inference in bayesian networksce.sharif.edu/courses/96-97/2/ce417-1/resources/root... ·...

Inference in Bayesian NetworksCE417: Introduction to Artificial Intelligence

Sharif University of Technology

Spring 2018

Soleymani

Slides are based on Klein and Abdeel, CS188, UC Berkeley.

Bayes’ Nets

Representation

Conditional Independences

Probabilistic Inference Enumeration (exact, exponential complexity)

Variable elimination (exact, worst-case

exponential complexity, often better)

Probabilistic inference is NP-complete

Sampling (approximate)

Learning Bayes’ Nets from Data

Recap: Bayes’ Net Representation

A directed, acyclic graph, one node per randomvariable

A conditional probability table (CPT) for each node

A collection of distributions over X, one for each combination ofparents’ values

Bayes’ nets implicitly encode joint distributions

As a product of local conditional distributions

To see what probability a BN gives to a full assignment, multiplyall the relevant conditionals together:

Example: Alarm Network

Burglary

Earthqk

B P(B)

+b 0.001

-b 0.999

E P(E)

+e 0.002

-e 0.998

B E A P(A|B,E)

+b +e +a 0.95

+b +e -a 0.05

+b -e +a 0.94

+b -e -a 0.06

-b +e +a 0.29

-b +e -a 0.71

-b -e +a 0.001

-b -e -a 0.999

A J P(J|A)

+a +j 0.9

+a -j 0.1

-a +j 0.05

-a -j 0.95

A M P(M|A)

+a +m 0.7

+a -m 0.3

-a +m 0.01

-a -m 0.99

[Demo: BN Applet]

Video of Demo BN Applet

B P(B)

+b 0.001

-b 0.999

E P(E)

+e 0.002

-e 0.998

B E A P(A|B,E)

+b +e +a 0.95

+b +e -a 0.05

+b -e +a 0.94

+b -e -a 0.06

-b +e +a 0.29

-b +e -a 0.71

-b -e +a 0.001

-b -e -a 0.999

A J P(J|A)

+a +j 0.9

+a -j 0.1

-a +j 0.05

-a -j 0.95

A M P(M|A)

+a +m 0.7

+a -m 0.3

-a +m 0.01

-a -m 0.99

B P(B)

+b 0.001

-b 0.999

E P(E)

+e 0.002

-e 0.998

B E A P(A|B,E)

+b +e +a 0.95

+b +e -a 0.05

+b -e +a 0.94

+b -e -a 0.06

-b +e +a 0.29

-b +e -a 0.71

-b -e +a 0.001

-b -e -a 0.999

A J P(J|A)

+a +j 0.9

+a -j 0.1

-a +j 0.05

-a -j 0.95

A M P(M|A)

+a +m 0.7

+a -m 0.3

-a +m 0.01

-a -m 0.99

Bayes’ Nets

Representation

Probabilistic Inference

Enumeration (exact, exponential complexity)

Variable elimination (exact, worst-case exponentialcomplexity, often better)

Inference is NP-complete

Examples:

Posterior probability

Most likely explanation:

Inference

Inference: calculating someuseful quantity from a jointprobability distribution

Inference by Enumeration

General case: Evidence variables:

Query* variable:

Hidden variables: All variables

* Works fine with multiple query variables, too

We want:

Step 1: Select the entries consistent with the evidence

Step 2: Sum out H to get joint of Query and evidence

Step 3: Normalize

Inference by Enumeration in Bayes’ Net

Given unlimited time, inference in BNs is easy

Reminder of inference by enumeration by example: B E

Burglary example: full joint probability

𝑃 𝑏 𝑗, ¬𝑚 =𝑃 𝑗, ¬𝑚, 𝑏

𝑃 𝑗, ¬𝑚= 𝐴 𝐸 𝑃 𝑗, ¬𝑚, 𝑏, 𝐴, 𝐸

𝐵 𝐴 𝐸 𝑃 𝑗, ¬𝑚, 𝑏, 𝐴, 𝐸

= 𝐴 𝐸 𝑃 𝑗 𝐴 𝑃 ¬𝑚 𝐴 𝑃 𝐴 𝑏, 𝐸 𝑃 𝑏 𝑃(𝐸)

𝐵 𝐴 𝐸 𝑃 𝑗 𝐴 𝑃 ¬𝑚 𝐴 𝑃 𝐴 𝐵, 𝐸 𝑃 𝐵 𝑃(𝐸)

𝑗: 𝐽𝑜ℎ𝑛𝐶𝑎𝑙𝑙𝑠 = 𝑇𝑟𝑢𝑒¬𝑏: 𝐵𝑢𝑟𝑔𝑙𝑎𝑟𝑦 = 𝐹𝑎𝑙𝑠𝑒

Short-hands

Inference by Enumeration?

Factor Zoo

Factor Zoo I

Joint distribution: P(X,Y)

Entries P(x,y) for all x, y

Sums to 1

Selected joint: P(x,Y)

A slice of the joint distribution

Entries P(x,y) for fixed x, all y

Sums to P(x)

Number of capitals = dimensionality of the table

hot sun 0.4

hot rain 0.1

cold sun 0.2

cold rain 0.3

cold sun 0.2

cold rain 0.3

Factor Zoo II

Single conditional: P(Y | x)

Entries P(y | x) for fixed x, all y

Sums to 1

Family of conditionals:

P(X |Y)

Multiple conditionals

Entries P(x | y) for all x, y

Sums to |Y|

hot sun 0.8

hot rain 0.2

cold sun 0.4

cold rain 0.6

cold sun 0.4

cold rain 0.6

Factor Zoo III

Specified family: P( y | X )

Entries P(y | x) for fixed y,

but for all x

Sums to … who knows!

hot rain 0.2

cold rain 0.6

Factor Zoo Summary

In general, when we write P(Y1 … YN | X1 … XM)

It is a “factor,” a multi-dimensional array

Its values are P(y1 … yN | x1 … xM)

Any assigned (=lower-case) X or Y is a dimension missing (selected) from the array

Example: Traffic Domain

RandomVariables

R: Raining

T:Traffic

L: Late for class! T

R+r 0.1-r 0.9

+r +t 0.8+r -t 0.2-r +t 0.1-r -t 0.9

+t +l 0.3+t -l 0.7-t +l 0.1-t -l 0.9

Inference by Enumeration: Procedural

Outline

Track objects called factors

Initial factors are local CPTs (one per node)

Any known values are selected

E.g. if we know , the initial factors are

Procedure: Join all factors, then eliminate all hidden variables

+r 0.1-r 0.9

+r +t 0.8+r -t 0.2-r +t 0.1-r -t 0.9

+t +l 0.3+t -l 0.7-t +l 0.1-t -l 0.9

+t +l 0.3-t +l 0.1

+r 0.1-r 0.9

+r +t 0.8+r -t 0.2-r +t 0.1-r -t 0.9

Operation 1: Join Factors

First basic operation: joining factors

Combining factors:

Just like a database join

Get all factors over the joining variable

Build a new factor over the union of the

variables involved

Example: Join on R

Computation for each entry: pointwise

products

+r 0.1-r 0.9

+r +t 0.8+r -t 0.2-r +t 0.1-r -t 0.9

+r +t 0.08+r -t 0.02-r +t 0.09-r -t 0.81

Example: Multiple Joins

R Join R

+r 0.1-r 0.9

+r +t 0.8+r -t 0.2-r +t 0.1-r -t 0.9

+t +l 0.3+t -l 0.7-t +l 0.1-t -l 0.9

+r +t 0.08+r -t 0.02-r +t 0.09-r -t 0.81

+t +l 0.3+t -l 0.7-t +l 0.1-t -l 0.9

R, T, L

+r +t +l 0.024

+r +t -l 0.056

+r -t +l 0.002

+r -t -l 0.018

-r +t +l 0.027

-r +t -l 0.063

-r -t +l 0.081

-r -t -l 0.729

Join T

Operation 2: Eliminate

Second basic operation: marginalization

Take a factor and sum out a variable

Shrinks a factor to a smaller one

A projection operation

Example:

+r +t 0.08+r -t 0.02-r +t 0.09-r -t 0.81

+t 0.17-t 0.83

Multiple Elimination

Sumout R

Sumout T

T, L LR, T, L

+r +t +l 0.024

+r +t -l 0.056

+r -t +l 0.002

+r -t -l 0.018

-r +t +l 0.027

-r +t -l 0.063

-r -t +l 0.081

-r -t -l 0.729

+t +l 0.051

+t -l 0.119

-t +l 0.083

-t -l 0.747

+l 0.134-l 0.886

Thus Far: Multiple Join, Multiple Eliminate (=

Inference by Enumeration)

Inference by Enumeration vs. Variable Elimination

Why is inference by enumeration so slow?

You join up the whole joint distribution beforeyou sum out the hidden variables

Idea: interleave joining and marginalizing!

Called “Variable Elimination”

Still NP-hard, but usually much faster than inference by enumeration

First we’ll need some new notation: factors

Traffic Domain

Inference by EnumerationT

Variable Elimination

Join on rJoin on r

Join on t

Eliminate r

Eliminate t

Eliminate r

Eliminate t

Marginalizing Early (= Variable Elimination)

Marginalizing Early! (aka VE)

Sum out R

+r +t 0.08+r -t 0.02-r +t 0.09-r -t 0.81

+t +l 0.3+t -l 0.7-t +l 0.1-t -l 0.9

+t 0.17-t 0.83

+t +l 0.3+t -l 0.7-t +l 0.1-t -l 0.9

+r 0.1-r 0.9

+r +t 0.8+r -t 0.2-r +t 0.1-r -t 0.9

+t +l 0.3+t -l 0.7-t +l 0.1-t -l 0.9

Join R

T, L L

+t +l 0.051

+t -l 0.119

-t +l 0.083

-t -l 0.747

+l 0.134-l 0.866

Join T Sum out T

Evidence

If evidence, start with factors that select that evidence

No evidence uses these initial factors:

Computing , the initial factors become:

We eliminate all vars other than query + evidence

+r 0.1-r 0.9

+r +t 0.8+r -t 0.2-r +t 0.1-r -t 0.9

+t +l 0.3+t -l 0.7-t +l 0.1-t -l 0.9

+r 0.1 +r +t 0.8+r -t 0.2

+t +l 0.3+t -l 0.7-t +l 0.1-t -l 0.9

Evidence II

Result will be a selected joint of query and evidence

E.g. for P(L | +r), we would end up with:

To get our answer, just normalize this!

That ’s it!

+l 0.26-l 0.74

+r +l 0.026+r -l 0.074

Normalize

Distribution of products on sums

Exploiting the factorization properties to allow sums and

products to be interchanged

𝑎 × 𝑏 + 𝑎 × 𝑐 needs three operations while 𝑎 × (𝑏 + 𝑐)requires two

General Variable Elimination

Query:

Start with initial factors:

Local CPTs (but instantiated by evidence)

While there are still hidden variables (not Qor evidence):

Pick a hidden variable H

Join all factors mentioning H

Eliminate (sum out) H

Join all remaining factors and normalize

Variable elimination: example

𝑃 𝑏, 𝑗 = 𝐴

𝑃 𝑏 𝑃 𝐸 𝑃 𝐴 𝑏, 𝐸 𝑃 𝑗 𝐴 𝑃 𝑀 𝐴

= 𝑃 𝑏

𝑃 𝐸

𝑃 𝐴 𝑏, 𝐸 𝑃 𝑗 𝐴 𝑀𝑃 𝑀 𝐴

𝑃 𝑏|𝑗 ∝ 𝑃(𝑏, 𝑗)Intermediate results are

probability distributions

Variable elimination: example

𝑃 𝐵, 𝑗 = 𝐴

𝑃 𝐵 𝑃 𝐸 𝑃 𝐴 𝐵, 𝐸 𝑃 𝑗 𝐴 𝑃 𝑀 𝐴

= 𝑃 𝐵

𝑃 𝐸

𝑃 𝐴 𝐵, 𝐸 𝑃 𝑗 𝐴 𝑀𝑃 𝑀 𝐴

𝒇4(𝐴)

𝒇7 𝐵, 𝐸 = 𝐴𝒇3(𝐴, 𝐵, 𝐸) × 𝒇4(𝐴) × 𝒇6(𝐴)

𝒇8 𝐵 = 𝐸𝒇2(𝐸) × 𝒇7 𝐵, 𝐸

𝑃 𝐵|𝑗 ∝ 𝑃(𝐵, 𝑗)

𝒇3(𝐴, 𝐵, 𝐸)𝒇1(𝐵) 𝒇2(𝐸) 𝒇5(𝐴,𝑀)

𝒇6(𝐴)

Intermediate results are

probability distributions

Variable elimination: Order of summations

An inefficient order:

𝑃 𝐵, 𝑗 = 𝑀

𝑃 𝐵 𝑃 𝐸 𝑃 𝐴 𝐵, 𝐸 𝑃 𝑗 𝐴 𝑃 𝑀 𝐴

= 𝑃 𝐵 𝑀 𝐸𝑃 𝐸

𝑃 𝐴 𝐵, 𝐸 𝑃 𝑗 𝐴 𝑃 𝑀 𝐴

𝒇(𝐴, 𝐵, 𝐸,𝑀)

Variable elimination:

Pruning irrelevant variables

Any variable that is not an ancestor of a query variable or

evidence variable is irrelevant to the query.

Prune all non-ancestors of query or evidence variables:

𝑃 𝑏, 𝑗

Burglary

John Calls

Earthquake

Variable elimination algorithm

Given: BN, evidence 𝑒, a query 𝑃(𝒀|𝒙𝒗)

Prune non-ancestors of {𝒀, 𝑿𝑽}

Choose an ordering on variables, e.g.,𝑋1,…,𝑋𝑛 For i = 1 to n, If 𝑋𝑖 ∉ {𝒀, 𝑿𝑽}

Collect factors 𝒇1, … , 𝒇𝑘 that include 𝑋𝑖 Generate a new factor by eliminating 𝑋𝑖 from these factors:

𝒈 = 𝑋𝑖

𝑗=1

𝒇𝑗

Normalize 𝑃(𝒀, 𝒙𝒗) to obtain 𝑃(𝒀|𝒙𝒗)

After this summation, 𝑋𝑖 is eliminated

Variable elimination algorithm

• Evaluating expressions in a proper order

• Storing intermediate results

• Summation only for those portions of the expression that

depend on that variable

Given: BN, evidence 𝑒, a query 𝑃(𝒀|𝒙𝒗)

Prune non-ancestors of {𝒀, 𝑿𝑽}

Choose an ordering on variables, e.g.,𝑋1,…,𝑋𝑛 For i = 1 to n, If 𝑋𝑖 ∉ {𝒀, 𝑿𝑽}

Collect factors 𝒇1, … , 𝒇𝑘 that include 𝑋𝑖 Generate a new factor by eliminating 𝑋𝑖 from these factors:

𝒈 = 𝑋𝑖

𝑗=1

𝒇𝑗

Normalize 𝑃(𝒀, 𝒙𝒗) to obtain 𝑃(𝒀|𝒙𝒗)

Variable elimination

Eliminates by summation non-observed non-query variables

one by one by distributing the sum over the product

Complexity determined by the size of the largest factor

Variable elimination can lead to significant costs saving but its

efficiency depends on the network structure .

there are still cases in which this algorithm we lead to exponential time.

Example

Choose A

Example

Choose E

Finish with B

Normalize

Same Example in Equations

marginal can be obtained from joint by summing out

use Bayes’ net joint distribution expression

use x*(y+z) = xy + xz

joining on a, and then summing out gives f1

use x*(y+z) = xy + xz

joining on e, and then summing out gives f2

All we are doing is exploiting uwy + uwz + uxy + uxz + vwy + vwz + vxy +vxz = (u+v)(w+x)(y+z) to improve computational efficiency!

Inference on a chain

𝑃 𝑑 = 𝐴 𝐵 𝐶𝑃(𝐴, 𝐵, 𝐶, 𝑑)

𝑃 𝑑 =

𝐵 𝐶𝑃 𝐴 𝑃 𝐵 𝐴 𝑃 𝐶 𝐵 𝑃(𝑑|𝐶)

A naïve summation needs to enumerate over an

exponential number of terms

𝐴 𝐵 𝐶 𝐷

Inference on a chain:

marginalization and elimination

𝑃 𝑑 =

𝐵 𝐶𝑃 𝐴 𝑃 𝐵 𝐴 𝑃 𝐶 𝐵 𝑃(𝑑|𝐶)

𝐵 𝐴𝑃 𝐴 𝑃 𝐵 𝐴 𝑃 𝐶 𝐵 𝑃(𝑑|𝐶)

𝑃(𝑑|𝐶) 𝐵𝑃 𝐶 𝐵

𝐴𝑃 𝐴 𝑃 𝐵 𝐴

In a chain of 𝑛 nodes each having 𝑘 values,𝑂(𝑛𝑘2) instead of 𝑂(𝑘𝑛)

𝑓(𝐵)

𝑓(𝐶)

𝐴 𝐵 𝐶 𝐷

Wampus example

𝑒𝑣𝑖𝑑𝑒𝑛𝑐𝑒 = ¬𝑏1,1 ∧ 𝑏1,2 ∧ 𝑏2,1 ∧ ¬𝑝1,1 ∧ ¬𝑝1,2 ∧ ¬𝑝2,1

𝑃 𝑃1,3 𝑒𝑣𝑖𝑑𝑒𝑛𝑐𝑒 =?

Wumpus example

Possible worlds with 𝑃1,3 = 𝑡𝑟𝑢𝑒 Possible worlds with 𝑃1,3 = 𝑓𝑎𝑙𝑠𝑒

𝑃 𝑃1,3 = 𝑇𝑟𝑢𝑒 𝑒𝑣𝑖𝑑𝑒𝑛𝑐𝑒 ∝ 0.2 × 0.2 × 0.2 + 0.2 × 0.8 + 0.8 × 0.2

𝑃 𝑃1,3 = 𝐹𝑎𝑙𝑠𝑒 𝑒𝑣𝑖𝑑𝑒𝑛𝑐𝑒 ∝ 0.8 × 0.2 × 0.2 + 0.2 × 0.8

⇒ 𝑃 𝑃1,3 = 𝑇𝑟𝑢𝑒 𝑒𝑣𝑖𝑑𝑒𝑛𝑐𝑒 = 0.31

Another Variable Elimination Example

Computational complexity critically depends on the largest factor being generated in this process. Size of factor = number of entries in table. In example above (assuming binary) all factors generated are of size 2 --- as they all only have one variable (Z, Z, and X3 respectively).

Variable Elimination Ordering

For the query P(Xn|y1,…,yn) work through the following two different

orderings as done in previous slide: Z, X1, …, Xn-1 and X1, …, Xn-1, Z.

What is the size of the maximum factor generated for each of the

orderings?

Answer: 2n+1 versus 22 (assuming binary)

In general: the ordering can greatly affect efficiency.

VE: Computational and Space Complexity

The computational and space complexity of variable elimination is

determined by the largest factor

The elimination ordering can greatly affect the size of the largest factor.

E.g., previous slide’s example 2n vs. 2

Does there always exist an ordering that only results in small factors?

Complexity of variable elimination algorithm

In each elimination step, the following computations are

required:

𝑓 𝑥, 𝑥1, … , 𝑥𝑘 = 𝑖=1𝑀 𝑔𝑖(𝑥, 𝒙𝑐𝑖)

𝑥 𝑓 𝑥, 𝑥1, … , 𝑥𝑘

We need:

(𝑀 − 1) × 𝑉𝑎𝑙(𝑋) × 𝑖=1𝑘 𝑉𝑎𝑙(𝑋𝑖) multiplications

For each tuple 𝑥, 𝑥1, … , 𝑥𝑘, we need 𝑀 − 1 multiplications

𝑉𝑎𝑙(𝑋) × 𝑖=1𝑘 𝑉𝑎𝑙(𝑋𝑖) additions

For each tuple 𝑥1, … , 𝑥𝑘, we need 𝑉𝑎𝑙(𝑋) additions

Complexity is exponential in number of variables in the intermediate factor

Size of the created factors is the dominant quantity in the complexity of VE

Example

Query: 𝑃(𝑋2|𝑋7 = 𝑥7)

𝑃 𝑋2 𝑥7 ∝ 𝑃 𝑋2, 𝑥7

𝑃 𝑥2, 𝑥7

𝑃 𝑥1, 𝑥2, 𝑥3, 𝑥4, 𝑥5, 𝑥6, 𝑥7, 𝑥8

Consider the elimination order 𝑋1, 𝑋3, 𝑋4, 𝑋5, 𝑋6, 𝑋8𝑃 𝑥2, 𝑥7

𝑃 𝑥1 𝑃 𝑥2 𝑃 𝑥3 𝑥1, 𝑥2 𝑃 𝑥4 𝑥3 𝑃 𝑥5 𝑥2 𝑃 𝑥6 𝑥3, 𝑥7 𝑃( 𝑥7|𝑥4, 𝑥5)𝑃 𝑥8 𝑥7

𝑋1 𝑋2

𝑃 𝑥2, 𝑥7 =

𝑃 𝑥2 𝑃 𝑥4 𝑥3 𝑃 𝑥5 𝑥2 𝑃 𝑥6 𝑥3, 𝑥7 𝑃( 𝑥7|𝑥4, 𝑥5)𝑃 𝑥8 𝑥7

𝑃 𝑥1 𝑃 𝑥3 𝑥1, 𝑥2

𝑃 𝑥2 𝑃 𝑥4 𝑥3 𝑃 𝑥5 𝑥2 𝑃 𝑥6 𝑥3, 𝑥7 𝑃 𝑥7 𝑥4, 𝑥5 𝑃 𝑥8 𝑥7 𝑚1(𝑥2, 𝑥3)

𝑃 𝑥2 𝑃 𝑥5 𝑥2 𝑃 𝑥7 𝑥4, 𝑥5 𝑃 𝑥8 𝑥7

𝑃 𝑥4 𝑥3 𝑃 𝑥6 𝑥3, 𝑥7 𝑚1(𝑥2, 𝑥3)

𝑃 𝑥2 𝑃 𝑥5 𝑥2 𝑃 𝑥7 𝑥4, 𝑥5 𝑃 𝑥8 𝑥7 𝑚3(𝑥2, 𝑥6, 𝑥4)

𝑃 𝑥2 𝑃 𝑥5 𝑥2 𝑃 𝑥8 𝑥7

𝑃 𝑥7 𝑥4, 𝑥5 𝑚3(𝑥2, 𝑥6, 𝑥4)

𝑃 𝑥2 𝑃 𝑥5 𝑥2 𝑃 𝑥8 𝑥7 𝑚4(𝑥2, 𝑥5, 𝑥6)

𝑃 𝑥2 𝑃 𝑥8 𝑥7

𝑃 𝑥5 𝑥2 𝑚4(𝑥2, 𝑥5, 𝑥6)

𝑃 𝑥2 𝑃 𝑥8 𝑥7 𝑚5(𝑥2, 𝑥6)

𝑃 𝑥2 𝑃 𝑥8 𝑥7

𝑚5(𝑥2, 𝑥6)

𝑃 𝑥2 𝑃 𝑥8 𝑥7 𝑚6(𝑥2) = 𝑚8(𝑥2)𝑚6(𝑥2)

Conditional probability

𝑃 𝑥2| 𝑥7 =𝑚8(𝑥2)𝑚6(𝑥2)

𝑥2𝑚8(𝑥2)𝑚6(𝑥2)

Graph elimination

Graph elimination is a simple unified treatment of inference

algorithms

Moralize the graph

Graph-theoretic property: the factors resulted during variable

elimination are captured by recording the elimination clique

The computational complexity of the Eliminate algorithm can

be reduced to purely graph-theoretic considerations

Graph elimination

Begin with the undirected GM or moralized BN

Choose an elimination ordering (query nodes should be last)

Eliminate a node from the graph and add edges (called fill

edges) between all pairs of its neighbors

Iterate until all non-query nodes are eliminated

Graph elimination

𝑋1 𝑋2

𝑋6𝑋7

𝑋1 𝑋2

𝑋6𝑋8

Removing a node from the graph and connecting the remaining

neighbors

Moralized

Summation ⇔ elimination

Intermediate term ⇔ elimination

clique

Graph elimination: elimination cliques

Induced dependency during marginalization is captured in

elimination cliques

A correspondence between maximal cliques in the induced

graph and maximal factors generated inVE algorithm

The complexity depends on the number of variables in the largest

elimination clique

The size of the maximal elimination clique in the induced

graph depends on the elimination ordering

Elimination order

Finding the best elimination ordering is NP-hard

Equivalent to finding the tree-width in the graph that is NP-

Tree-width: one less than the smallest achievable size of the

largest elimination clique, ranging over all possible elimination

ordering

Good elimination orderings lead to small cliques and

thus reduce complexity

What is the optimal order for trees?

Polytrees

A polytree is a directed graph with no undirected cycles

For poly-trees you can always find an ordering that is efficient

Try it!!

Cut-set conditioning for Bayes’ net inference

Choose set of variables such that if removed only a polytree remains

Exercise:Think about how the specifics would work out!

Worst Case Complexity?

If we can answer P(z) equal to zero or not, we answered whether the 3-SAT problem has a solution.

Hence inference in Bayes’ nets is NP-hard. No known efficient probabilistic inference in general.

Variable Elimination: summary

Interleave joining and marginalizing

dk entries computed for a factor over kvariables with domain sizes d

Ordering of elimination of hiddenvariables can affect size of factorsgenerated

Worst case: running time exponentialin the size of the Bayes’ net

Bayes’ Nets Representation

Probabilistic Inference

Enumeration (exact, exponential complexity)

Variable elimination (exact, worst-caseexponential complexity, often better)

Inference is NP-complete

Approximate Inference: Sampling

Sampling

Sampling is a lot like repeated simulation

Predicting the weather, basketball games, …

Basic idea

Draw N samples from a sampling distribution S

Compute an approximate posterior probability

Show this converges to the true probability P

Why sample?

Learning: get samples from a distribution you don’t know

Inference: getting a sample is faster than computing the right answer (e.g. with variable elimination)

Sampling

Sampling from given distribution

Step 1: Get sample u from uniform distribution over [0, 1)

E.g. random() in python

Step 2: Convert this sample u into an outcome for the given distribution by having each

outcome associated with a sub-interval of [0,1) with sub-interval size equal to probability

of the outcome

If random() returns u = 0.83, then our sample is C = blue

E.g, after sampling 8 times:

C P(C)

red 0.6

green 0.1

blue 0.3

Sampling in Bayes’ Nets

Prior Sampling

Rejection Sampling

Likelihood Weighting

Gibbs Sampling

Prior Sampling

Cloudy

Sprinkler Rain

WetGrass

Cloudy

Sprinkler Rain

WetGrass

+c 0.5-c 0.5

+c +s 0.1-s 0.9

-c +s 0.5-s 0.5

+c +r 0.8-r 0.2

-c +r 0.2-r 0.8

+s +r +w 0.99-w 0.01

-r +w 0.90-w 0.10

-s +r +w 0.90-w 0.10

-r +w 0.01-w 0.99

Samples:

+c, -s, +r, +w-c, +s, -r, +w…

Prior Sampling

For i=1, 2,…, n

Sample xi from P(Xi | Parents(Xi))

Return (x1, x2,…, xn)

Prior Sampling

This process generates samples with probability:

…i.e. the BN’s joint probability

Let the number of samples of an event be

I.e., the sampling procedure is consistent

Example

We’ll get a bunch of samples from the BN:

+c, -s, +r, +w

+c, +s, +r, +w

-c, +s, +r, -w

+c, -s, +r, +w

-c, -s, -r, +w

If we want to know P(W)

We have counts <+w:4, -w:1>

Normalize to get P(W) = <+w:0.8, -w:0.2>

This will get closer to the true distribution with more samples

Can estimate anything else, too

What about P(C| +w)? P(C| +r, +w)? P(C| -r, -w)?

Fast: can use fewer samples if less time (what’s the drawback?)

Rejection Sampling

+c, -s, +r, +w+c, +s, +r, +w-c, +s, +r, -w+c, -s, +r, +w-c, -s, -r, +w

Rejection Sampling

Let’s say we want P(C)

No point keeping all samples around

Just tally counts of C as we go

Let’s say we want P(C| +s)

Same thing: tally C outcomes, but ignore(reject) samples which don’t have S=+s

This is called rejection sampling

It is also consistent for conditional probabilities(i.e., correct in the limit)

Rejection Sampling

IN: evidence instantiation

For i=1, 2,…, n

If xi not consistent with evidence

Reject: Return, and no sample is generated in this cycle

Return (x1, x2,…, xn)

Idea: fix evidence variables and sample the rest Problem: sample distribution not consistent!

Solution: weight by probability of evidence given parents

Problem with rejection sampling:

If evidence is unlikely, rejects lots of samples

Evidence not exploited as you sample

Consider P(Shape|blue)

Shape ColorShape Color

pyramid, greenpyramid, redsphere, bluecube, redsphere, green

pyramid, bluepyramid, bluesphere, bluecube, bluesphere, blue

+c 0.5-c 0.5

+c +s 0.1-s 0.9

-c +s 0.5-s 0.5

+c +r 0.8-r 0.2

-c +r 0.2-r 0.8

+s +r +w 0.99-w 0.01

-r +w 0.90-w 0.10

-s +r +w 0.90-w 0.10

-r +w 0.01-w 0.99

Samples:

+c, +s, +r, +w…

Cloudy

Sprinkler Rain

WetGrass

Cloudy

Sprinkler Rain

WetGrass

IN: evidence instantiation

w = 1.0

for i=1, 2,…, n

if Xi is an evidence variable

Xi = observation xi for Xi

Set w = w * P(xi | Parents(Xi))

return (x1, x2,…, xn), w

Sampling distribution if z sampled and e fixed evidence

Now, samples have weights

Together, weighted sampling distribution is consistent

Cloudy

Likelihood weighting is good

We have taken evidence into account as we generate the sample

E.g. here, W’s value will get picked based on the evidence values of S, R

More of our samples will reflect the state of the world suggested by the evidence

Likelihood weighting doesn’t solve all our problems

Evidence influences the choice of downstream variables, but not upstream ones (C isn’tmore likely to get a value matching the evidence)

We would like to consider evidence when we sample every variable

Gibbs sampling

Gibbs Sampling

Procedure: keep track of a full instantiation x1, x2,…, xn.

Start with an arbitrary instantiation consistent with the evidence.

Sample one variable at a time, conditioned on all the rest, but keep evidence fixed.

Keep repeating this for a long time.

Property: in the limit of repeating this infinitely many times the resulting sample is

coming from the correct distribution

Rationale: both upstream and downstream variables condition on evidence.

In contrast: likelihood weighting only conditions on upstream evidence, and hence

weights obtained in likelihood weighting can sometimes be very small.

Sum of weights over all samples is indicative of how many “effective” samples were obtained, so

want high weight.

Gibbs Sampling Example: P( S | +r)

Step 1: Fix evidence

R = +r

Step 2: Initialize other variables

Randomly

Steps 3: Repeat

Choose a non-evidence variable X

Resample X from P( X | all other variables)

Gibbs Sampling

How is this better than sampling from the full joint?

In a Bayes’ Net, sampling a variable given all the other variables

(e.g. P(R|S,C,W)) is usually much easier than sampling from the

full joint distribution Only requires a join on the variable to be sampled (in this case, a join on R)

The resulting factor only depends on the variable’s parents, its children, and its children’s

parents (this is often referred to as its Markov blanket)

Efficient Resampling of One Variable

Sample from P(S | +c, +r, -w)

Many things cancel out – only CPTs with S remain!

More generally: only CPTs that have resampled variable need to be

considered, and joined together

Bayes’ Net Sampling Summary

Prior Sampling P

Likelihood Weighting P( Q | e)

Rejection Sampling P( Q | e )

Gibbs Sampling P( Q | e )

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