true)
• 100 samples
• 73 samples gt Sprinkler false
• 27 samples gt Sprinkler true
• 8 samples gt Rain true
• 19 samples gt Rain false

• P(RainSprinkler true) NORMALIZE(8,19)
  0.296,0.704.
• The true answer is 0.3,0.7
• As more samples are collected, the estimate will
  converge to the true answer

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Rejection sampling (4) Problems

• Problems
• Rejection sampling rejects too many samples.
• The fraction of samples consistent with the
  evidence e drops exponentially as the number of
  evidence variable drops.
• Procedure unusable for complex problems

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Likelihood weighting (1)

• Avoids the inefficiency of rejection sampling by
  generating only events that are consistent with
  the evidence e.
• That is, It fixes the values for the evidence
  variable E and samples only the remaining
  variables X and Y.
• Each event generated is consistent with the
  evidence
• Also, it weights each sample by product of
  conditional probabilities of evidence variables,
  given its parents

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Likelihood weighting (2)Example for one sample

• Example
• Query P(RainSprinklertrue, WetGrass true)
• The weight is set to 1.0
• 1. Sample from P(Cloudy) 0.5,0.5 gt true
• 2. Sprinkler is an evidence variable with value
  true
• w lt- w P(Sprinklertrue Cloudy true) 0.1
• 3. Sample from P(RainCloudytrue)0.8,0.2 gt
  true
• 4. WetGrass is an evidence variable with value
  true
• w lt- w P(WetGrasstrue Sprinklertrue, Rain
  true) 0.1

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Likelihood-Weighting(3) algorithm

• function LIKELIHOOD-WEIGHTING(X,e,bn,N) returns
  an estimate of P(Xe)
• inputs X, the query variable
• e, evidence specified as an event
• bn, a Bayesian Network
• N, the total number of samples to be generated
• local variables W, a vector of weighted counts
  over X, initially zero
• for j 1 to N do
• x, w lt- Weighted-Sample(bn)
• Wx lt- Wx w where x is the value of X in
  x
• return NORMALIZE(WX)
• __________________________________________________
  _______________________________
• function WEIGHTED-SAMPLE(e,bn) returns an event
  and a weight
• x lt- an event with n elements w lt- 1
• for i 1 to n do
• if Xi has a value xi in e
• then w lt- w P(Xixi parents(Xi))
• else xi lt- a random sample from
  P(Xiparents(Xi))

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Likelihood weighting (4)Example for one query
P(B).05
FamilyOut
P(F).2
BowelProblems
LightsOn
DogOut
D P(HD) t .6 f .25
F B P(HD) t t .994 t f .88 f t .96 f f .2
HearDogBark
F P(LF) t .99 f .1
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Likelihood weighting (4)Example for one query

• For N100000, obtain
• Correct

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Markov Chain Monte Carlo algorithm(1)

• Some vocabulary
• A node is conditionally independent of all other
  nodes in the network, given its parents,
  children, and childrens parents that is, given
  its Markov Blanket. For example, Burglary is
  independent of JohnCalls and MaryCalls, given
  alarm and Earthquake.
• A sequence of discrete random variables X0, X1,
  is called a Markov chain with state space S iff
• Thus, Xn is conditionally independent of all
  other variables given Xn-1

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Markov Chain Monte Carlo algorithm(2)

• function MCMC-Ask(X,e,bn,N) returns an estimate
  of P(Xe)
• local variables NX, a vector of counts over
  X, initially zero
• Z, the nonevidence variables in bn.
• x, the current state of the network,
  initially copied from e
• initialize x with random values for the
  variables in Z
• for j 1 to N do
• Nx lt- Nx 1 where x is the value of X in x
• for each Zi in Z do
• sample the value of Zi in x from P(Zimb(Zi))
  given the values of MB(Zi) in x
• return NORMALIZE(NX)

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Markov Chain Monte Carlo algorithm(3)

• Lets 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

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Markov Chain Monte Carlo Example1

• Query P(RainSprinkler 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(CloudySprinkler true,
  Rainfalse)
• 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(RainCloudyfalse,Sprinkler
  true, Rainfalse)
• Suppose the result is Rain true.
• Then current state is false, true, true, true
• After all the iterations, lets 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

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Markov Chain Monte Carlo Example2

• Query P(Fl,d)
• Start with arbitrary non-evidence settings
  (f,b,h)
• Pick F, sample from P(Fl,d,b), obtain f
• Pick B, sample from P(B f,d), obtain b
• Pick H, sample from P(Hd), obtain d
• Iterate last three steps 50000 times, keep last
  10000 states
• Obtain P(fl,d) 0.9016(correct 0.90206)

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• 1. Inference in Multiply Connected Network
• Clustering algorithms
• Cutset conditioning methods
• 2. Approximate inference (Monte Carlo Algorithms)
• Direct sampling methods
• Rejection sampling in Bayesian networks
• Likelihood weighting
• Markov Chain Monte Carlo algorithm
• 3. Implementation tools and applications of
  Bayesian network

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Implementation Tool Microsoft Belief Networks

• MSBNx is a component-based Windows application
  for creating, assessing, and evaluating Bayesian
  Networks, created at Microsoft Research
• MSBNx http//www.research.microsoft.com/adapt/MSB
  Nx/
• Why MSBNx?
• Its free and available on the net.
• Its easy to learn how to use it.
• We can specify full and causally independent
  probability distributions
• If we have sufficient data and use machine
  learning tools to create Bayesian Networks, we
  can use MSBNx to edit and evaluate the results

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Implementation Tool Microsoft Belief Networks

• How to use it?
• First, we make the nodes on the window
• Then, we specify all the probabilities
• Finally, we do what we want to do with the
  network

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Implementation Tool Microsoft Belief Networks
(example) making the network
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Implementation Tool Microsoft Belief Networks
(example) assessing probabilities(1)
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Implementation Tool Microsoft Belief Networks
(example) assessing probabilities(2)
CPT
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Implementation Tool Microsoft Belief Networks
(example) Obtaining results without evidence(1)
To obtain results
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Implementation Tool Microsoft Belief Networks
(example) Obtaining results without evidence(2)
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Implementation Tool Microsoft Belief Networks
(example) Obtaining results with evidence(1)
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Implementation Tool Microsoft Belief Networks
(example) Obtaining results with evidence(2)
Evidence
Evidence
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Implementation Tool Microsoft Belief Networks
(example) Obtaining results with evidence(3)
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