" A wise man considers which side is
supported by the greatest number of experiments: To that side he
inclines, with doubt and hesitation; and when at last he fixes his
judgment, the evidence exceeds not what we properly call probability". David Hume,
Philosophical Essays concerning Human Understanding, 1748, A. Millar,
London.
Socrates
and Aristotle were among the first to explore formalized ways of
thinking.
Plato's dialogue Theaetetus examines methods of
thinking. In the
next two thousand years, philosophers became increasingly confident in
their
method for pursuing knowledge. Whitehead and Russel's
Principia
Mathematica appeared in 1910, when
logic and
the foundations of mathematics were seen to provide a firm foundation
for all
reasoning. Analytical philosophy developed this strand of thinking
further. It
has been defining for Anglo-Saxon thinking ever since. Its emphasis on
the
analysis of language and the classes used in language provide a
linguistic
solution for dealing with uncertainty. Wittgenstein wrote On
Certainty, and there are philosophical writings on Vagueness
(Keefe, Williamson). Logic itself has tried to incorporate
reasoning under uncertainty. It has not been successful at this.
Inductive
reasoning, for example, is not well analyzed in logic.
Statistics developed as an aide to gambling. With counting of
combinations of
events, it led to the analysis of frequencies. This was the first
framework for
dealing with uncertainty. It is useful for the analysis of experiments,
where
a-priori knowledge is a bias that has to be avoided. I have always
relied on Probability,
Random
Variables, and Stochastic Processes by Athanansios
Papoulis. Introductory
Statistics
by Sheldon M. Ross and Introduction
to
Probability Models by
Sheldon M. Ross are also excellent references. These books go beyond
basic
statistics, of course, and are grounded in the modern theory of
probabilities.
Strategy is only one aspect of playing a game. Equally important is the
uncertainty about what your opponent will do. The theory of games also
developed out of gambling and games of chance. It has developed a
framework for
decision making under uncertainty. My reference for this is Game Theory by D Fudenberg
and J Tirole,
complemented by The Theory of
Learning in
Games by D Fudenberg and D Levine. Learning and adaptation
are essential
for
dealing with uncertainty.
Equilibria in games are related to the equilibria in micro-economic systems.
Microeconomics deals
well with the use of utilities to model preferences of agents. You will
find a
good review of choice, utility, and games in one of the standard
microeconomics
texts, Microeconomic
Theory by Andreu Mas-Colell,
Michael D. Whinston,
and Jerry R. Green. Utilities are overly deterministic for dealing with
human
preferences. George Shackle's work and Prospect Theory (Kahneman
and Tversky) have tried to correct this,
and neuroeconomics takes this further, see
below.
If you extend the complexity of economic agents beyond classical
economy, you
end up simulating mutli-agent systems of
economic
decision makers. This has led to the field of agent-based computational
economics, ACE. Leigh Tesfatsion is a
major
contributor, and her webpage is full of
useful info. ACE also leads to agent
ecosystems. Ecosystems with an uncertain fitness function are a
challenging
research topic.
Linguistic classes are a major tool
humans use to
deal with uncertainty. Linguistics will not help you much with a
computational
implementation of classes, but there is a vast
literature on artificial classifiers. Pattern
Classification by Richard O. Duda,
Peter E. Hart, and David G. Stork is a reference by authors who have
been
working in this field for a long time. Pattern
Recognition and
Machine Learning by
Christopher M. Bishop is up-to-date, and Information
Theory,
Inference and Learning Algorithms by David J. C. MacKay makes the
necessary link with information
theory. Much of this field used to be called neural networks, but this
is a
misnomer. Modern pattern classification has nothing to do with the
structure of
the brain. For a wider view on classifiers in artificial intelligence,
see Machine
Learning by
Tom M. Mitchell. Support vector machines as
especially useful classifiers, and Learning with
Kernels by
B Scholkopf is a
clearly written introduction.
Bayesian inference, Markov decision processes, and
reinforcement learning all have an intuitive appeal, and link to
cognitive
science. Machine
Learning by
Tom M. Mitchell is a good introduction.
Most books on pattern classification take a Bayesian approach to
statistical
inference. Most of the Bayesian practitioners are outright hostile to
fuzzy
logic. This should not be: fuzzy logic is an alternative approach, and
was the
first to take a distance from modelling
uncertainty
using probability distributions. It is well founded, and Fuzzy Logic
with
Engineering Applications
by Timothy J. Ross is a good introduction, with many practical
examples. A more
fundamental work is Fundamentals
of Fuzzy Sets
by Didier Dubois and Henri Prade.
Dempster-Shafer
theory also fits into this programme
of relaxing
constraints on probability measures.
The notion of probability can be expanded without relying on fuzzy
logic. If a
probability is dependent on another random variable, it is just a
conditional
probability, of course. Savage, and now Jeffries have worked on
subjective
probabilities, see Subjective
Probability
by Richard Jeffrey. Indeterminate probabilities
(Nau), unreliable probabilities (Gardenfors),
and imprecise probabilities (Walley) have
all been
defined and explored. Recently fuzzy logic has also introduced
uncertainty of
membership functions, type-2 fuzzy logic.
Psychology has spawned cognitive psychology, cognitive science, and
cognitive
neuroscience. Decision making under uncertainty is an active research
topic in
these fields. They are fast absorbing and extending behavioural
economics and Kahneman and Tversky's
Prospect Theory. Brain imaging is providing data on decision making
under
uncertainty. A Bayesian interpretation is possible, see Bayesian
Brain:
Probabilistic Approaches to Neural Coding by K Doya.
We need to
understand better which part of Bayesian decision making is learned,
and which
part is hardwired in the brain. Neuroeconomics: Decision
Making and the
Brain by
Paul W. Glimcher, Colin Camerer, Russell
Alan Poldrack, and Ernst Fehr will give a
fair outline
of what we know, and what we don't know.
I try to add two approaches to this varied and extensive list of
attempts to
formalize dealing with uncertainty as a computational process. First is
a
theory of fuzzy
games. This combines the
achievements of
game theory with fuzzy utilities and uncertainty about the moves your
opponents
are making. It is complementary to Bayesian equilibria
in games.
Second, is a pico-economic approach to uncertainty. This takes into account the interconnected
networks of neurons, astrocytes and blood
vessels in
the brain. Classification, generalization, and inductive reasoning
are
emergent properties of neurons, astrocytes
and blood
vessels. This research takes into account processes at micron scales
that are
not visible to current neuroimaging. It's
the detail
that neuroeconomics is missing.
I have discussed these issues in the Vloesberghs lectures in Brussels in 2010.