Kevin H Knuth Game Theory 2009 Automating the Processes of Inference and Inquiry Kevin H. Knuth University at Albany

Kevin H Knuth Game Theory 2009 Describing the World

Kevin H Knuth Game Theory 2009 applebananacherry states of a piece of fruit picked from my grocery basket States

Kevin H Knuth Game 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 subset inclusion Statements (States of Knowledge)

Kevin H Knuth Game 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 Knuth Game 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 in one of three states {a, b, c} implies knowing that it is in some other set of states Inference

Kevin H Knuth Game Theory 2009 Quantification

Kevin H Knuth Game 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 Knuth Game Theory 2009 Local Consistency Any general rule must hold for special cases Look at special cases to constrain general rule We enforce local consistency This implies that: x y x  y I

Kevin H Knuth Game Theory 2009 Associativity of Join V Write the same element two different ways This implies that:

Kevin H Knuth Game Theory 2009 Associativity of Join V Write the same element two different ways This implies that: The general solution (Aczel) is: DERIVATION OF THE SUMMATION AXIOM IN MEASURE THEORY! (Knuth, 2003, 2009)

Kevin H Knuth Game Theory 2009 Valuation xy x  y I VALUATION

Kevin H Knuth Game Theory 2009 General Case xy x  y x  y z

Kevin H Knuth Game Theory 2009 General Case xy x  y x  y z

Kevin H Knuth Game Theory 2009 General Case xy x  y zx  y

Kevin H Knuth Game Theory 2009 General Case xy x  y zx  y

Kevin H Knuth Game Theory 2009 SUM RULE symmetric form (self-dual)

Kevin H Knuth Game Theory 2009 Lattice Products x = Direct (Cartesian) product of two spaces

Kevin H Knuth Game Theory 2009 The lattice product is associative After the sum rule, the only freedom left is rescaling DIRECT PRODUCT RULE

Kevin H Knuth Game Theory 2009 Context and Bi-Valuations Valuation Bi-Valuation Measure of x with respect to Context i is implicit Context i is explicit Bi-valuations generalize lattice inclusion to degrees of inclusion BI-VALUATION I

Kevin H Knuth Game Theory 2009 Context Explicit Sum Rule Direct Product Rule

Kevin H Knuth Game Theory 2009 Associativity of Context =

Kevin H Knuth Game Theory 2009 CHAIN RULE a c b

Kevin H Knuth Game Theory 2009 Extending the Chain Rule Since x  x and x  x  y, w(x | x) =1 and w(x  y | x)=1 x y x  y x  y

Kevin H Knuth Game Theory 2009 Extending the Chain Rule y x z x  yy  z x  y  z

Kevin H Knuth Game Theory 2009 Extending the Chain Rule y x z x  yy  z x  y  z

Kevin H Knuth Game Theory 2009 Extending the Chain Rule y x z x  yy  z x  y  z

Kevin H Knuth Game Theory 2009 Extending the Chain Rule y x z x  yy  z x  y  z

Kevin H Knuth Game Theory 2009 Extending the Chain Rule y x z x  yy  z x  y  z

Kevin H Knuth Game Theory 2009 Constraint Equations Sum Rule Direct Product Rule Product Rule (Knuth, MaxEnt 2009)

Kevin H Knuth Game Theory 2009 Commutativity leads to Bayes Theorem… Bayes Theorem involves a change of context.

Kevin H Knuth Game Theory 2009 Automated Learning

Kevin H Knuth Game Theory 2009 Application to Statements Applied to the lattice of statements our bi-valuation quantifies degrees of implication M represents a statement about our MODEL D represents a statement about our observed DATA T is the TRUISM (what we assume to be true)

Kevin H Knuth Game Theory 2009 Change of Context = Learning Re-arranging the terms highlights the learning process Updated state of knowledge about the MODEL Initial state of knowledge about the MODEL DATA dependent term

Kevin H Knuth Game Theory 2009 Information Gain

Kevin H Knuth Game Theory 2009 Predict the measurement value D e we would expect to obtain by measuring at some position (x e, y e ). We rely on our previous data D, and hypothesized model M: Using the product rule Predict a Measurement Value

Kevin H Knuth Game 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 Select an Experiment

Kevin H Knuth Game Theory 2009 By writing the joint entropy of the model M and the predicted measurement D e, 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!) Maximum Information Gain

Kevin H Knuth Game 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 Knuth Game Theory 2009 Initial Stage BLUE: Inference Engine generates samples from space of polygons / circles COPPER: 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 Knuth Game Theory 2009 After Several Black Measurements With several black measurements, the hypothesized shapes become smaller. Exploration is naturally focused on unexplored regions

Kevin H Knuth Game Theory 2009 After One White Measurement A positive result naturally focuses exploration around promising region

Kevin H Knuth Game Theory 2009 After Two White Measurements A second positive result naturally focuses exploration around the edges

Kevin H Knuth Game Theory 2009 After Many Measurements Edge exploration becomes more pronounced as data accumulates. This is all handled naturally by the entropy!

Kevin H Knuth Game Theory 2009 John Skilling Janos Aczél Ariel Caticha Keith Earle Philip Goyal Steve Gull Jeffrey Jewell Carlos Rodriguez Phil Erner Scott Frasso Rotem Gutman Nabin Malakar A.J. Mesiti Special Thanks to: