Download - Bayesian Networks
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Bayesian Networks
Tamara Berg
CS 590-133 Artificial Intelligence
Many slides throughout the course adapted from Svetlana Lazebnik, Dan Klein, Stuart Russell, Andrew Moore, Percy Liang, Luke Zettlemoyer, Rob Pless, Killian Weinberger, Deva Ramanan
Announcements
• HW3 will be released tonight– Written questions only (no programming)– Due Tuesday, March 18, 11:59pm
Review: Probability
• Random variables, events• Axioms of probability• Atomic events• Joint and marginal probability distributions• Conditional probability distributions• Product rule• Independence and conditional independence• Inference
Bayesian decision making
• Suppose the agent has to make decisions about the value of an unobserved query variable X based on the values of an observed evidence variable E
• Inference problem: given some evidence E = e, what is P(X | e)?
• Learning problem: estimate the parameters of the probabilistic model P(X | E) given training samples {(x1,e1), …, (xn,en)}
Bayesian networks (BNs)
A type of graphical model A BN states conditional independence
relationships between random variables Compact specification of full joint distributions
Random Variables
Random variables
be a realization of Let
A random variable is some aspect of the world about which we (may) have uncertainty.
Random variables can be:Binary (e.g. {true,false}, {spam/ham}), Take on a discrete set of values
(e.g. {Spring, Summer, Fall, Winter}), Or be continuous (e.g. [0 1]).
Joint Probability Distribution
Random variables
Joint Probability Distribution:
be a realization of Let
Also written
Gives a real value for all possible assignments.
Queries
Joint Probability Distribution:
Also written
Given a joint distribution, we can reason about unobserved variables given observations (evidence):
Stuff you care about Stuff you already know
Representation
One way to represent the joint probability distribution for discrete is as an n-dimensional table, each cell containing the probability for a setting of X. This would have entries if each ranges over values.
Joint Probability Distribution:
Also written
Graphical Models!
Representation
Graphical models represent joint probability distributions more economically, using a set of “local” relationships among variables.
Joint Probability Distribution:
Also written
Graphical Models
Graphical models offer several useful properties:
1. They provide a simple way to visualize the structure of a probabilistic model and can be used to design and motivate new models.
2. Insights into the properties of the model, including conditional independence properties, can be obtained by inspection of the graph.
3. Complex computations, required to perform inference and learning in sophisticated models, can be expressed in terms of graphical manipulations.
from Chris Bishop
Main kinds of models
• Undirected (also called Markov Random Fields) - links express constraints between variables.
• Directed (also called Bayesian Networks) - have a notion of causality -- one can regard an arc from A to B as indicating that A "causes" B.
Syntax Directed Acyclic Graph (DAG) Nodes: random variables
Can be assigned (observed)or unassigned (unobserved)
Arcs: interactions An arrow from one variable to another indicates
direct influence Encode conditional independence
Weather is independent of the other variables Toothache and Catch are conditionally independent
given Cavity Must form a directed, acyclic graph
Weather Cavity
Toothache Catch
Example: Naïve Bayes document model
Random variables: X: document class W1, …, Wn: words in the document
W1 W2 Wn…
X
Example: Burglar Alarm
I have a burglar alarm that is sometimes set off by minor earthquakes. My two neighbors, John and Mary, promise to call me at work if they hear the alarm
Example inference task: suppose Mary calls and John doesn’t call. What is the probability of a burglary?
What are the random variables? Burglary, Earthquake, Alarm, JohnCalls, MaryCalls
What are the direct influence relationships? A burglar can set the alarm off An earthquake can set the alarm off The alarm can cause Mary to call The alarm can cause John to call
Directed Graphical Models
Directed Graph, G = (X,E)
Nodes
Edges
Each node is associated with a random variable
Definition of joint probability in a graphical model:
where are the parents of
Conditional Independence
By Chain Rule (using the usual arithmetic ordering)
Missing variables in the local conditional probability functions correspond to missing edges in the underlying graph.
Removing an edge into node i eliminates an argument from the conditional probability factor
Joint distribution from the example graph:
Semantics A BN represents a full joint distribution in a compact way.
We only need to specify a conditional probability distribution for each node given its parents: P (X | Parents(X))
Z1 Z2 Zn
X
…
P (X | Z1, …, Zn)
Example: Alarm Network
Burglary Earthqk
Alarm
John calls
Mary calls
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
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 turns out to be faster to answer queries (coming)
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The joint probability distribution
For example, P(j, m, a, ¬b, ¬e)
= P(¬b) P(¬e) P(a | ¬b, ¬e) P(j | a) P(m | a)
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:
– 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?
X Y Z
Independence
Key properties:a) Each node is conditionally independent of its non-descendants given its parentsb) A node is conditionally independent of all other nodes in the graph given it’s Markov blanket (it’s parents, children, and children’s other parents)
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:
– 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?
X Y Z
Causal Chains
• This configuration is a “causal chain”
– Is Z independent of X given Y?
– Evidence along the chain “blocks” the influence
X Y Z
Yes!
X: Project due
Y: No office hours
Z: Students panic
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Common Cause
• Another basic configuration: two effects of the same cause– Are X and Z independent?
– Are X and Z independent given Y?
– Observing the cause blocks influence between effects.
X
Y
Z
Yes!
Y: Homework due
X: Full attendance
Z: Students sleepy
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Common Effect
• Last configuration: two causes of one effect (v-structures)– Are X and Z 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 Z independent given Y?• 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
Y
Z
X: Raining
Z: Ballgame
Y: Traffic
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The General Case
• Any complex example can be analyzed using these three canonical cases
• General question: in a given BN, are two variables independent (given evidence)?
• Solution: analyze the graph40
Causal Chain
Common Cause
(Unobserved)Common Effect
Reachability (D-Separation)• Question: Are X and Y
conditionally independent given evidence vars {Z}?– Yes, if X and Y “separated” by Z– Look for active paths from X to Y– No active paths = independence!
• A 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 descendants is observed
All it takes to block a path is a single inactive segment
Active Triples Inactive Triples
Bayes Ball
• Shade all observed nodes. Place balls at the starting node, let them bounce around according to some rules, and ask if any of the balls reach any of the goal node.
• We need to know what happens when a ball arrives at a node on its way to the goal node.
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Constructing Bayesian networks
1. Choose an ordering of variables X1, … , Xn
2. For i = 1 to n add Xi to the network select parents from X1, … ,Xi-1 such that
P(Xi | Parents(Xi)) = P(Xi | X1, ... Xi-1)
Suppose we choose the ordering M, J, A, B, E
P(J | M) = P(J)? No
P(A | J, M) = P(A)?
P(A | J, M) = P(A | J)?
P(A | J, M) = P(A | M)?
Example
Suppose we choose the ordering M, J, A, B, E
P(J | M) = P(J)? No
P(A | J, M) = P(A)? No
P(A | J, M) = P(A | J)? No
P(A | J, M) = P(A | M)? No
Example
Suppose we choose the ordering M, J, A, B, E
P(J | M) = P(J)? No
P(A | J, M) = P(A)? No
P(A | J, M) = P(A | J)? No
P(A | J, M) = P(A | M)? No
P(B | A, J, M) = P(B)?
P(B | A, J, M) = P(B | A)?
Example
Suppose we choose the ordering M, J, A, B, E
P(J | M) = P(J)? No
P(A | J, M) = P(A)? No
P(A | J, M) = P(A | J)? No
P(A | J, M) = P(A | M)? No
P(B | A, J, M) = P(B)? No
P(B | A, J, M) = P(B | A)? Yes
Example
Suppose we choose the ordering M, J, A, B, E
P(J | M) = P(J)? No
P(A | J, M) = P(A)? No
P(A | J, M) = P(A | J)? No
P(A | J, M) = P(A | M)? No
P(B | A, J, M) = P(B)? No
P(B | A, J, M) = P(B | A)? Yes
P(E | B, A ,J, M) = P(E)?
P(E | B, A, J, M) = P(E | A, B)?
Example
Suppose we choose the ordering M, J, A, B, E
P(J | M) = P(J)? No
P(A | J, M) = P(A)? No
P(A | J, M) = P(A | J)? No
P(A | J, M) = P(A | M)? No
P(B | A, J, M) = P(B)? No
P(B | A, J, M) = P(B | A)? Yes
P(E | B, A ,J, M) = P(E)? No
P(E | B, A, J, M) = P(E | A, B)? Yes
Example
Example contd.
Deciding conditional independence is hard in noncausal directions
The causal direction seems much more natural, but is not mandatory
Network is less compact