kevin h knuth game theory 2009 automating the processes of inference and inquiry kevin h. knuth...
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Kevin H KnuthGame Theory
2009
Automating the Processes of
Inference and Inquiry
Kevin H. KnuthUniversity at Albany
Kevin H KnuthGame Theory
2009
Describing the World
Kevin H KnuthGame Theory
2009
apple banana cherry
states of a piece of fruit picked from my grocery basket
States
Kevin H KnuthGame Theory
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)
Kevin H KnuthGame Theory
2009
ordering encodes implication
{ a } { b } { c }
{ a, b } { a, c } { b, c }
{ a, b, c }
statements about a piece of fruit
implies
Implication
Kevin H KnuthGame Theory
2009
{ a } { b } { c }
{ a, b } { a, c } { b, c }
{ a, b, c }
statements about a piece of fruit
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
Kevin H KnuthGame Theory
2009
Quantification
Kevin H KnuthGame Theory
2009
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!
Kevin H KnuthGame Theory
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
Kevin H KnuthGame Theory
2009
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),
Kevin H KnuthGame Theory
2009
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)
Kevin H KnuthGame Theory
2009
Valuation
v(y)v(x)y)v(x
x y
x y
RLx:v IVALUATIONv(x) v(y) thenxy If
Kevin H KnuthGame Theory
2009
General Case
x y
x y
x y
z
Kevin H KnuthGame Theory
2009
General Case
x y
x y
x y
z
v(z)y)v(xv(y)
Kevin H KnuthGame Theory
2009
General Case
x y
x y
v(z)v(x)y)v(x
zx y
v(z)y)v(xv(y)
Kevin H KnuthGame Theory
2009
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
Kevin H KnuthGame Theory
2009
SUM RULE
y)v(xv(y)v(x)y)v(x
y)v(xy)v(xv(y)v(x)
symmetric form (self-dual)
Kevin H KnuthGame Theory
2009
Lattice Products
x =
Direct (Cartesian) product of two spaces
Kevin H KnuthGame Theory
2009
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
Kevin H KnuthGame Theory
2009
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
Kevin H KnuthGame Theory
2009
i)|yw(xi)|yw(xi)|w(yi)|w(x
Context Explicit
j)|w(bi)|w(aj))(i,|b)w((a,
Sum Rule
Direct Product Rule
Kevin H KnuthGame Theory
2009
Associativity of Context
=
Kevin H KnuthGame Theory
2009
c)|w(bb)|w(ac)|w(a
CHAIN RULE
a
c
b
Kevin H KnuthGame Theory
2009
Extending the Chain Rulex)|yw(xx)|yw(xx)|w(yx)|w(x
Since xx and x xy, w(x|x)=1 and w(xy |x)=1
x)|yw(xx)|w(y x y
x y
x y
Kevin H KnuthGame Theory
2009
Extending the Chain Rule
yx z
x y y z
x y z
y)x|zyx)w(x|yw(xx)|zyw(x
Kevin H KnuthGame Theory
2009
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
Kevin H KnuthGame Theory
2009
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
Kevin H KnuthGame Theory
2009
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
Kevin H KnuthGame Theory
2009
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
Kevin H KnuthGame Theory
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)
Kevin H KnuthGame Theory
2009
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
Kevin H KnuthGame Theory
2009
Automated Learning
Kevin H KnuthGame Theory
2009
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)
Kevin H KnuthGame Theory
2009
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
Kevin H KnuthGame Theory
2009
Information Gain
Kevin H KnuthGame Theory
2009
)),,(,|,()),,(,|( 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
Kevin H KnuthGame Theory
2009
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
Kevin H KnuthGame Theory
2009
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
Kevin H KnuthGame Theory
2009
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
Kevin H KnuthGame Theory
2009
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
Kevin H KnuthGame Theory
2009
After Several Black Measurements
With several black measurements, the hypothesized shapes become smaller. Exploration is naturally focused on unexplored regions
Kevin H KnuthGame Theory
2009
After One White Measurement
A positive result naturally focuses exploration around promising region
Kevin H KnuthGame Theory
2009
After Two White Measurements
A second positive result naturally focuses exploration around the edges
Kevin H KnuthGame Theory
2009
After Many Measurements
Edge exploration becomes more pronounced as data accumulates.This is all handled naturally by the entropy!
Kevin H KnuthGame Theory
2009
John Skilling
Janos AczélAriel Caticha Keith EarlePhilip GoyalSteve GullJeffrey JewellCarlos Rodriguez
Phil ErnerScott FrassoRotem GutmanNabin MalakarA.J. Mesiti
Special Thanks to: