Slides Set 11 (part a): Sampling Techniques for Probabilistic and - - PowerPoint PPT Presentation

Algorithms for Reasoning with graphical models

Slides Set 11 (part a):

Rina Dechter

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Sampling Techniques for Probabilistic and Deterministic Graphical models

(Reading” Darwiche chapter 15, related papers)

Sampling Techniques for Probabilistic and Deterministic Graphical models

ICS 276, Spring 2018 Bozhena Bidyuk

Reading” Darwiche chapter 15, related papers

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Overview

• 1. Basics of sampling
• 2. Importance Sampling
• 3. Markov Chain Monte Carlo: Gibbs Sampling
• 4. Sampling in presence of Determinism
• 5. Rao‐Blackwellisation, cutset sampling

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Overview

• 1. Basics of sampling
• 2. Importance Sampling
• 3. Markov Chain Monte Carlo: Gibbs Sampling
• 4. Sampling in presence of Determinism
• 5. Rao‐Blackwellisation, cutset sampling

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 Sum-Inference  Max-Inference  Mixed-Inference

Types of queries

• NP-hard: exponentially many terms
• We will focus on approximation algorithms

– Anytime: very fast & very approximate ! Slower & more accurate

Harder

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Monte Carlo estimators

• Most basic form: empirical estimate of probability
• Relevant considerations

– Able to sample from the target distribution p(x)? – Able to evaluate p(x) explicitly, or only up to a constant?

• “Any‐time” properties

– Unbiased estimator,

• r asymptotically unbiased,

– Variance of the estimator decreases with m

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Monte Carlo estimators

• Most basic form: empirical estimate of probability
• Central limit theorem

– p(U) is asymptotically Gaussian:

• Finite sample confidence intervals

– If u(x) or its variance are bounded, e.g., probability concentrates rapidly around the expectation:

m=1: m=5: m=15:

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A Sample

• Given a set of variables X={X1,...,Xn}, a sample,

denoted by St is an instantiation of all variables:

) ,..., , (

t n t t t

x x x S

2 1



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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

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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 or

Forward Sampling.

) ,..., | ( ... ) | ( ) ( ) (

1 1 1 2 1 

   

n n

X X X P X X P X P X P

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Sampling in Bayes nets (Forward Sampling)

• No evidence: “causal” form makes sampling easy

– Follow variable ordering defined by parents – Starting from root(s), sample downward – When sampling each variable, condition on values of parents

A B C D

Sample:

[e.g., Henrion 1988]

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Froward 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 xi

t from P(xi | pai)

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Forward Sampling (example)

X1 X4 X2 X3

) (

1

X P

) | (

1 2 X

X P

) | (

1 3 X

X P

) | ( from Sample . 4 ) | ( from Sample . 3 ) | ( from Sample . 2 ) ( from Sample . 1 sample generate // Evidence No

3 3 , 2 2 4 4 1 1 3 3 1 1 2 2 1 1

x X x X x P x x X x P x x X x P x x P x k    

) , | ( ) | ( ) | ( ) ( ) , , , (

3 2 4 1 3 1 2 1 4 3 2 1

X X X P X X P X X P X P X X X X P    

) , | (

3 2 4

X X X P

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Forward 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 xi

t from P(xi | pai)

4. If Xi in E and Xi  xi, reject sample: 5. Goto Step 1.

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Forward Sampling (example)

X1 X4 X2 X3

) ( 1 x P

) | (

1 2 x

x P ) , | (

3 2 4

x x x P

) | (

1 3 x

x P

) | ( from Sample 5.

• therwise

1, from start and sample reject 0, If . 4 ) | ( from Sample . 3 ) | ( from Sample . 2 ) ( from Sample . 1 sample generate // : Evidence

3 , 2 4 4 3 1 3 3 1 2 2 1 1 3

x x x P x x x x P x x x P x x P x k X  

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Expected value: Given a probability distribution P(X) and a function g(X) defined over a set of variables X = {X1, X2, … Xn}, the expected value of g w.r.t. P is Variance: The variance of g w.r.t. P is:

How to answer queries with sampling? Expected value and Variance

) ( ) ( )] ( [ x P x g x g E

x P





 

) ( )] ( [ ) ( )] ( [

2

x P x g E x g x g Var

x P P



 

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Many queries can be phrased as computing expectation of some functions

Monte Carlo Estimate

• Estimator:

– An estimator is a function of the samples. – It produces an estimate of the unknown parameter of the sampling distribution.

 

 

T t t P

S g T g P

1 T 2 1

1 : by given is [g(x)] E

• f

estimate carlo Monte the , from drawn S , S , S samples i.i.d. Given ) ( ˆ

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Example: Monte Carlo estimate

• Given:

– A distribution P(X) = (0.3, 0.7). – g(X) = 40 if X equals 0 = 50 if X equals 1.

• Estimate EP[g(x)]=(40x0.3+50x0.7)=47.
• Generate k samples from P: 0,1,1,1,0,1,1,0,1,0

46 10 6 50 4 40 1 50 40            samples X samples X samples g # ) ( # ) ( # ˆ

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Bayes Nets with Evidence

• Estimating posterior probabilities, P[A = a | E=e]?
• Rejection sampling

– Draw x ~ p(x), but discard if E e – Resulting samples are from p(x | E=e); use as before – Problem: keeps only P[E=e] fraction of the samples! – Performs poorly when evidence probability is small

• Estimate the ratio: P[A=a,E=e] / P[E=e]

– Two estimates (numerator & denominator) – Good finite sample bounds require low relative error! – Again, performs poorly when evidence probability is small – What bounds can we get?

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Bayes Nets With Evidence

• Estimating the probability of evidence, P[E=e]:

– Finite sample bounds: u(x) ∈ [0,1] – Relative error bounds [Dagum & Luby 1997] [e.g., Hoeffding]

What if the evidence is unlikely? P[E=e]=1e‐6 ) could estimate U = 0!

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So, if U, the probability of evidence is very small we would need many many samples Tht are not rejected.

Overview

• 1. Basics of sampling
• 2. Importance Sampling
• 3. Markov Chain Monte Carlo: Gibbs Sampling
• 4. Sampling in presence of Determinism
• 5. Rao‐Blackwellisation, cutset sampling

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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 carlo estimator (average).

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Importance Sampling

• Basic empirical estimate of probability:
• Importance sampling:

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Importance Sampling

• Basic empirical estimate of probability:
• Importance sampling:

“importance weights”

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Estimating P(E) and P(X|e)

Importance Sampling For P(e)

) ( , ) ( ) ( ˆ : )] ( [ ) ( ) , ( ) ( ) ( ) , ( ) , ( ) ( ) ( ) , ( , \ Z Q z w T e P z w E z Q e z P E z Q z Q e z P e z P e P z Q e z P E X Z Let

t T t t Q Q z z

               

  



z where 1 estimate Carlo Monte : as P(e) rewrite can we Then, satisfying

• n,

distributi (proposal) a be Q(Z) Let

1

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Properties of IS Estimate of P(e)

• Convergence: by law of large numbers
• Unbiased.
• Variance:

    

 

T for ) ( ) ( 1 ) ( ˆ

. . 1

e P z w T e P

s a T i i

 

T z w Var z w T Var e P Var

Q N i i Q Q

)] ( [ ) ( 1 ) ( ˆ

1

       

 

) ( )] ( ˆ [ e P e P EQ 

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Properties of IS Estimate of P(e)

• Mean Squared Error of the estimator

 

   

   

T x w Var e P Var e P Var e P E e P e P e P E e P MSE

Q Q Q Q Q Q

)] ( [ ) ( ˆ ) ( ˆ )] ( ˆ [ ) ( ) ( ) ( ˆ ) ( ˆ             

2 2

This quantity enclosed in the brackets is zero because the expected value of the estimator equals the expected value of g(x)

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Estimating P(E) and P(X|e)

Estimating P(Xi|e)

 

) | ( ) | ( E : biased is Estimate , , ) ( ˆ ) , ( ˆ ) | ( : estimate Ratio IS. by r denominato and numerator Estimate : Idea ) ( ) , ( ) ( ) , ( ) ( ) , ( ) , ( ) ( ) ( ) , ( ) | (

• therwise.

and x contains z if 1 is which function, delta

• dirac

a be (z) Let

T 1 k T 1 k i xi

e x P e x P e) w(z e) )w(z (z e P e x P e x P z Q e z P E z Q e z P z E e z P e z P z e P e x P e x P

i i k k k x i i Q x Q z z x i i

i i i

                 

   

 

   

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Properties of the IS estimator for P(Xi|e)

• Convergence: By Weak law of large numbers
• Asymptotically unbiased
• Variance

– Harder to analyze – Liu suggests a measure called “Effective sample size”

   T as ) | ( ) | ( e x P e x P

i i

) | ( )] | ( [ lim e x P e x P E

i i P T



 

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Effective Sample Size

possible. as small as be must weights the

• f

variance the Therefore, Q. from samples T) ESS(Q, worth are P from samples T Thus ) , ( )] | ( [ )] | ( ˆ [ )] ( [ var 1 ) , ( : 1 ˆ : using e) | P(x estimate can we e), | P(z from samples Given ) | ( ) ( ) | (

1 i

T Q ESS T e x P Var e x P Var z w T T Q ESS Define ) (z g T |e) (x P e z P z g e x P

i Q i P Q T j t x i z x i

i i

    

 



Ideal estimator Measures how much the estimator deviates from the ideal one.

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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:

– Z1Q(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)

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Summary: IS for Common Queries

• Partition function

– Ex: MRF, or BN with evidence – Unbiased; only requires evaluating unnormalized function f(x)

• General expectations wrt p(x) / f(x)?

– E.g., marginal probabilities, etc.

Only asymptotically unbiased… Estimate separately

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More on Properties of IS

• Importance sampling:
• IS is unbiased and fast if q(.) is easy to sample from
• IS can be lower variance if q(.) is chosen well

– Ex: q(x) puts more probability mass where u(x) is large – Optimal: q(x) ∝ |u(x) p(x)|

• IS can also give poor performance

– If q(x) << u(x) p(x): rare but very high weights! – Then, empirical variance is also unreliable! – For guarantees, need to analytically bound weights / variance…

How to get a good proposal?

Outline

• Definitions and Background on Statistics
• Theory of importance sampling
• Likelihood weighting
• State‐of‐the‐art importance sampling

techniques

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Likelihood Weighting

(Fung and Chang, 1990; Shachter and Peot, 1990)

Works well for likely evidence!

“Clamping” evidence+ logic sampling+ weighing samples by evidence likelihood Is an instance of importance sampling!

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Likelihood Weighting: Sampling

e e e e e e e e e

Sample in topological order over X ! Clamp evidence, Sample xi P(Xi|pai), P(Xi|pai) is a look‐up in CPT!

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Likelihood Weighting: Proposal Distribution

    

    

            

E E j j E X X i i E X X E E j j i i n E X X i i

j i i j i

pa e P e pa x P pa e P e pa x P x Q e x P w x x x Weights x X X X P X P e pa X P E X Q ) | ( ) , | ( ) | ( ) , | ( ) ( ) , ( ) ,.., ( : : ) , | ( ) ( ) , | ( ) \ (

\ \ \

sample a Given ) X , Q(X . x X Evidence and ) X , X | P(X ) X | P(X ) P(X ) X , X , P(X : network Bayesian a Given : Example

1 2 2 1 3 1 3 1 2 2 2 1 3 1 2 1 3 2 1

Notice: Q is another Bayesian network

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Likelihood Weighting: Estimates







T t t

w T e P

1 ) (

1 ) ( ˆ

Estimate P(e):

• therwise

zero equals and x if 1 ) ( ) ( ) ( ˆ ) , ( ˆ ) | ( ˆ

i ) ( 1 ) ( ) ( 1 ) ( t i t x T t t t x T t t i i

x x g w x g w e P e x P e x P

i i

   

 

 

Estimate Posterior Marginals:

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Properties of 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

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Outline

• Definitions and Background on Statistics
• Theory of importance sampling
• Likelihood weighting
• State‐of‐the‐art importance sampling

techniques

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Proposal selection

• One should try to select a proposal that is as

close as possible to the posterior distribution.

 

) | ( ) ( ) ( ) ( ) , ( estimator variance

• zero

a have to , ) ( ) ( ) , ( ) ( ) ( ) ( ) , ( 1 )] ( [ ) ( ˆ

2

e z P z Q z Q e P e z P e P z Q e z P z Q e P z Q e z P N T z w Var e P Var

Z z Q Q

                





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Proposal Distributions used in Literature

• AIS‐BN (Adaptive proposal)
• Cheng and Druzdzel, 2000
• Iterative Belief Propagation
• Changhe and Druzdzel, 2003
• Iterative Join Graph Propagation (IJGP) and

variable ordering

• Gogate and Dechter, 2005

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Perfect sampling using Bucket Elimination

• Algorithm:

– Run Bucket elimination on the problem along an

• rdering 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.

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Exact Sampling using Bucket Elimination

• Algorithm:

– Run Bucket elimination on the problem along an

• rdering o=(X1,..,XN).

– Sample along the reverse ordering – At each branch point, recover the edge probabilities by performing a constant‐time table lookup! – Complexity: O(Bucket‐elimination)+O(M*n)

• M is the number of solution samples and n is the

number of variables

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Downward message normalizes bucket; ratio is a conditional distribution

E: C: D: B: A:

How to sample from a Markov network? Exact sampling via inference

• Draw samples from P[A|E=e] directly?

– Model defines un‐normalized p(A,…,E=e) – Build (oriented) tree decomposition & sample

Z Work: O(exp(w)) to build distribution O(n d) to draw each sample

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Bucket Elimination

) , ( ) | (    e a P e a P





 

d e b c

c b e P b a d P a c P a b P a P e a P

, , ,

) , | ( ) , | ( ) | ( ) | ( ) ( ) , (

   





d c b e

b a d P c b e P a b P a c P a P ) , | ( ) , | ( ) | ( ) | ( ) (

Elimination Order: d,e,b,c Query:

D: E: B: C: A:





d D

b a d P b a f ) , | ( ) , ( ) , | ( b a d P ) , | ( c b e P ) , | ( ) , ( c b e P c b fE  





b E D B

c b f b a f a b P c a f ) , ( ) , ( ) | ( ) , ( ) ( ) ( ) , ( a f A p e a P

C

  ) (a P ) | ( a c P





c B C

c a f a c P a f ) , ( ) | ( ) ( ) | ( a b P

D,A,B E,B,C B,A,C C,A A ) , ( b a fD ) , ( c b fE ) , ( c a fB ) (a fC

A A D D E E C C B B

Bucket Tree

D E B C A

Original Functions Messages Time and space exp(w*) slides11a 828X 2019

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Bucket elimination (BE)



b

Elimination operator

P(e)

bucket B: P(a) P(C|A) P(B|A) P(D|B,A) P(e|B,C) bucket C: bucket D: bucket E: bucket A: B C D E A

e) (A, hD

(a) hE

e) C, D, (A, hB

e) D, (A, hC

A A D D E E C C B B

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Sampling from the output of BE

(Dechter 2002) bucket B: P(A) P(C|A) P(B|A) P(D|B,A) P(e|B,C) bucket C: bucket D: bucket E: bucket A:

e) (A, hD

(A) hE

e) C, D, (A, hB e) D, (A, hC

Q(A) a A : (A) h P(A) Q(A)

E

    Sample ignore : bucket Evidence

e) D, (a, h e) a, | Q(D d D : Sample bucket in the a A Set

C

   

e) C, d, (a, h ) | ( d) e, a, | Q(C c C : Sample bucket the in d D a, A Set

B

      A C P ) , | ( ) , | ( ) | ( d) e, a, | Q(B b B : Sample bucket the in c C d, D a, A Set c b e P a B d P a B P      

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57

Mini‐Bucket Elimination

bucket A: bucket E: bucket D: bucket C: bucket B:

ΣB

P(B|A) P(D|B,A) hE(A) hB(A,D) P(e|B,C)

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 (Yedidia et 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 one of the best performing IS

scheme!

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Example: IJGP‐Sampling

• Run IJGP

A B C D E Sampling Order Approx #Solutions (i=2)

CD AD BCD BE DE ABE E

A C E D B

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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.

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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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 

k ) ( ˆ Re ' ) ( Q Update ) ( N 1 ) ( ˆ e) (E P ˆ Q z ,..., z samples Generate do k to 1 i For ) ( ˆ )) ( | ( ... )) ( | ( ) ( ) ( Q Proposal Initial

1 k 1 N 1 2 2 1 1

e E P turn End Q Q k Q z w e E P from e E P Z pa Z Q Z pa Z Q Z Q Z

k k i N j k k n n

               

 





Adaptive Importance Sampling

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Adaptive Importance Sampling

• General case
• Given k proposal distributions
• Take N samples out of each distribution
• Approximate P(e)

 

1 ) ( ˆ

1





   

k j

proposal jth weight Avg k e P

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Estimating Q'(z)

sampling importance by estimated is ) Z ,.., Z | (Z Q' each where )) ( | ( ' ... )) ( | ( ' ) ( ' ) ( Q

1

• i

1 i 2 2 1 ' n n

Z pa Z Q Z pa Z Q Z Q Z    

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Choosing a proposal (wmb‐IS)

• Can use WMB upper bound to define a proposal q(x):

E: C: D: B: A:

mini‐buckets

U = upper bound

…

Weighted mixture: use minibucket 1 with probability w1

• r, minibucket 2 with probability w2 = 1 ‐ w1

where Key insight: provides bounded importance weights! [Liu, Fisher, Ihler 2015]

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WMB‐IS Bounds

• Finite sample bounds on the average
• Compare to forward sampling

101 102 103 104 105 Sample Size (m) 101 102 103 104 105 106 Sample Size (m) BN_6 BN_11 ‐58.4 ‐53 ‐63 ‐39.4 ‐34 ‐44

“Empirical Bernstein” bounds

[Liu, Fisher, Ihler 2015]

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Other Choices of Proposals

• Belief propagation

– BP‐based proposal [Changhe & Druzdzel 2003] – Join‐graph BP proposal [Gogate & Dechter 2005] – Mean field proposal [Wexler & Geiger 2007] E: C: D: B: A:

Join graph:

{B|A,C} {B|D,E} {C|A,E} {D|A,E} {E|A} {A}

{B} {D,E} {A} {A} {A,E} {A,C}

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Other Choices of Proposals

• Belief propagation

– BP‐based proposal [Changhe & Druzdzel 2003] – Join‐graph BP proposal [Gogate & Dechter 2005] – Mean field proposal [Wexler & Geiger 2007]

• Adaptive importance sampling

– Use already‐drawn samples to update q(x) – Rates vt and ´t adapt estimates, proposal – Ex: [Cheng & Druzdzel 2000] [Lapeyre & Boyd 2010] … – Lose “iid”‐ness of samples

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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

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Outline

• Definitions and Background on Statistics
• Theory of importance sampling
• Likelihood weighting
• Error estimation
• State‐of‐the‐art importance sampling

techniques

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Logic Sampling –How many samples?

Theorem: Let s(y) be the estimate of P(y) resulting

from a randomly chosen sample set S with T samples. Then, to guarantee relative error at most  with probability at least 1‐ it is enough to have:

  1 ) (

2 

  y P c T

Derived from Chebychev’s Bound.

 

2

2

2 ] ) ( , ) ( [ ) ( Pr



 

N

e y P y P y P



   

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Logic Sampling: Performance

Advantages:

• P(xi | pa(xi)) is readily available
• Samples are independent !

Drawbacks:

• If evidence E is rare (P(e) is low), then we will reject

most of the samples!

• Since P(y) in estimate of T is unknown, must estimate

P(y) from samples themselves!

• If P(e) is small, T will become very big!

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Bucket Elimination Overview

A C E D B F(B,C) F(B,D) F(B,E) F’(B,D,E) F’(B,D,E) F(A,E) F(A,D) F(A,B) F(C,D) F’(C,D,E) F(D,E) F’(D,E) F’(C,D,E) F’(D,E) F’(E) F’(E) D: C: B: E: A:

Sampling Direction

Complexity: Exp (3) or n3

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