Post on 20-May-2020
Overview
1. Probabilistic Reasoning/Graphical models
2. Importance Sampling
3. Markov Chain Monte Carlo: Gibbs Sampling
4. Sampling in presence of Determinism
5. Rao-Blackwellisation
6. AND/OR importance sampling
Overview
1. Probabilistic Reasoning/Graphical models
2. Importance Sampling
3. Markov Chain Monte Carlo: Gibbs Sampling
4. Sampling in presence of Determinism
5. Cutset-based Variance Reduction
6. AND/OR importance sampling
Probabilistic Reasoning;Graphical models
• Graphical models:
– Bayesian network, constraint networks, mixed network
• Queries
• Exact algorithm
– using inference,
– search and hybrids
• Graph parameters:
– tree-width, cycle-cutset, w-cutset
Queries
• Probability of evidence (or partition function)
• Posterior marginal (beliefs):
• Most Probable Explanation
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Approximation
• Since inference, search and hybrids are too expensive when graph is dense; (high treewidth) then:
• Bounding inference:• mini-bucket and mini-clustering• Belief propagation
• Bounding search:• Sampling
• Goal: an anytime scheme
Overview
1. Probabilistic Reasoning/Graphical models
2. Importance Sampling
3. Markov Chain Monte Carlo: Gibbs Sampling
4. Sampling in presence of Determinism
5. Rao-Blackwellisation
6. AND/OR importance sampling
Outline
• Definitions and Background on Statistics
• Theory of importance sampling
• Likelihood weighting
• State-of-the-art importance sampling techniques
7
A sample
• Given a set of variables X={X1,...,Xn}, a sample, denoted by St is an instantiation of all variables:
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How to draw a sample ?Univariate distribution
• Example: Given random variable X having domain {0, 1} and a distribution P(X) = (0.3, 0.7).
• Task: Generate samples of X from P.
• How?
– draw random number r [0, 1]
– If (r < 0.3) then set X=0
– Else set X=1
9
How to draw a sample?Multi-variate distribution
• Let X={X1,..,Xn} be a set of variables
• Express the distribution in product form
• Sample variables one by one from left to right, along the ordering dictated by the product form.
• Bayesian network literature: Logic sampling
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Sampling for Prob. InferenceOutline
• Logic Sampling
• Importance Sampling
– Likelihood Sampling
– Choosing a Proposal Distribution
• Markov Chain Monte Carlo (MCMC)
– Metropolis-Hastings
– Gibbs sampling
• Variance Reduction
Logic Sampling:
No Evidence (Henrion 1988)
Input: Bayesian network
X= {X1,…,XN}, N- #nodes, T - # samples
Output: T samples
Process nodes in topological order – first process the
ancestors of a node, then the node itself:
1. For t = 0 to T
2. For i = 0 to N
3. Xi sample xit from P(xi | pai)
12
Logic sampling (example)
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Logic Sampling w/ Evidence
Input: Bayesian network
X= {X1,…,XN}, N- #nodes
E – evidence, T - # samples
Output: T samples consistent with E
1. For t=1 to T
2. For i=1 to N
3. Xi sample xit from P(xi | pai)
4. If Xi in E and Xi xi, reject sample:
5. Goto Step 1.
14
Logic Sampling (example)
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Expected value: Given a probability distribution P(X)
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Variance: The variance of g w.r.t. P is:
Expected value and Variance
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Monte Carlo Estimate
• Estimator:
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– It produces an estimate of the unknown parameter of the sampling distribution.
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= 50 if X equals 1.
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Outline
• Definitions and Background on Statistics
• Theory of importance sampling
• Likelihood weighting
• State-of-the-art importance sampling techniques
19
Importance sampling: Main idea
• Express query as the expected value of a random variable w.r.t. to a distribution Q.
• Generate random samples from Q.
• Estimate the expected value from the generated samples using a monte carloestimator (average).
20
Importance sampling for P(e)
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• Convergence: By Weak law of large numbers
• Asymptotically unbiased
• Variance
– Harder to analyze
– Liu suggests a measure called “Effective sample size”
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Generating samples from Q
• No restrictions on “how to”
• Typically, express Q in product form:
– Q(Z)=Q(Z1)xQ(Z2|Z1)x….xQ(Zn|Z1,..Zn-1)
• Sample along the order Z1,..,Zn
• Example:
– Z1Q(Z1)=(0.2,0.8)
– Z2 Q(Z2|Z1)=(0.1,0.9,0.2,0.8)
– Z3 Q(Z3|Z1,Z2)=Q(Z3)=(0.5,0.5)
Outline
• Definitions and Background on Statistics
• Theory of importance sampling
• Likelihood weighting
• State-of-the-art importance sampling techniques
27
Likelihood Weighting(Fung and Chang, 1990; Shachter and Peot, 1990)
28
Works well for likely evidence!
“Clamping” evidence+logic sampling+weighing samples by evidence likelihood
Is an instance of importance sampling!
Likelihood Weighting: Sampling
29
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Sample in topological order over X !
Clamp evidence, Sample xi P(Xi|pai), P(Xi|pai) is a look-up in CPT!
Likelihood Weighting: Proposal Distribution
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Likelihood Weighting: Estimates
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Likelihood Weighting
• Converges to exact posterior marginals
• Generates Samples Fast
• Sampling distribution is close to prior (especially if E Leaf Nodes)
• Increasing sampling variance
Convergence may be slow
Many samples with P(x(t))=0 rejected
32
Outline
• Definitions and Background on Statistics
• Theory of importance sampling
• Likelihood weighting
• Error estimation
• State-of-the-art importance sampling techniques
33
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Outline
• Definitions and Background on Statistics
• Theory of importance sampling
• Likelihood weighting
• State-of-the-art importance sampling techniques
38
Proposal selection
• One should try to select a proposal that is as close as possible to the posterior distribution.
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Perfect sampling using Bucket Elimination
• Algorithm:
– Run Bucket elimination on the problem along an ordering o=(XN,..,X1).
– Sample along the reverse ordering: (X1,..,XN)
– At each variable Xi, recover the probability P(Xi|x1,...,xi-1) by referring to the bucket.
41
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Bucket elimination (BE) Algorithm elim-bel (Dechter 1996)
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Sampling from the output of BE(Dechter 2002)
bucket B:
P(A)
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bucket C:
bucket D:
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Mini-buckets: “local inference”
• Computation in a bucket is time and space exponential in the number of variables involved
• Therefore, partition functions in a bucket into “mini-buckets” on smaller number of variables
• Can control the size of each “mini-bucket”, yielding polynomial complexity.
SP2 44
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Mini-Bucket Elimination
bucket A:
bucket E:
bucket D:
bucket C:
bucket B:
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Mini-buckets
ΣB
P(C|A) hB(C,e)
hD(A)
hC(A,e)
Approximation of P(e)
Space and Time constraints:Maximum scope size of the new function generated should be bounded by 2
BE generates a function having scope size 3. So it cannot be used.
P(A)
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Sampling from the output of MBE
bucket A:
bucket E:
bucket D:
bucket C:
bucket B: P(B|A) P(D|B,A)
hE(A)
hB(A,D)
P(e|B,C)
P(C|A) hB(C,e)
hD(A)
hC(A,e)Sampling is same as in BE-sampling except that now we construct Q from a randomly selected “mini-bucket”
IJGP-Sampling (Gogate and Dechter, 2005)
• Iterative Join Graph Propagation (IJGP)
– A Generalized Belief Propagation scheme (Yedidiaet al., 2002)
• IJGP yields better approximations of P(X|E) than MBE
– (Dechter, Kask and Mateescu, 2002)
• Output of IJGP is same as mini-bucket “clusters”
• Currently the best performing IS scheme!
Current Research question
• Given a Bayesian network with evidence or a Markov network representing function P, generate another Bayesian network representing a function Q (from a family of distributions, restricted by structure) such that Q is closest to P.
• Current approaches– Mini-buckets– Ijgp– Both
• Experimented, but need to be justified theoretically.
Algorithm: Approximate Sampling
1) Run IJGP or MBE
2) At each branch point compute the edge probabilities by consulting output of IJGP or MBE
• Rejection Problem:
– Some assignments generated are non solutions
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Adaptive Importance Sampling
• General case
• Given k proposal distributions
• Take N samples out of each distribution
• Approximate P(e)
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Overview
1. Probabilistic Reasoning/Graphical models
2. Importance Sampling
3. Markov Chain Monte Carlo: Gibbs Sampling
4. Sampling in presence of Determinism
5. Rao-Blackwellisation
6. AND/OR importance sampling
Markov Chain
• A Markov chain is a discrete random process with the property that the next state depends only on the
current state (Markov Property):
54
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• If P(Xt|xt-1) does not depend on t (time homogeneous) and state space is finite, then it is often expressed as a transition function (aka transition matrix) 1)(
x
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Example: Drunkard’s Walk• a random walk on the number line where, at
each step, the position may change by +1 or −1 with equal probability
55
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Multi-Variable System
• state is an assignment of values to all the variables
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Bayesian Network System
• Bayesian Network is a representation of the joint probability distribution over 2 or more variables
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Stationary DistributionExistence
• If the Markov chain is time-homogeneous, then the vector (X) is a stationary distribution (aka invariant or equilibrium distribution, aka “fixed point”), if its entries sum up to 1 and satisfy:
• Finite state space Markov chain has a unique stationary distribution if and only if:– The chain is irreducible
– All of its states are positive recurrent
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Irreducible
• A state x is irreducible if under the transition rule one has nonzero probability of moving from x to any other state and then coming back in a finite number of steps
• If one state is irreducible, then all the states must be irreducible
(Liu, Ch. 12, pp. 249, Def. 12.1.1)
61
Recurrent
• A state x is recurrent if the chain returns to x
with probability 1
• Let M(x ) be the expected number of steps to return to state x
• State x is positive recurrent if M(x ) is finiteThe recurrent states in a finite state chain are positive recurrent .
Stationary Distribution Convergence
• Consider infinite Markov chain:nnn PPxxPP 00)( )|(
• Initial state is not important in the limit
“The most useful feature of a “good” Markov chain is its fast forgetfulness of its past…”
(Liu, Ch. 12.1)
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• If the chain is both irreducible and aperiodic, then:
Aperiodic
• Define d(i) = g.c.d.{n > 0 | it is possible to go from i to i in n steps}. Here, g.c.d. means the greatest common divisor of the integers in the set. If d(i)=1 for i, then chain is aperiodic
• Positive recurrent, aperiodic states are ergodic
63
Markov Chain Monte Carlo
• How do we estimate P(X), e.g., P(X|e) ?
64
• Generate samples that form Markov Chain with stationary distribution =P(X|e)
• Estimate from samples (observed states):
visited states x0,…,xn can be viewed as “samples” from distribution
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MCMC Summary
• Convergence is guaranteed in the limit
• Samples are dependent, not i.i.d.
• Convergence (mixing rate) may be slow
• The stronger correlation between states, the slower convergence!
• Initial state is not important, but… typically, we throw away first K samples - “burn-in”
65
Gibbs Sampling (Geman&Geman,1984)
• Gibbs sampler is an algorithm to generate a sequence of samples from the joint probability distribution of two or more random variables
• Sample new variable value one variable at a time from the variable’s conditional distribution:
• Samples form a Markov chain with stationary distribution P(X|e)
66
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The process of Gibbs sampling can be understood as a random walk in the space of all instantiations of X=x (remember drunkard’s walk):
In one step we can reach instantiations that differ from current one by value assignment to at most one variable (assume randomized choice of variables Xi).
Ordered Gibbs Sampler
Generate sample xt+1 from xt :
In short, for i=1 to N:
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Transition Probabilities in BN
Markov blanket:
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69
Computation is linear in the size of Markov blanket!
Ordered Gibbs Sampling Algorithm(Pearl,1988)
Input: X, E=e
Output: T samples {xt }
Fix evidence E=e, initialize x0 at random1. For t = 1 to T (compute samples)
2. For i = 1 to N (loop through variables)
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Gibbs Sampling Example - BN
71
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Gibbs Sampling Example - BN
72
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Answering Queries P(xi |e) = ?• Method 1: count # of samples where Xi = xi (histogram estimator):
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• Method 2: average probability (mixture estimator):
• Mixture estimator converges faster (consider estimates for the unobserved values of Xi; prove via Rao-Blackwell theorem)
Rao-Blackwell Theorem
Rao-Blackwell Theorem: Let random variable set X be composed of two groups of variables, R and L. Then, for the joint distribution (R,L) and function g, the following result applies
74
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for a function of interest g, e.g., the mean or covariance (Casella&Robert,1996, Liu et. al. 1995).
• theorem makes a weak promise, but works well in practice!• improvement depends the choice of R and L
Importance vs. Gibbs
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Gibbs Sampling: Convergence
76
• Sample from `P(X|e)P(X|e)
• Converges iff chain is irreducible and ergodic
• Intuition - must be able to explore all states:– if Xi and Xj are strongly correlated, Xi=0 Xj=0,
then, we cannot explore states with Xi=1 and Xj=1
• All conditions are satisfied when all probabilities are positive
• Convergence rate can be characterized by the second eigen-value of transition matrix
Gibbs: Speeding Convergence
Reduce dependence between samples (autocorrelation)
• Skip samples
• Randomize Variable Sampling Order
• Employ blocking (grouping)
• Multiple chains
Reduce variance (cover in the next section)
77
Blocking Gibbs Sampler
• Sample several variables together, as a block
• Example: Given three variables X,Y,Z, with domains of size 2, group Y and Z together to form a variable W={Y,Z} with domain size 4. Then, given sample (xt,yt,zt), compute next sample:
+ Can improve convergence greatly when two variables are strongly correlated!
- Domain of the block variable grows exponentially with the #variables in a block!
78
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xZYPwzy
wPzyXPx
Gibbs: Multiple Chains
• Generate M chains of size K
• Each chain produces independent estimate Pm:
79
M
i
mPM
P1
1ˆ
K
t
i
t
iim xxxPK
exP1
)\|(1
)|(
Treat Pm as independent random variables.
• Estimate P(xi|e) as average of Pm (xi|e) :
Gibbs Sampling Summary
• Markov Chain Monte Carlo method(Gelfand and Smith, 1990, Smith and Roberts, 1993, Tierney, 1994)
• Samples are dependent, form Markov Chain
• Sample from which converges to
• Guaranteed to converge when all P > 0
• Methods to improve convergence:– Blocking
– Rao-Blackwellised
80
)|( eXP )|( eXP
Overview
1. Probabilistic Reasoning/Graphical models
2. Importance Sampling
3. Markov Chain Monte Carlo: Gibbs Sampling
4. Sampling in presence of Determinism
5. Rao-Blackwellisation
6. AND/OR importance sampling
Sampling: Performance
• Gibbs sampling
– Reduce dependence between samples
• Importance sampling
– Reduce variance
• Achieve both by sampling a subset of variables and integrating out the rest (reduce dimensionality), aka Rao-Blackwellisation
• Exploit graph structure to manage the extra cost
82
Smaller Subset State-Space
• Smaller state-space is easier to cover
83
},,,{ 4321 XXXXX },{ 21 XXX
64)( XD 16)( XD
Smoother Distribution
84
00
01
10
11
0
0.1
0.2
0001
1011
P(X1,X2,X3,X4)
0-0.1 0.1-0.2 0.2-0.26
0
1
0
0.1
0.2
0
1
P(X1,X2)
0-0.1 0.1-0.2 0.2-0.26
Speeding Up Convergence
• Mean Squared Error of the estimator:
PVarBIASPMSE QQ 2
2
2
][ˆ]ˆ[]ˆ[ PEPEPVarPMSE QQQQ
• Reduce variance speed up convergence !
• In case of unbiased estimator, BIAS=0
85
Rao-Blackwellisation
86
)}(~{]}|)([{)}({
)}(ˆ{
]}|)([{)}({
]}|)({var[]}|)([{)}({
]}|)([]|)([{1
)(~
)}()({1
)(ˆ
1
1
xgVarT
lxhEVar
T
xhVarxgVar
lxgEVarxgVar
lxgElxgEVarxgVar
lxhElxhET
xg
xhxhT
xg
LRX
T
T
Liu, Ch.2.3
Rao-Blackwellisation
• X=RL
• Importance Sampling:
• Gibbs Sampling: – autocovariances are lower (less correlation
between samples)
– if Xi and Xj are strongly correlated, Xi=0 Xj=0, only include one fo them into a sampling set
87
})(
)({}
),(
),({
RQ
RPVar
LRQ
LRPVar QQ
Liu, Ch.2.5.5
“Carry out analytical computation as much as possible” - Liu
Blocking Gibbs Sampler vs. Collapsed
• Standard Gibbs:
(1)
• Blocking:
(2)
• Collapsed:
(3)
88
X Y
Z
),|(),,|(),,|( yxzPzxyPzyxP
)|,(),,|( xzyPzyxP
)|(),|( xyPyxP
FasterConvergence
Collapsed Gibbs Sampling
Generating Samples
Generate sample ct+1 from ct :
89
),\|(
),,...,,|(
...
),,...,,|(
),,...,,|(
1
1
1
1
2
1
1
1
3
1
12
1
22
321
1
11
ecccPcC
eccccPcC
eccccPcC
eccccPcC
i
t
i
t
ii
t
K
tt
K
t
KK
t
K
ttt
t
K
ttt
from sampled
In short, for i=1 to K:
Collapsed Gibbs Sampler
Input: C X, E=e
Output: T samples {ct }
Fix evidence E=e, initialize c0 at random1. For t = 1 to T (compute samples)
2. For i = 1 to N (loop through variables)
3. cit+1 P(Ci | ct\ci)
4. End For
5. End For
Calculation Time
• Computing P(ci| ct\ci,e) is more expensive (requires inference)
• Trading #samples for smaller variance:
– generate more samples with higher covariance
– generate fewer samples with lower covariance
• Must control the time spent computing sampling probabilities in order to be time-effective!
91
Exploiting Graph Properties
Recall… computation time is exponential in the adjusted induced width of a graph
• w-cutset is a subset of variable s.t. when they are observed, induced width of the graph is w
• when sampled variables form a w-cutset , inference is exp(w) (e.g., using Bucket Tree
Elimination)
• cycle-cutset is a special case of w-cutset
92
Sampling w-cutset w-cutset sampling!
What If C=Cycle-Cutset ?
93
}{},{ 9
0
5
0
2
0 XE,xxc
X1
X7
X5 X4
X2
X9 X8
X3
X6
X1
X7
X4
X9 X8
X3
X6
P(x2,x5,x9) – can compute using Bucket Elimination
P(x2,x5,x9) – computation complexity is O(N)
Computing Transition Probabilities
94
),,1(:
),,0(:
932
932
xxxPBE
xxxPBE
X1
X7
X5 X4
X2
X9 X8
X3
X6
),,1()|1(
),,0()|0(
),,1(),,0(
93232
93232
932932
xxxPxxP
xxxPxxP
xxxPxxxP
Compute joint probabilities:
Normalize:
Cutset Sampling-Answering Queries
• Query: ci C, P(ci |e)=? same as Gibbs:
95
computed while generating sample tusing bucket tree elimination
compute after generating sample tusing bucket tree elimination
T
t
i
t
ii ecccPT
|ecP1
),\|(1
)(ˆ
T
t
t
ii ,ecxPT
|e)(xP1
)|(1
• Query: xi X\C, P(xi |e)=?
Cutset Sampling vs. Cutset Conditioning
96
)|()|(
)()|(
)|(1
)(
)(
1
ecPc,exP
T
ccountc,exP
,ecxPT
|e)(xP
CDc
i
CDc
i
T
t
t
ii
• Cutset Conditioning
• Cutset Sampling
)|()|()(
ecPc,exP|e)P(xCDc
ii
Cutset Sampling Example
97
)(
)(
)(
3
1)|(
)(
)(
)(
9
2
52
9
1
52
9
0
52
92
9
2
52
3
2
9
1
52
2
2
9
0
52
1
2
,x| xxP
,x| xxP
,x| xxP
xxP
,x| xxP x
,x| xxP x
,x| xxP x
X1
X7
X6 X5 X4
X2
X9 X8
X3
Estimating P(x2|e) for sampling node X2 :
Sample 1
Sample 2
Sample 3
Cutset Sampling Example
98
),,|(},{
),,|(},{
),,|(},{
9
3
5
3
23
3
5
3
2
3
9
2
5
2
23
2
5
2
2
2
9
1
5
1
23
1
5
1
2
1
xxxxPxxc
xxxxPxxc
xxxxPxxc
Estimating P(x3 |e) for non-sampled node X3 :
X1
X7
X6 X5 X4
X2
X9 X8
X3
),,|(
),,|(
),,|(
3
1)|(
9
3
5
3
23
9
2
5
2
23
9
1
5
1
23
93
xxxxP
xxxxP
xxxxP
xxP
CPCS54 Test Results
99
MSE vs. #samples (left) and time (right)
Ergodic, |X|=54, D(Xi)=2, |C|=15, |E|=3
Exact Time = 30 sec using Cutset Conditioning
CPCS54, n=54, |C|=15, |E|=3
0
0.001
0.002
0.003
0.004
0 1000 2000 3000 4000 5000
# samples
Cutset Gibbs
CPCS54, n=54, |C|=15, |E|=3
0
0.0002
0.0004
0.0006
0.0008
0 5 10 15 20 25
Time(sec)
Cutset Gibbs
CPCS179 Test Results
100
MSE vs. #samples (left) and time (right) Non-Ergodic (1 deterministic CPT entry)|X| = 179, |C| = 8, 2<= D(Xi)<=4, |E| = 35
Exact Time = 122 sec using Cutset Conditioning
CPCS179, n=179, |C|=8, |E|=35
0
0.002
0.004
0.006
0.008
0.01
0.012
100 500 1000 2000 3000 4000
# samples
Cutset Gibbs
CPCS179, n=179, |C|=8, |E|=35
0
0.002
0.004
0.006
0.008
0.01
0.012
0 20 40 60 80
Time(sec)
Cutset Gibbs
CPCS360b Test Results
101
MSE vs. #samples (left) and time (right)
Ergodic, |X| = 360, D(Xi)=2, |C| = 21, |E| = 36
Exact Time > 60 min using Cutset Conditioning
Exact Values obtained via Bucket Elimination
CPCS360b, n=360, |C|=21, |E|=36
0
0.00004
0.00008
0.00012
0.00016
0 200 400 600 800 1000
# samples
Cutset Gibbs
CPCS360b, n=360, |C|=21, |E|=36
0
0.00004
0.00008
0.00012
0.00016
1 2 3 5 10 20 30 40 50 60
Time(sec)
Cutset Gibbs
Random Networks
102
MSE vs. #samples (left) and time (right)
|X| = 100, D(Xi) =2,|C| = 13, |E| = 15-20
Exact Time = 30 sec using Cutset Conditioning
RANDOM, n=100, |C|=13, |E|=15-20
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.0035
0 200 400 600 800 1000 1200
# samples
Cutset Gibbs
RANDOM, n=100, |C|=13, |E|=15-20
0
0.0002
0.0004
0.0006
0.0008
0.001
0 1 2 3 4 5 6 7 8 9 10 11
Time(sec)
Cutset Gibbs
Coding NetworksCutset Transforms Non-Ergodic Chain to Ergodic
103
MSE vs. time (right)
Non-Ergodic, |X| = 100, D(Xi)=2, |C| = 13-16, |E| = 50
Sample Ergodic Subspace U={U1, U2,…Uk}
Exact Time = 50 sec using Cutset Conditioning
x1 x2 x3 x4
u1 u2 u3 u4
p1 p2 p3 p4
y4y3y2y1
Coding Networks, n=100, |C|=12-14
0.001
0.01
0.1
0 10 20 30 40 50 60
Time(sec)
IBP Gibbs Cutset
Non-Ergodic Hailfinder
104
MSE vs. #samples (left) and time (right)
Non-Ergodic, |X| = 56, |C| = 5, 2 <=D(Xi) <=11, |E| = 0
Exact Time = 2 sec using Loop-Cutset Conditioning
HailFinder, n=56, |C|=5, |E|=1
0.0001
0.001
0.01
0.1
1
1 2 3 4 5 6 7 8 9 10
Time(sec)
Cutset Gibbs
HailFinder, n=56, |C|=5, |E|=1
0.0001
0.001
0.01
0.1
0 500 1000 1500
# samples
Cutset Gibbs
CPCS360b - MSE
105
cpcs360b, N=360, |E|=[20-34], w*=20, MSE
0
0.000005
0.00001
0.000015
0.00002
0.000025
0 200 400 600 800 1000 1200 1400 1600
Time (sec)
Gibbs
IBP
|C|=26,fw=3
|C|=48,fw=2
MSE vs. Time
Ergodic, |X| = 360, |C| = 26, D(Xi)=2
Exact Time = 50 min using BTE
Cutset Importance Sampling
• Apply Importance Sampling over cutset C
T
t
tT
tt
t
wTcQ
ecP
TeP
11
1
)(
),(1)(ˆ
T
t
tt
ii wccT
ecP1
),(1
)|(
T
t
tt
ii wecxPT
exP1
),|(1
)|(
where P(ct,e) is computed using Bucket Elimination
(Gogate & Dechter, 2005) and (Bidyuk & Dechter, 2006)
Likelihood Cutset Weighting (LCS)
• Z=Topological Order{C,E}
• Generating sample t+1:
107
For End
If End
),...,|(
Else
If
:do For
1
1
11
1
1
t
i
t
i
t
i
ii
t
i
i
i
zzZPz
e ,zzz
E Z
ZZ • computed while generating sample tusing bucket tree elimination
• can be memoized for some number of instances K (based on memory available
KL[P(C|e), Q(C)+ ≤ KL*P(X|e), Q(X)]
Pathfinder 1
108
Pathfinder 2
109
Link
110
Summary
• i.i.d. samples
• Unbiased estimator
• Generates samples fast
• Samples from Q
• Reject samples with zero-weight
• Improves on cutset
• Dependent samples
• Biased estimator
• Generates samples slower
• Samples from `P(X|e)
• Does not converge in presence of constraints
• Improves on cutset
111
Importance Sampling Gibbs Sampling
CPCS360b
112
cpcs360b, N=360, |LC|=26, w*=21, |E|=15
1.E-05
1.E-04
1.E-03
1.E-02
0 2 4 6 8 10 12 14
Time (sec)
MS
ELW
AIS-BN
Gibbs
LCS
IBP
LW – likelihood weightingLCS – likelihood weighting on a cutset
CPCS422b
113
1.0E-05
1.0E-04
1.0E-03
1.0E-02
0 10 20 30 40 50 60
MS
E
Time (sec)
cpcs422b, N=422, |LC|=47, w*=22, |E|=28 LW
AIS-BN
Gibbs
LCS
IBP
LW – likelihood weightingLCS – likelihood weighting on a cutset
Coding Networks
114
coding, N=200, P=3, |LC|=26, w*=21
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
0 2 4 6 8 10
Time (sec)
MS
ELWAIS-BNGibbsLCSIBP
LW – likelihood weightingLCS – likelihood weighting on a cutset