The Foundations of Inference

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In October 2009, Kevin Knuth (President, Autonomous Exploration, Inc.) was invited to present a talk titled “Foundations of Inference” at the Perimeter Institute in Waterloo, CANADA.  This theoretical talk discussed the mathematical foundation of inference and its relation to order theory and quantum mechanics.  Due to the Quantum to Cosmos Festival which ran at the same time, this talk was not videotaped.

An abstract follows below:

Foundations of Inference

Kevin H. Knuth
Associate Professor of Physics and Informatics, University at Albany
President, Autonomous Exploration, Inc.

In our mutual desire to develop an intuitive understanding of quantum mechanics, John Skilling and I have been working to clarify the foundations of inference.

In this talk I introduce lattice theory as a new foundation for a rational description of the world around us.  I will show that fundamental mathematical fields such as measure theory, probability theory and information theory are based on basic symmetries of the lattice and can be easily derived.

Our rational description relies on a partially ordered set (poset) of states.  The process of expansion (lattice exponentiation) generates a distributive lattice from a poset.  I show that successive expansions applied to the poset of states results in a lattice of all the possible statements that can be made about the system, and a lattice of all the possible questions that can be asked of a system.

Quantification is performed by introducing real-valued functions called valuations.  The symmetries of the lattice lead to a unique calculus that obeys a summation property, which is the basis of addition as the fundamental operation in measure theory.  Introduction of the notion of context via functions called bi-valuations leads to a product rule for context-based measures.  On the lattice of statements, this unique calculus is the probability calculus with its sum and product rule.  An additional constraint relating the relevance of a question to the probabilities of the statements that answer it leads to entropy as the basic measure of a question.  The unique calculus on the lattice of questions is a new context-based generalization of information theory.

I will finish by introducing a new picture of quantum mechanics where experimental states are partially ordered to form a poset.  The valuations on this poset are the quantum amplitudes with Feynman path integrals representing the summation property.  Expansion of the state poset to a lattice of statements leads to probabilities whose values depend on the quantum amplitudes.  In this way probability theory naturally enters the quantum picture as inference.

Automating the Processes of Inference and Inquiry

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In September 2009, Kevin Knuth (President, Autonomous Exploration, Inc.) was invited to present a talk titled “Automating the Processes of Inference and Inquiry” at the From Game Theory to Game Engineering Workshop, Oxford-Man Institute of Quantitative Finance, Oxford, UK.  This talk discussed the foundation of automated inference and inquiry and its application to game theory.

The talk involved a demonstration of an autonomous robot that designs its own experiments using Bayesian Adaptive Exploration.

An abstract follows below:

Automating the Processes of Inference and Inquiry

Kevin H. Knuth
Associate Professor of Physics and Informatics, University at Albany
President, Autonomous Exploration, Inc.

Most decision making processes must be performed in the face of uncertain or incomplete information.  It is critical to be able to make the best inferences one can under these situations, as well as to ask for additional information in an intelligent manner.  In this talk, I will introduce a unified framework for automated inference and inquiry based on Bayesian probability theory and Bayesian Adaptive Exploration.

I will begin by considering a system described by a finite set of possible states, and from this I will derive the rules of rational inference by considering the lattice of possible statements that can be made about the system.  These rules result in Bayesian probability theory, but seen from a new perspective.  I will then introduce Bayesian Adaptive Exploration, which can be used to decide which new data to collect given what is currently known.

I will demonstrate the theory with a Bayesian machine (robot) that locates and characterizes shapes on a playing field.  The intelligent aspects of this machine rely on two systems: an inference engine and an inquiry engine. The inference engine simultaneously performs Bayesian model estimation and Bayesian parameter estimation by relying on the nested sampling algorithm. Meanwhile the inquiry engine performs experimental design by identifying the most relevant measurement given the current state of knowledge of the machine.  This is accomplished by quantifying the entropy of the possible experimental outcomes. This research represents a real-time implementation of a unified framework of inference and inquiry.

Autonomous Sensor Placement

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In November 2008, Kevin Knuth (President, Autonomous Exploration, Inc.) and Julian Center (CEO, Autonomous Exploration, Inc.) presented a talk titled “Autonomous Sensor Placement” at the 2008 IEEE International Conference on Technologies for Practical Robot Applications in Woburn MA.  In addition to the talk, a demonstration of a robot employing autonomous sensor placement was performed.

An abstract follows below:

Autonomous Sensor Placement

Kevin H. Knuth
Associate Professor of Physics and Informatics, University at Albany
President, Autonomous Exploration, Inc.

Julian Center
CEO, Autonomous Exploration, Inc.

With an increasing reliance on robotic platforms to perform scientific exploration in remote or hostile environments, it is becoming crucial that robotic systems be able to perform autonomous intelligent sensor placement as well as autonomous experimental design.  Such a system requires encoding of scientific knowledge, the ability to make inferences from data, and the ability to identify the most relevant question to ask given both the instrument’s prior knowledge and the issue it is designed to address.  This requires implementation of two computational engines: the inference engine and the inquiry engine. Here we demonstrate our first efforts to develop intelligent instruments that rely on autonomous sensor placement.

Automating the Scientific Method

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In February 2008, Kevin Knuth (President, Autonomous Exploration, Inc.) was invited to present a talk titled “Automating the Scientific Method” at the New York Workshop on Computer, Earth, and Space Sciences (CESS 2009) held at the NASA Goddard Institute for Space Studies in New York City NY.  This talk discussed the computational foundation of automated inference and inquiry.  An abstract follows below:

Automating the Scientific Method

Kevin H. Knuth
Associate Professor of Physics and Informatics, University at Albany
President, Autonomous Exploration, Inc.


In the last decade we have seen computer science, statistics, and the earth,
space, life and social sciences come together with a new synergy based on the
common goal of data analysis. These multi-disciplinary interactions have
become necessary as we pursue both high quality data analysis as well as
analysis of extremely large data sets. However, the ultimate goal is more
fundamental than mere data analysis. We aim to automate the scientific method
itself.

The scientific method relies on the cyclic application of three activities:
hypothesis generation, inquiry (experimental design) and inference
(data analysis). The majority of our efforts at this point have been focused
on the process of automating inference. However, little attention has been
paid to automating the processes of inquiry and hypothesis generation.

The most scientifically-useful approach to data analysis is model-based.
I will briefly review the methodology behind automating model-based inference
with a focus on Bayesian probability theory. I will then introduce a new
related methodology called the inquiry calculus, which enables the automation
of model-based inquiry. Automated hypothesis (model) generation will be left
for another day, as it is the least advanced of the three technologies. I will
demonstrate the application of automated inference and inquiry with a robotic
scientist that performs its own experiments and analyzes its own data.