Download Advanced Mean Field Methods: Theory and Practice by Manfred Opper, David Saad PDF

By Manfred Opper, David Saad

A huge challenge in sleek probabilistic modeling is the large computational complexity fascinated about commonplace calculations with multivariate likelihood distributions while the variety of random variables is huge. simply because specified computations are infeasible in such instances and Monte Carlo sampling strategies could succeed in their limits, there's a want for ways that permit for effective approximate computations. one of many easiest approximations relies at the suggest box approach, which has a protracted heritage in statistical physics. the tactic is prevalent, really within the becoming box of graphical models.Researchers from disciplines comparable to statistical physics, computing device technological know-how, and mathematical facts are learning how you can increase this and comparable equipment and are exploring novel program components. best techniques comprise the variational method, which matches past factorizable distributions to accomplish systematic advancements; the faucet (Thouless-Anderson-Palmer) process, which contains correlations by means of together with potent response phrases within the suggest box idea; and the extra basic equipment of graphical models.Bringing jointly rules and strategies from those assorted disciplines, this e-book covers the theoretical foundations of complex suggest box equipment, explores the relation among the various techniques, examines the standard of the approximation received, and demonstrates their software to numerous components of probabilistic modeling.

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An important advantage of this procedure is that it does not require the concept of a free energy. T herefore, it can be applied to arbitrary probability distributions, such as arising in asymmetric stochastic neural networks and graphical models. 1 Introduction During the last few years, the use of probabilistic methods in artificial intelligence and machine learning has gained enormous popularity. In particular, probabilis­ tic graphical models have become the preferred method for knowledge represen­ tation and reasoning [4].

For each a, ¢o:( xO:) is a probability table, ie. a table of numbers indexed by O:x . 2 Thus, ¢ can be thought of as a vector of probability tables. This manifold contains values of ¢ for which p( l x¢) is factorized. Let ¢q be such a value, then = = (14) 0: q( x) denotes this factorized distribution. We can therefore use the mean field approach outlined in section 2 where the coordinates of the manifold ( (}, w) are given by {¢o:( xO:), a 1, ... ,m } and the q factorized point ( (} , O) is given by log q( x).

J. Phys. A 15, 1971, 1982. [25] Sherrington D. , Phys. Rev. Lett. 35, 1792, 1975. W. , Phil. Mag. 35, 593, 1977. , Bayesian Belief Propagation for Image Understanding, available at homepage, 1999. , 1992 Lectures in Complex Systems, L. Nadel and D. , Addison-Wesley, 299, 1993. T. com/reports/TR2000-26/ . 2000. S. , J. Phys. A 23, 2165, 1990. , Spin Glasses and Random Fields, World Scientific, ed. 1998. Yair Weiss's 4 Mean Field Theory for Graphical Models Hilbert J. Kappen and Wim J. Wiegerinck In this chapter, mean field theory is introduced from an information theoretic view point.

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