kevin h knuth game theory 2009 automating the processes of inference and inquiry kevin h. knuth...

Kevin H KnuthGame Theory

2009

Automating the Processes of

Inference and Inquiry

Kevin H. KnuthUniversity at Albany


2009

Describing the World


2009

apple banana cherry

states of a piece of fruit picked from my grocery basket

States


2009

statements describe potential states

powerset

a b c

states of a piece of fruit

{ a } { b } { c }

{ a, b } { a, c } { b, c }

{ a, b, c }

statements about a piece of fruit

subsetinclusion

Statements (States of Knowledge)


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ordering encodes implication

{ a } { b } { c }

{ a, b } { a, c } { b, c }

{ a, b, c }


implies

Implication


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{ a } { b } { c }

{ a, b } { a, c } { b, c }

{ a, b, c }


inference works backwards

Quantify to what degree knowing that the system is inone of three states {a, b, c}

implies knowing that it is in some other set of states

Inference


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Quantification


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Quantification

quantify the partial order = assign real numbers to the elements

{ a } { b } { c }

{ a, b } { a, c } { b, c }

{ a, b, c }

Any quantification must be consistent with the lattice structure.

Otherwise, it does not quantify the partial order!


2009

Local ConsistencyAny general rule must hold for special cases

Look at special cases to constrain general rule

We enforce local consistency

f(y)andf(x)y)f(x

f(y)]S[f(x),y)f(x This implies that:x y

x y

RLx:f I


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Associativity of Join VWrite the same element two different ways

This implies that:

zy)(xz)(yx

f(z)]f(y)],S[S[f(x),f(z)]]S[f(y),S[f(x),


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Associativity of Join VWrite the same element two different ways

This implies that:

v(y)v(x)y)v(x

The general solution (Aczel) is:F(f(y))F(f(x))f(y)])F(S[f(x),

DERIVATION OF THE SUMMATION AXIOM IN MEASURE THEORY!

zy)(xz)(yx

f(z)]f(y)],S[S[f(x),f(z)]]S[f(y),S[f(x),

(Knuth, 2003, 2009)


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Valuation

v(y)v(x)y)v(x

x y

x y

RLx:v IVALUATIONv(x) v(y) thenxy If


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General Case

x y

x y

x y

z


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General Case

x y

x y

x y

z

v(z)y)v(xv(y)


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General Case

x y

x y

v(z)v(x)y)v(x

zx y

v(z)y)v(xv(y)


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General Case

y)v(xv(y)v(x)y)v(x

v(z)y)v(xv(y) v(z)v(x)y)v(x

x y

x y

zx y


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SUM RULE

y)v(xv(y)v(x)y)v(x

y)v(xy)v(xv(y)v(x)

symmetric form (self-dual)


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Lattice Products

x =

Direct (Cartesian) product of two spaces


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The lattice product is associativeCB)(AC)(BA

After the sum rule, the only freedom left is rescaling

v(b)v(a)b))v((a,

DIRECT PRODUCT RULE


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Context and Bi-Valuations

Valuation

Bi-Valuation v(x)i)|w(x (x)vi

Measure of xwith respect to

Context i

Context iis implicit

Context iis explicit

Bi-valuations generalize lattice inclusion to

degrees of inclusion

BI-VALUATION RLix,:w I


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Associativity of Context

=


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c)|w(bb)|w(ac)|w(a

CHAIN RULE

a

c

b


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Extending the Chain Rule

yx z

x y y z

x y z

y)x|zyx)w(x|yw(xx)|zyw(x


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Extending the Chain Rule

yx z

x y y z

x y z

y)x|zyx)w(x|yw(xx)|zyw(x

y)x|x)w(z|w(yx)|zw(y


2009

i)|yw(xi)|yw(xi)|w(yi)|w(x

Constraint Equations

j)|w(bi)|w(aj))(i,|b)w((a,

Sum Rule

Direct Product Rule

y)x|x)w(z|w(yx)|zw(y Product Rule

(Knuth, MaxEnt 2009)


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CommutativityCommutativityleads to Bayes Theorem…

Bayes Theorem involves a change of context.

yxyx

i)|w(y

i)|w(xi)x|w(yi)y|w(x

i)|w(yi)|w(x

x)|w(yy)|w(x


2009

Automated Learning


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Application to Statements

T)|p(D

T)|p(MM)|p(DD)|p(M

Applied to the lattice of statements our bi-valuation quantifies degrees of implication

M represents a statement about our MODELD represents a statement about our observed DATA

T is the TRUISM (what we assume to be true)


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Change of Context = Learning

T)|p(D

M)|p(DT)|p(MD)|p(M

Re-arranging the terms highlights the learning process

Updated state of knowledgeabout the MODEL

Initial state of knowledgeabout the MODEL

DATA dependent term


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Information Gain


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)),,(,|,()),,(,|( IyxDpdIyxDp eeeeee DMMD

Predict the measurement value De we would expect to obtain by measuring at some position (xe, ye). We rely on our previous data D, and hypothesized model M:

)),,(,|()),,(,,|( IyxpIyxDpd eeeee DMDMM

Using the product rule

)),,(,|()),,(,|( IyxpIyxDpd eeeee DMMM

Predict a Measurement Value


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Probability theory is not sufficient to select an optimal experiment. Instead, we rely on decision theory, where U(.) is an utility function

Using the Shannon Information as the Utility function

)),(,()),,(,|()ˆ,ˆ( eeeeeeeee yxDUIyxDpdDyx D

)),,(,,|(log)),,(,,|()),(,( IyxDpIyxDpdyxDU eeeeeeeee DMDMM

Select an Experiment


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By writing the joint entropy of the model M and the predicted measurement De, two different ways, one can show that(Loredo 2003)

We choose the experiment that maximizes the entropy of the distribution of predicted measurements.Other cost functions will lead to other results (GOOD FOR ROBOTICS!)

)(maxarg

)),,(,|(log)),,(,|(maxarg

)ˆ,ˆ(

),(

),(

eyx

eeeeeeeyx

ee

DH

IyxDpIyxdpdD

yx

ee

ee

DD

Maximum Information Gain


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This robot is equipped with a light sensor.

It is to locate and characterize a white circle on a black playing field with as few measurements as possible.

Robotic Scientists


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Initial StageBLUE: Inference Engine generates samples from space of polygons / circlesCOPPER: Inquiry Engine computes entropy map of predicted measurement results

With little data, the hypothesized shapes are extremely varied and it is good to look just about anywhere


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After Several Black Measurements

With several black measurements, the hypothesized shapes become smaller. Exploration is naturally focused on unexplored regions


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After One White Measurement

A positive result naturally focuses exploration around promising region


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After Two White Measurements

A second positive result naturally focuses exploration around the edges


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After Many Measurements

Edge exploration becomes more pronounced as data accumulates.This is all handled naturally by the entropy!


2009

John Skilling

Janos AczélAriel Caticha Keith EarlePhilip GoyalSteve GullJeffrey JewellCarlos Rodriguez

Phil ErnerScott FrassoRotem GutmanNabin MalakarA.J. Mesiti

Special Thanks to:

kevin h knuth game theory 2009 automating the processes of inference and inquiry kevin h. knuth...

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