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Bayesian Inference:

MAP and Max Likelihood

Lecture 29 of 41

Lecture Outline

• Overview of Bayesian Learning
  • Framework: using probabilistic criteria to generate hypotheses of all kinds
  • Probability: foundations
• Bayes’s Theorem
  • Definition of conditional (posterior) probability
  • Ramifications of Bayes’s Theorem
    • Answering probabilistic queries
    • MAP hypotheses
• Generating Maximum A Posteriori (MAP) Hypotheses
• Generating Maximum Likelihood Hypotheses
• Next Week: Sections 6.6-6.13, Mitchell; Roth; Pearl and Verma
  • More Bayesian learning: MDL, BOC, Gibbs, Simple (Naïve) Bayes
  • Learning over text

Markov Blanket

Constructing Bayesian Networks:

The Chain Rule of Inference

Automated Reasoning using Probabilistic Models:

Inference Tasks

• Fusion
  • Methods for combining multiple beliefs
  • Theory more precise than for fuzzy, ANN inference
  • Data and sensor fusion
    • Resolving conflict (vote-taking, winner-take-all, mixture estimation)
    • Paraconsistent reasoning
• Propagation
  • Modeling process of evidential reasoning by updating beliefs
  • Source of parallelism
  • Natural object-oriented (message-passing) model
    • Communication: asynchronous – dynamic workpool management problem
    • Concurrency: known Petri net dualities
• Structuring
  • Learning graphical dependencies from scores, constraints
  • Two parameter estimation problems: structure learning, belief revision

Fusion, Propagation, and Structuring

Two Roles for Bayesian Methods

• Practical Learning Algorithms
  • Naïve Bayes ( aka simple Bayes)
  • Bayesian belief network (BBN) structure learning and parameter estimation
  • Combining prior knowledge (prior probabilities) with observed data
    • A way to incorporate background knowledge (BK), aka domain knowledge
    • Requires prior probabilities (e.g., annotated rules)
• Useful Conceptual Framework
  • Provides “gold standard” for evaluating other learning algorithms
    • Bayes Optimal Classifier (BOC)
    • Stochastic Bayesian learning: Markov chain Monte Carlo (MCMC)
  • Additional insight into Occam’s Razor (MDL)

Choosing Hypotheses

arg maxf  x

x Ω

• Bayes’s Theorem
• MAP Hypothesis
  • Generally want most probable hypothesis given the training data
  • Define:  the value of x in the sample space  with the highest f ( x )
  • Maximum a posteriori hypothesis, hMAP
• ML Hypothesis
  • Assume that p ( hi ) = p ( hj ) for all pairs i , j (uniform priors, i.e., PH ~ Uniform)
  • Can further simplify and choose the maximum likelihood hypothesis, hML

argmaxPD |hP  h

P D

PD|hP h argmax

h argmaxP h|D

h H

hH

hH MAP







PD 

P h D

P D

P D|hP h P h|D

 i

h H

hML argmaxPD|h i 

Basic Formulas for Probabilities

• Product Rule (Alternative Statement of Bayes’s Theorem)
  • Proof: requires axiomatic set theory, as does Bayes’s Theorem
• Sum Rule
  • Sketch of proof (immediate from axiomatic set theory)
    • Draw a Venn diagram of two sets denoting events A and B
    • Let A  B denote the event corresponding to A  B …
• Theorem of Total Probability
  • Suppose events A 1 , A 2 , …, An are mutually exclusive and exhaustive
    • Mutually exclusive: i  j  Ai  Aj = 
    • Exhaustive:  P ( Ai ) = 1
  • Then
  • Proof: follows from product rule and 3 rd Kolmogorov axiom

PB 

P A B

P A|B

P  AB P A PB  P AB

     i

n

i

P B  PB|Ai P A

 1

A B

Bayesian Learning Example:

Unbiased Coin [1]

• Coin Flip
  • Sample space:  = { Head , Tail }
  • Scenario: given coin is either fair or has a 60% bias in favor of Head
    • h 1  fair coin: P ( Head ) = 0.
    • h 2  60% bias towards Head : P ( Head ) = 0.
  • Objective: to decide between default (null) and alternative hypotheses
• A Priori ( aka Prior) Distribution on H
  • P ( h 1 ) = 0.75, P ( h 2 ) = 0.
  • Reflects learning agent’s prior beliefs regarding H
  • Learning is revision of agent’s beliefs
• Collection of Evidence
  • First piece of evidence: d  a single coin toss, comes up Head
  • Q: What does the agent believe now?
  • A: Compute P ( d ) = P ( d | h 1 ) P ( h 1 ) + P ( d | h 2 ) P ( h 2 )

• Start with Uniform Priors
  • Equal probabilities assigned to each hypothesis
  • Maximum uncertainty (entropy), minimum prior information
• Evidential Inference
  • Introduce data (evidence) D 1 : belief revision occurs
    • Learning agent revises conditional probability of inconsistent hypotheses to 0
    • Posterior probabilities for remaining h  VSH,D revised upward
  • Add more data (evidence) D 2 : further belief revision

Evolution of Posterior Probabilities

P ( h )

Hypotheses

P ( h | D 1 )

Hypotheses

P ( h | D 1 , D 2 )

Hypotheses

• Problem Definition
  • Target function: any real-valued function f
  • Training examples < xi , yi > where yi is noisy training value
    • yi = f ( xi ) + ei
    • ei is random variable (noise) i.i.d. ~ Normal (0,  ), aka Gaussian noise
  • Objective: approximate f as closely as possible
• Solution
  • Maximum likelihood hypothesis hML
  • Minimizes sum of squared errors (SSE)

Maximum Likelihood:

Learning A Real-Valued Function [1]

y

x

f ( x )

hML e

2

1





m

i

i i hH

hML argmin d h x

Terminology

• Introduction to Bayesian Learning
  • Probability foundations
    • Definitions: subjectivist, frequentist, logicist
    • (3) Kolmogorov axioms
• Bayes’s Theorem
  • Prior probability of an event
  • Joint probability of an event
  • Conditional (posterior) probability of an event
• Maximum A Posteriori (MAP) and Maximum Likelihood (ML) Hypotheses
  • MAP hypothesis: highest conditional probability given observations (data)
  • ML: highest likelihood of generating the observed data
  • ML estimation (MLE): estimating parameters to find ML hypothesis
• Bayesian Inference: Computing Conditional Probabilities (CPs) in A Model
• Bayesian Learning: Searching Model (Hypothesis) Space using CPs

Summary Points

• Introduction to Bayesian Learning
  • Framework: using probabilistic criteria to search H
  • Probability foundations
    • Definitions: subjectivist, objectivist; Bayesian, frequentist, logicist
    • Kolmogorov axioms
• Bayes’s Theorem
  • Definition of conditional (posterior) probability
  • Product rule
• Maximum A Posteriori (MAP) and Maximum Likelihood (ML) Hypotheses
  • Bayes’s Rule and MAP
  • Uniform priors: allow use of MLE to generate MAP hypotheses
  • Relation to version spaces, candidate elimination
• Next Week: 6.6-6.10, Mitchell; Chapter 14-15, Russell and Norvig; Roth
  • More Bayesian learning: MDL, BOC, Gibbs, Simple (Naïve) Bayes
  • Learning over text