By Manfred Opper, David Saad
An incredible challenge in glossy probabilistic modeling is the massive computational complexity concerned with general calculations with multivariate chance distributions while the variety of random variables is huge. simply because precise computations are infeasible in such instances and Monte Carlo sampling ideas could succeed in their limits, there's a want for tactics that permit for effective approximate computations. one of many least difficult approximations is predicated at the suggest box procedure, which has an extended historical past in statistical physics. the tactic is typical, really within the starting to be box of graphical models.Researchers from disciplines similar to statistical physics, desktop technological know-how, and mathematical records are learning how one can enhance this and comparable tools and are exploring novel program components. top ways contain the variational strategy, which works past factorizable distributions to accomplish systematic advancements; the faucet (Thouless-Anderson-Palmer) method, which contains correlations through together with powerful response phrases within the suggest box conception; and the extra common equipment of graphical models.Bringing jointly rules and methods from those diversified disciplines, this e-book covers the theoretical foundations of complicated suggest box tools, explores the relation among the several techniques, examines the standard of the approximation got, and demonstrates their software to varied components of probabilistic modeling.
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Additional resources for Advanced Mean Field Methods: Theory and Practice
Otherwise, however, we must settle for approximations. If we want to make a distinction between the exact marginal probabilities and approximate ones (something physicists do not usually bother doing explicitly), then we can call the approximation of the exact marginal probability Pi(Xi) the "belief" bi(Xi), and similarly we call the approximation of the exact two-node marginal probability Pij(Xi,Xj) the belief bij(Xi,Xj). The mathematical problem we will focus on for the rest of this chapter is as follows: given some arbitrary Markov network defined as in equation (1), compute as accurately as possible any desired beliefs.
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The Boltzmann-Gibbs distributions is widely used in physics, and mean field theory has been known for these distributions for a long time. For instance, for the Ising model on a square lattice, it is known as the Bragg-Williams approximation  and it is generalized to other models in the Landau theory . One can show that the above lower bound corresponds to the first term in a Taylor series expansion of the free energy around a factorized model. This Taylor series can be continued and the second order term is known as the Thouless Anderson Palmer ( TAP ) correction [16; 13; 6; 7].
Advanced Mean Field Methods: Theory and Practice by Manfred Opper, David Saad