By Manfred Opper, David Saad
An incredible challenge in smooth probabilistic modeling is the massive computational complexity fascinated by ordinary calculations with multivariate likelihood distributions whilst the variety of random variables is huge. simply because designated computations are infeasible in such situations and Monte Carlo sampling options might succeed in their limits, there's a desire for ways that permit for effective approximate computations. one of many easiest approximations is predicated at the suggest box procedure, which has a protracted background in statistical physics. the strategy is usual, rather within the becoming box of graphical models.Researchers from disciplines comparable to statistical physics, machine technological know-how, and mathematical statistics are learning how you can enhance this and similar tools and are exploring novel software components. major ways contain the variational strategy, which fits past factorizable distributions to accomplish systematic advancements; the faucet (Thouless-Anderson-Palmer) technique, which contains correlations by means of together with powerful response phrases within the suggest box conception; and the extra common equipment of graphical models.Bringing jointly rules and strategies from those varied disciplines, this publication covers the theoretical foundations of complex suggest box equipment, explores the relation among the various methods, examines the standard of the approximation acquired, and demonstrates their program to numerous components of probabilistic modeling.
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Additional info for Advanced Mean Field Methods: Theory and Practice
Assume that we have some system which can be in, say, K different states. The probability of each state is some number Pa where L �=l Pa = 1. Let there be some quantity Xa ( like the energy) which depends on which state the system is in, and introduce the notation for the mean value K (X ) == L PaXa· a=l (8) Then by the convexity of the exponential function, we can prove that (9) Now consider the partition function Z= L exp (-Ea/ T). (10) Let us introduce some arbitrary "trial" energy function E�.
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. T he mean field approximation is defined as the factorized distribution that is closest to the target distribution. When using the KL divergence to define closeness, this factorized distribution must have equal marginals as the target distribution.
26lSherrington D. ,Phys. Rev. Lett. 35,1792 (1975). ohio-state. edu/ - talagran/. ,this book. W. ,Solution of a 'Solvable Model of a Spin Glass',Phil. Mag. 35,593 (1977). ,this book. ,Li S. ,this book. ,this book. , Quantum Field Theory and Critical Phenomena, Clarendon Press, Oxford (1989). 3 An Idiosyncratic Journey Beyond Mean Field Theory Jonathan S. Yedidia The connecting thread between the different methods described here is the Gibbs free energy. After introducing the inference problem we are interested in analyzing, I will define the Gibbs free energy, and describe how to derive a mean field approximation to it using a vari ational approach.
Advanced Mean Field Methods: Theory and Practice by Manfred Opper, David Saad