  This paper presents a novel way to approximate a distribution governing a
system of coupled particles with a product of independent distributions. The
approach is an extension of mean field theory that allows the independent
distributions to live in a different space from the system, and thereby capture
statistical dependencies in that system. It also allows different Hamiltonians
for each independent distribution, to facilitate Monte Carlo estimation of
those distributions. The approach leads to a novel energy-minimization
algorithm in which each coordinate Monte Carlo estimates an associated
spectrum, and then independently sets its state by sampling a Boltzmann
distribution across that spectrum. It can also be used for high-dimensional
numerical integration, (constrained) combinatorial optimization, and adaptive
distributed control. This approach also provides a simple, physics-based
derivation of the powerful approximate energy-minimization algorithms
semi-formally derived in \cite{wowh00, wotu02c, wolp03a}. In addition it
suggests many improvements to those algorithms, and motivates a new (bounded
rationality) game theory equilibrium concept.
