CIS 471/571(Fall 2020): Introduction to Artificial Intelligence
Lecture 14: Bayes Nets – Independence
Thanh H. Nguyen
Source: http://ai.berkeley.edu/home.html
Announcement
§Homework 4: Bayes Nets and HMMs §Will be posted today (Nov 12, 2020) §Deadline: Nov 24, 2020
Thanh H. Nguyen
11/11/20
2
Probability Recap
§Conditional probability §Product rule
§Chain rule
§X, Y independent if and only if:
§X and Y are conditionally independent given Z if and only if:
Bayes’ Nets
§A Bayes’ net is an efficient encoding of a probabilistic model of a domain
§Questions we can ask:
§ Inference: given a fixed BN, what is P(X | e)?
§ Representation: given a BN graph, what kinds of distributions can it encode? § Modeling: what BN is most appropriate for a given domain?
Bayes’ Net Semantics
§ A directed, acyclic graph, one node per random variable § A conditional probability table (CPT) for each node
§ A collection of distributions over X, one for each combination of parents’ 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, multiply all the relevant conditionals together:
Example: Alarm Network
BE A
JM
B
+b
-b
P(B)
0.001
0.999
E
+e
-e
P(E)
0.002
0.998
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
+b
+b
+b
+b
-b
-b
-b
-b
E
+e
+e
-e
-e
+e
+e
-e
-e
A
+a
-a
+a
-a
+a
-a
+a
-a
P(A|B,E)
0.95
0.05
0.94
0.06
0.29
0.71
0.001
0.999
Example: Alarm Network
BE A
JM
B
+b
-b
P(B)
0.001
0.999
E
+e
-e
P(E)
0.002
0.998
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
+b
+b
+b
+b
-b
-b
-b
-b
E
+e
+e
-e
-e
+e
+e
-e
-e
A
+a
-a
+a
-a
+a
-a
+a
-a
P(A|B,E)
0.95
0.05
0.94
0.06
0.29
0.71
0.001
0.999
Size of a Bayes’ Net
§How big is a joint distribution over N Boolean variables?
2N
§ How big is an N-node net if nodes have up to k parents?
O(N * 2k+1)
§ Both give you the power to calculate
§ BNs: Huge space savings!
§ Also easier to elicit local CPTs
§ Also faster to answer queries (coming)
Bayes’ Nets
§Representation
§Conditional Independences §Probabilistic Inference §Learning Bayes’ Nets from Data
Conditional Independence
§X and Y are independent if
§X and Y are conditionally independent given Z
§(Conditional) independence is a property of a distribution §Example:
Bayes Nets: Assumptions
§Assumptions we are required to make to define the Bayes net when given the graph:
P (xi|x1 · · · xi 1) = P (xi|parents(Xi))
§Beyond above “chain ruleàBayes net” conditional
independence assumptions
§ Often additional conditional independences § They can be read off the graph
§Important for modeling: understand assumptions made when choosing a Bayes net graph
Example
§ Conditional independence assumptions directly from simplifications in chain rule:
XYZW
§Additional implied conditional independence assumptions?
Independence in a BN
§Important question about a BN:
§Are two nodes independent given certain evidence? § If yes, can prove using algebra (tedious in general) § If no, can prove with a counter example
§ Example:
XYZ
§ Question: are X and Z necessarily independent?
§ Answer: no. Example: low pressure causes rain, which causes traffic.
§ X can influence Z, Z can influence X (via Y)
§ Addendum: they could be independent: how?
D-separation: Outline
D-separation: Outline
§Study independence properties for triples
§Analyze complex cases in terms of member triples
§D-separation: a condition / algorithm for answering such queries
Causal Chains
§ This configuration is a “causal chain”
§ Guaranteed X independent of Z ? No!
§ One example set of CPTs for which X is not independent of Z is sufficient to show this independence is not guaranteed.
X: Low pressure
Y: Rain
Z: Traffic
traffic,
high pressure causes no rain causes no traffic
§ Example:
§ Low pressure causes rain causes
§ In numbers:
P( +y | +x ) = 1, P( -y | – x ) = 1, P( +z | +y ) = 1, P( -z | -y ) = 1
Causal Chains
§Thisconfigurationisa“causalchain” §GuaranteedXindependentofZgiven Y?
X: Low pressure Y: Rain Z: Traffic
Yes!
§ Evidence along the chain “blocks” the influence
Common Cause
§ This configuration is a “common cause” § Guaranteed X independent of Z ? No!
X: Forums busy
Z: Lab full
Y: Project due
§ One example set of CPTs for which X is not independent of Z is sufficient to show this independence is not guaranteed.
§ Example:
§ Project due causes both forums busy
and lab full § In numbers:
P( +x | +y ) = 1, P( -x | -y ) = 1, P( +z | +y ) = 1, P( -z | -y ) = 1
Common Cause
§ This configuration is a “common cause” § Guaranteed X and Z independent given Y?
Y: Project due
X: Forums busy
Z: Lab full
Yes!
§ Observing the cause blocks influence between effects.
Common Effect
§Last configuration: two causes of one effect (v-structures)
§ Are X and Y independent?
§ Yes: the ballgame and the rain cause traffic,
but they are not correlated
§ Still need to prove they must be (try it!)
§ Are X and Y independent given Z? § No: seeing traffic puts the rain and the
ballgame in competition as explanation.
§ This is backwards from the other cases
§ Observing an effect activates influence between possible causes.
X: Raining
Y: Ballgame
Z: Traffic
The General Case
The General Case
§General question: in a given BN, are two variables independent (given evidence)?
§Solution: analyze the graph
§Any complex example can be broken
into repetitions of the three canonical cases
Active / Inactive Paths
§ Question: Are X and Y conditionally independent given evidence variables {Z}?
Active Triples
Inactive Triples
§ § §
§ A §
§ §
Yes, if X and Y “d-separated” by Z Consider all (undirected) paths from X to Y No active paths = independence!
path is active if each triple is active:
Causal chain A ® B ® C where B is unobserved (either direction)
Common cause A ¬ B ® C where B is unobserved Common effect (aka v-structure)
A ® B ¬ C where B or one of its descendents is observed
§ All it takes to block a path is a single inactive segment
D-Separation
§ Query:
§Check all (undirected!) paths between
?
Xi Xj|{Xk1,…,Xkn}
§ If one or more active, then independence not guaranteed
and
Xi Xj|{Xk1,…,Xkn} § Otherwise (i.e. if all paths are inactive),
then independence is guaranteed
Xi Xj|{Xk1,…,Xkn}
Example
Yes
RB
T
Tʼ
Example
Yes Yes
L RB
DT Tʼ
Yes
Example
§Variables: §R: Raining §T: Traffic §D: Roof drips §S: I’m sad
§Questions:
Yes
R TD
S
Structure Implications
§Given a Bayes net structure, can run d- separation algorithm to build a complete list of conditional independences that are necessarily true of the form
Xi Xj|{Xk1,…,Xkn}
§This list determines the set of probability distributions that can be represented
Computing All Independences
Y
XZ Y
XZ XZ
Y Y
XZ
Topology Limits Distributions
§Given some graph topologyG,onlycertain joint distributions can be encoded
{X Y,X Z,Y Z, X Z|Y,X Y |Z,Y Z|X}
Y
{ X Z | Y } Y
XZ Y
XZ Y
XZ
{}
YY XZXZ YY XZXZ
§ The graph structure X guarantees certain
§ (There might be more independence)
§ Adding arcs increases the set of distributions, but has several costs
§ Full conditioning can encode any distribution
Z
(conditional) independences
Y XZ Y XZ
Bayes Nets Representation Summary
§Bayes nets compactly encode joint distributions §Guaranteed independencies of distributions can be
§D-separation gives precise conditional independence guarantees from graph alone
§A Bayes’ net’s joint distribution may have further (conditional) independence that is not detectable until you inspect its specific distribution
deduced from BN graph structure
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