introduction to bayesian inference part...
TRANSCRIPT
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INTRODUCTION TOBAYESIAN INFERENCE – PART 1
CHRIS BISHOP
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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
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Second Generation
Neural networks, support vector machines
Difficult to incorporate complex domain knowledge
General theme: black-box statistical models
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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, ...
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Probability Theory
Apples and Oranges
Fruit is orange, what is probability that box was blue?
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The Rules of Probability
Sum rule
Product rule
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Bayes’ Theorem
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Oranges and Apples
Suppose
Suppose we select an orange
Then
and hence
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Probability Densities
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Bayesian Inference
Consistent use of probability to quantify uncertainty
Predictions involve marginalisation, e.g.
posterior likelihood function prior
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Why is prior knowledge important?
?
x
y
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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
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Decomposition
Consider an arbitrary joint distribution
By successive application of the product rule:
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Directed Graphs
Arrows indicate causal relationships
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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, ...
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{0,1} {0,1} {0,1} {0,1}
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21
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Factor Graphs
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From Directed Graph to Factor Graph
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Inference on Graphs
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Factor Trees: Separation
v w x
f1(v,w) f2(w,x)
y
f3(x,y)
z
f4(x,z)
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Messages: From Factors To Variables
w x
f2(w,x)
y
f3(x,y)
z
f4(x,z)
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Messages: From Variables To Factors
x
f2(w,x)
y
f3(x,y)
z
f4(x,z)
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What if the graph is not a tree?
Keep iterating the messages:
loopy belief propagation
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What if marginalisations are not tractable?
True distribution Monte Carlo VMP / Loopy BP / EP
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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
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Two Player Match Outcome Model
y12
1 2
s1 s2
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Two Team Match Outcome Model
y12
t1 t2
s2 s3s1 s4
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Multiple Team Match Outcome Model
s1 s2 s3 s4
t1
y12
t2 t3
y23
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s1 s2 s3 s4
t1
y12
t2 t3
y23
Gaussian Prior Factors
Ranking Likelihood Factors
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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™)
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Skill Dynamics
y12
¼1 ¼2
y12
¼1’ ¼2’
s1 s2 s1’ s2’
’
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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
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John Winn
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research.microsoft.com/infernet
Tom Minka
John Winn
John Guiver
Anitha Kannan
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Infer.Net demonstration
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Different?
True False
Prob. Cure
(Treated)Prob. Cure
(Placebo)
Prob. Cure
(All)