Approximate Inference

transcript

Slides by Nir Friedman

When can we hope to approximate?

Two situations: Highly stochastic distributions

“Far” evidence is discarded “Peaked” distributions

improbable values are ignored

Stochasticity & Approximations

Consider a chain:

P(Xi+1 = t | Xi = t) = 1- P(Xi+1 = f | Xi = f) = 1-

Computing the probability of Xn+1 given X1 , we get

X1 X2 X3Xn+1

121211

ntXfXP

ntXtXP

Even # of flips:

Odd # of flips:

Plot of P(Xn = t | X1 = t)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

n = 10

n = 20

Stochastic Processes

This behavior of a chain (a Markov Process) is called Mixing.

In general Bayes nets there is a similar behavior. If probabilities are far from 0 & 1, then effect of

“far” evidence vanishes (and so can be discarded in approximations).

“Peaked” distributions If the distribution is “peaked”, then most of the

mass is on few instances If we can focus on these instances, we can

ignore the rest

Probability

Instances

Global conditioning

Fixing value of A & B

a b c d le

ledcbaPmP...

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

Fixing values in the beginning of the summation can decrease tables formed by variable elimination. This way space is traded with time. Special case: choose to fix a set of nodes that “break all loops”. This method is called cutset-conditioning.

a b b a

Bounded conditioning

Fixing value of A & B

By examining only the probable assignment of A & B, we perform several simple computations instead of a complex one.

Bounded conditioning

Choose A and B so that P(Y,e |a,b) can be computed easily. E.g., a cycle cutset.

Search for highly probable assignments to A,B. Option 1--- select a,b with high P(a,b). Option 2--- select a,b with high P(a,b | e).

We need to search for such high mass values and that can be hard.

obasbleba

b)P(ab)|ayYP)yYPPr,

,,,(,( ee

Bounded Conditioning

Advantages: Combines exact inference within approximation Continuous: more time can be used to examine more cases Bounds: unexamined mass

used to compute error-bars

Possible problems: P(a,b) is prior mass not the posterior. If posterior is significantly different P(a,b| e), Computation

can be wasted on irrelevant assignments

obableba

b)P(aPr,

Network Simplifications

In these approaches, we try to replace the original network with a simpler one

the resulting network allows fast exact methods

Network Simplifications

Typical simplifications: Remove parts of the network Remove edges Reduce the number of values (value abstraction) Replace a sub-network with a simpler one

(model abstraction) These simplifications are often w.r.t. to the

particular evidence and query

Stochastic Simulation

Suppose our goal is the compute the likelihood of evidence P(e) where e is an assignment to some variables in {X1,…,Xn}.

Assume that we can sample instances <x1,…,xn> according to the distribution P(x1,…,xn).

What is then the probability that a random sample <x1,…,xn> satisfies e?

Answer: simply P(e) which is what we wish to compute.

Each sample simulates the tossing of a biased coin with probability P(e) of “Heads”.

Stochastic Sampling

Intuition: given a sufficient number of samples x[1],…,x[N], we can estimate

Law of large number implies that as N grows, our estimate will converge to p with high probability

[i])|P

NHeads

Zeros or ones

How many samples do we need to get a reliable estimation?

We will not discuss this issue here.

Sampling a Bayesian Network

If P(X1,…,Xn) is represented by a Bayesian network, can we efficiently sample from it?

Idea: sample according to structure of the network Write distribution using the chain rule, and then

sample each variable given its parents

Samples:

B E A C R

Logic sampling

P(b) 0.03P(e) 0.001

b e b e b e b e

0.98 0.40.7 0.01

0.8 0.05

0.3 0.001

Earthquake

Burglary

Samples:

B E A C R

Logic sampling

P(b) 0.03P(e) 0.001

b e b e b e b e

0.98 0.40.7 0.01

0.8 0.05

0.3 0.001

Earthquake

Burglary

Samples:

B E A C R

Logic sampling

P(b) 0.03P(e) 0.001

b e b e b e b e

0.98 0.40.7 0.01

0.8 0.05

0.3 0.001

Earthquake

Burglary

Samples:

B E A C R

Logic sampling

P(b) 0.03P(e) 0.001

b e b e b e b e

0.98 0.40.7 0.01

0.8 0.05

0.3 0.001

e a cb

Earthquake

Burglary

Samples:

B E A C R

Logic sampling

P(b) 0.03P(e) 0.001

b e b e b e b e

0.98 0.40.7 0.01

0.8 0.05

0.3 0.001

e a cb

Earthquake

Burglary

Samples:

B E A C R

Logic sampling

P(b) 0.03P(e) 0.001

b e b e b e b e

0.98 0.40.7 0.01

0.8 0.05

0.3 0.001

e a cb r

Earthquake

Burglary

Logic Sampling

Let X1, …, Xn be order of variables consistent with arc direction

for i = 1, …, n do sample xi from P(Xi | pai ) (Note: since Pai {X1,…,Xi-1}, we already

assigned values to them) return x1, …,xn

Logic Sampling

Sampling a complete instance is linear in number of variables Regardless of structure of the network

However, if P(e) is small, we need many samples to get a decent estimate

Can we sample from P(Xi|e) ?

If evidence e is in roots of the Bayes network, easily If evidence is in leaves of the network, we have a

problem: Our sampling method proceeds according to the

order of nodes in the network.

Likelihood Weighting

Can we ensure that all of our sample satisfy e? One simple (but wrong) solution:

When we need to sample a variable Y that is assigned value by e, use its specified value.

For example: we know Y = 1 Sample X from P(X) Then take Y = 1

Is this a sample from P(X,Y |Y = 1) ? NO.

Problem: these samples of X are from P(X) Solution:

Penalize samples in which P(Y=1|X) is small

We now sample as follows: Let xi be a sample from P(x) Let wi= P(Y = 1|X = xi )

)|XPw)YxXP

xx(1|(

Let X1, …, Xn be order of variables consistent with arc direction

w = 1 for i = 1, …, n do

if Xi = xi has been observedw w P(Xi = xi | pai )

elsesample xi from P(Xi | pai )

return x1, …,xn, and w

Samples:

B E A C R

P(b) 0.03P(e) 0.001

b e b e b e b e

0.98 0.40.7 0.01

0.8 0.05

0.3 0.001

Earthquake

Burglary

Weight

Samples:

B E A C R

P(b) 0.03P(e) 0.001

b e b e b e b e

0.98 0.40.7 0.01

0.8 0.05

0.3 0.001

Earthquake

Burglary

Weight

= r = a

Samples:

B E A C R

P(b) 0.03P(e) 0.001

b e b e b e b e

0.98 0.40.7 0.01

0.8 0.05

0.3 0.001

Earthquake

Burglary

Weight

= r = a

Samples:

B E A C R

P(b) 0.03P(e) 0.001

b e b e b e b e

0.98 0.40.7 0.01

0.8 0.05

0.3 0.001

Earthquake

Burglary

0.05Weight

= r = a

Samples:

B E A C R

P(b) 0.03P(e) 0.001

b e b e b e b e

0.98 0.40.7 0.01

0.8 0.05

0.3 0.001

e cb r

Earthquake

Burglary

Weight

= r = a

a 0.6*0.3

Why does this make sense? When N is large, we expect to sample NP(X = x)

samples with x[i] = x Thus,

)xXPPN

PN)x|XP

i 1Y|(1)Y(

1)Y, x X(

[i]) x X|1 P(Y

x[i]([i]) x X|1 P(Y

Summary

Approximate inference is needed for large pedigrees. We have seen a few methods today. Some could fit genetic linkage analysis and some do not. There are many other approximation algorithms: Variational methods, MCMC, and others.

In next semester’s project of Bioinformatics (236524), we will offer projects that seek to implement some approximation methods and embed them in the superlink software.

Approximate Inference

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