INTRODUCTION TO BAYESIAN INFERENCE PART 1mlg.eng.cam.ac.uk/mlss09/mlss_slides/Bishop-MLSS-09-1.pdfINTRODUCTION TO BAYESIAN INFERENCE ... knowledge and statistical learning ... Bayesian

INTRODUCTION TOBAYESIAN INFERENCE – PART 1

CHRIS BISHOP

http://research.microsoft.com/~cmbishop

First Generation

“Artificial Intelligence” (GOFAI)

Within a generation ... the problem of creating ‘artificial intelligence’ will largely be solved

Marvin Minsky (1967)

Expert Systems (1980s)

knowledge-based AI

rules elicited from humans

Combinatorial explosion

General theme: hand-crafted rules

Second Generation

Neural networks, support vector machines

Difficult to incorporate complex domain knowledge

General theme: black-box statistical models

Third Generation

General theme: deep integration of domainknowledge and statistical learning

Bayesian framework

Probabilistic graphical models

Fast inference using local message-passing

Origins: Bayesian networks, decision theory, HMMs, Kalman filters, MRFs, mean field theory, ...

Probability Theory

Apples and Oranges

Fruit is orange, what is probability that box was blue?

The Rules of Probability

Sum rule

Product rule

Bayes’ Theorem

Oranges and Apples

Suppose

Suppose we select an orange

Then

and hence

Probability Densities

Bayesian Inference

Consistent use of probability to quantify uncertainty

Predictions involve marginalisation, e.g.

posterior likelihood function prior

Why is prior knowledge important?

?

x

y

Probabilistic Graphical Models

Combine probability theory with graphs

new insights into existing models

framework for designing new models

Graph-based algorithms for calculation and computation (c.f. Feynman diagrams in physics)

efficient software implementation

Directed graphs to specify the model

Factor graphs for inference and learning

Decomposition

Consider an arbitrary joint distribution

By successive application of the product rule:

Directed Graphs

Arrows indicate causal relationships

MAAS

Manchester Asthma and Allergies Study

Goal: discover environmental and genetic causes of asthma

1,186 children monitored since birth

640k SNPs per child

Many environment and physiological measurements:

skin and IgE blood tests at age 1, 3, 5, and 8

wheezing, methacholine response,

pets, parental smoking, day-care, breast feeding, ...

{0,1} {0,1} {0,1} {0,1}

21

Factor Graphs

From Directed Graph to Factor Graph

Inference on Graphs

Factor Trees: Separation

v w x

f1(v,w) f2(w,x)

y

f3(x,y)

z

f4(x,z)

Messages: From Factors To Variables

w x

f2(w,x)

y

f3(x,y)

z

f4(x,z)

Messages: From Variables To Factors

x

f2(w,x)

y

f3(x,y)

z

f4(x,z)

What if the graph is not a tree?

Keep iterating the messages:

loopy belief propagation

What if marginalisations are not tractable?

True distribution Monte Carlo VMP / Loopy BP / EP

Illustration: Bayesian Ranking

Goal: global ranking from noisy partial rankings

Conventional approach: Elo (used in chess)

maintains a single strength value for each player

cannot handle team games, or > 2 players

Ralf Herbrich

Tom Minka

Thore Graepel

Two Player Match Outcome Model

y12

1 2

s1 s2

Two Team Match Outcome Model

y12

t1 t2

s2 s3s1 s4

Multiple Team Match Outcome Model

s1 s2 s3 s4

t1

y12

t2 t3

y23

s1 s2 s3 s4

t1

y12

t2 t3

y23

Gaussian Prior Factors

Ranking Likelihood Factors

0

5

10

15

20

25

30

35

40Level

0 100 200 300 400

Number of Games

char (Elo)

SQLWildman (Elo)

char (TrueSkill™)

SQLWildman (TrueSkill™)

Skill Dynamics

y12

¼1 ¼2

y12

¼1’ ¼2’

s1 s2 s1’ s2’

’

TrueSkillTM

Xbox 360 Live: launched September 2005

TrueSkillTM for ranking and to match players

10M active users, 2.5M matches per day

“Planet-scale” application of Bayesian methods

John Winn

research.microsoft.com/infernet

Tom Minka

John Winn

John Guiver

Anitha Kannan

Infer.Net demonstration

Different?

True False

Prob. Cure

(Treated)Prob. Cure

(Placebo)

Prob. Cure

(All)