Bayesian Networks in Probabilistic Reasoning

Artificial Intelligence: Representation and Problem SolvingProbabilistic Reasoning (2): Bayesian Networks 15-381 / 681 Instructors: Fei Fang (This Lecture) and Dave Touretzky feifang@cmu.edu Wean Hall 4126

Recap • Probability Models • Joint probability distribution of random variables • Probabilistic Inference • Compute Marginal probability or Conditional probability • Chain Rule, Independence, Bayes’ Rule • Full joint distribution is hard to estimate and too big to represent explicitly Fei Fang

Outline • Bayesian Networks • Independence in Bayes’ Net • Construct a Bayes’ Net • Exact Inference in Bayes’ Net • Applications of Bayes’ Net Fei Fang

Bayesian Network • Bayesian Network (Bayes’ net) Overview • A compact way to represent knowledge in an uncertain domain • Describe full joint probability distributions using simple, local distributions • A probabilistic graphical model • A directed acyclic graph • Each node corresponds to a random variable • Each direct edge represent is a parent of • Each node has a conditional probability distribution (≈ “directly influences”) Way to Commute to Work Weather Sleeping Quality Fei Fang

Example: Alarm • I’m at work, both of my neighbors John and Mary call to say my alarm for burglary is ringing. Sometimes it’s set off by minor earthquakes. Is there a burglary? • How do we model this scenario? How can we represent our knowledge in such a domain with uncertainty? Fei Fang

Example: Alarm • Random Variables: , , , , • Domain • Knowledge base: Full joint probability distribution • How big is the table? • Task: Compute Fei Fang

Example: Alarm • Recall Independence • Can be represented much more efficiently! • Are all the random variables independent in this example? Fei Fang

Example: Alarm • However, there are some intuitive independence relationships based on our causal knowledge! • Causal knowledge – • A burglary 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 Burglary Earthquake Alarm MaryCalls JohnCalls Fei Fang

Example: Alarm • and are independent with each other • Conditioned on the value of , and are independent • Similar independence assumptions for and • Conditioned on the value of , and are independent with each other Burglary (B), Earthquake (E), Alarm (A), JohnCalls (J), MaryCalls (M)

Example: Alarm • Given these independence relationships, • We don’t need fill the full joint probability table anymore to represent our knowledge! • Only need to provide these conditional probabilities • Is this better or worse? Fei Fang

Example: Alarm How many numbers we need here? ! Recall we need for the original table. burglary (B), Earthquake (E), Alarm (A), JohnCalls (J), MaryCalls (M)

Example: Alarm Enrich the network with more links: more realistic, less compact Fei Fang

Bayesian Network • Bayesian Network: A compact way to represent knowledge in an uncertain domain Fei Fang

Bayesian Network • Bayesian Network: Describe full joint probability distributions using simple, local distributions Global semantics Have to be equivalent! Fei Fang

Bayesian Network • Bayesian Network: Describe full joint probability distributions using simple, local distributions • What is ? means Have to be equivalent! Fei Fang

Bayesian Network • Bayesian Network: A probabilistic graphical model • A directed acyclic graph • Node – random variable; Edge – parent-child relationship • Conditional probability distribution • Often represented by CPT (conditional probability table) A Bayes’ Net = topology (graph) + local conditional probabilities Fei Fang

Quiz 1 • At least how many entries are needed for a general CPT (conditional probability table) for the node “Way to Commute to Work”? • A: 18 • B: 12 • C: 6 • D: 3 Weather Way to Commute to Work Sleeping Quality Fei Fang

Quiz 1 Weather Way to Commute to Work Sleeping Quality Fei Fang

Another Perspective of Bayes’ Net • Assume you have no “causal knowledge” but someone gives you the full joint probability table • You observe there is a valid factorization of the full joint probability distribution • You further observe that such factorization can be represented using a DAG • You can prove ’s equal to “conditional probabilities” • Now you get a Bayes’ Net Fei Fang

Independence in Bayes’ Net • Given a Bayes’ Net, which variables are independent? • Each node is conditionally independent of its non-descendants given its parents Local semantics Given , is independent of Fei Fang

Independence in Bayes’ Net • Each node is conditionally independent of all others given its Markov blanket: parents + children + children’s parents Fei Fang

Example • List all the independence relationships Local Semantics: Each node is conditionally independent of its non-descendants given its parents Each node is conditionally independent of all others given its Markov blanket: parents + children + children’s parents Fei Fang

Quiz 2 • Which of the following statements of independence are true given the Bayes’ Net based on local semantics and Markov blankets? • A: • B: • C: • D: • E: B C A D E G F Local Semantics: Each node is conditionally independent of its non-descendants given its parents Each node is conditionally independent of all others given its Markov blanket: parents + children + children’s parents H Fei Fang

Is Bayes’ Net Expressive Enough? • Any full joint probability table can be represented by a Bayes’ Net Fei Fang

Is Bayes’ Net Unique? • One (full joint probability distribution)-to-many (Bayes’ Net) mapping Fei Fang

Construct a Bayes’ Net • Construct a (ideally simple) Bayes’ Net systematically As a knowledge engineer or domain expert Choose an ordering of variables For Add to the network Select a minimal subset of variables from , denoted such that Add edges from nodes in to , write down the conditional probability table (CPT) This process guarantees Fei Fang

Construct a Bayes’ Net • Construct a (ideally simple) Bayes’ Net systematically • Ordering of variables matters • Exploit domain knowledge to determine the ordering: intuitively, parent of a node should contain all nodes that directly influences Fei Fang

Probabilistic Inference in Bayes’ Net • Recap: Probabilistic inference: • No evidence: Marginal probability • With evidence: Posterior / Conditional probability • Inference with full joint probability distribution: Marginalization, Bayes’ Rule • Exact Inference in Bayes’ Net: (1) Enumeration; (2) Variable Elimination Fei Fang

General Inference Procedure • Partition the set of random variables in the model • Evidence variables , and be e the list of observed values from them • Remaining unobserved / hidden variables • Query variables • The query can be answered by Fei Fang

Inference in Bayes’ Net • Inference with full joint probability distribution table available: Read the joint probability from the table • Inference in Bayes’ Net: compute joint probability through conditional probability table Fei Fang

Example: Alert • I’m at work, both of my neighbors John and Mary call to say my alarm for burglary is ringing. Sometimes it’s set off by minor earthquakes. Is there a burglary? Fei Fang

Example: Alert • Evaluate through depth-first recursion of the following expression tree Top-down DF evaluation: × Values along each path + at the branching nodes Fei Fang

Example: Alert • Normalize Fei Fang

Exact Inference in Bayes’ Net: Enumeration Fei Fang

Exact Inference in Bayes’ Net: Variable Elimination • Avoid repeated computation of subexpressions in the enumeration algorithm • Similar to dynamic programming Fei Fang

Bayes’ Net as a Model of Real World • Bayes’ Net represents knowledge in an uncertain domain • View it as a way to model the real world based on domain knowledge • Is your model (Bayes’ Net) for a real-world problem correct? Not necessarily. Fei Fang

Bayes’ Net as a Model of Real World • "All models are wrong“ • Acommon aphorism in statistics • Generally attributed to the statistician George Box "Essentially, all models are wrong, but some are useful". https://en.wikipedia.org/wiki/All_models_are_wrong Fei Fang

Use of Bayes’ Net • Diagnosis: (cause | symptom)? • Prediction: (symptom | cause)? • Classification: (class | data) • Decision-making (given a cost function) Fei Fang

Use of Bayes’ Net Russel and Novig Fei Fang

Summary • Bayes’ Net • Graphical model • Decompose full joint probability distributions into interpretable, simple, local distributions • Independence in Bayes’ Net • Local semantics • Markov Blanket • Construct a Bayes Net • Exact Inference in Bayes’ Net • Applications of Bayes’ Net Fei Fang

Acknowledgment • Some slides are borrowed from previous slides made by Tai Sing Lee Fei Fang

Backup Slides Fei Fang

Conditional Independence • Example graph (1) Fei Fang

D-Separation for Conditional Independence • is valid in general if and only if all the paths from any node in to any node in are blocked • A path is blocked if and only if it includes a node such that either one of the following statements are true • The rows on the path meet head-to-tail or tail-to-tail at the node, and the node is in the set • The rows on the path meet head-to-head at the node, and neither the node nor any of its descendants, is in the set Head-to-tail at Tail-to-tail at Head-to-head at Fei Fang

Bayesian Networks in Probabilistic Reasoning