By Manfred Opper, David Saad
An immense challenge in smooth probabilistic modeling is the large computational complexity thinking about normal calculations with multivariate likelihood distributions whilst the variety of random variables is big. simply because targeted computations are infeasible in such situations and Monte Carlo sampling concepts may perhaps achieve their limits, there's a desire for ways that let for effective approximate computations. one of many easiest approximations is predicated at the suggest box technique, which has an extended historical past in statistical physics. the strategy is everyday, fairly within the starting to be box of graphical models.Researchers from disciplines equivalent to statistical physics, machine technological know-how, and mathematical statistics are learning how one can increase this and similar tools and are exploring novel software components. prime ways contain the variational technique, which fits past factorizable distributions to accomplish systematic advancements; the faucet (Thouless-Anderson-Palmer) strategy, which includes correlations by means of together with powerful response phrases within the suggest box concept; and the extra basic equipment of graphical models.Bringing jointly rules and methods from those various disciplines, this publication covers the theoretical foundations of complex suggest box tools, explores the relation among the several methods, examines the standard of the approximation bought, and demonstrates their software to varied components of probabilistic modeling.
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Additional info for Advanced Mean Field Methods: Theory and Practice
This book. J. ,this book. J. ,Efficient Learning in Boltzmann Machines Using Linear Response Theory,Neural Computation 10,1137 (1998). [12lKabashima Y. ,Belief propagation vs. TAP for decoding corrupted messages, Europhys. Lett. 44, 668 (1998) [13lKabashima Y. ,this book. ,The Space of interactions in Neural Networks: Gardner's Computation with the Cavity Method, J. Phys. A (Math. Gen. 22,2181 (1989). [15lMezard M. ,Mean Field Theory of Randomly Frustrated Systems with Finite Connectivity, Europhys.
Of course, the suitability of these equations as an algorithm was not appreciated. Recently, Y. Kabashima and D. Saad [13; 14) have shown that for a number of other specific disordered models, the TAP approach and belief propagation give rise to identical equations, and speculated that this might be true in general. Freeman, Weiss and I have shown that this identity does in fact hold in gen eral [29). To prove it for general Markov networks, you simply need to identify the following relationship between the Lagrange multipliers Aij(Xj) that we introduced in the last section and the messages Mij(xj) : Aij(Xj) = TIn II Mkj(xj) kEN(j)\i (42) Using this relation, one can easily show that equations (36) and (37) derived for the Bethe approximation in the last section are equivalent to the belief propagation equations (38) and (39).
T. , 7th International Conference Computer Vision, 1182, 1999. J. Graphical Models for Machine Learning and Digital Communication, Cambridge: MIT Press, 1998. , Phys. Rev. Lett. 64, 2937, 1990.  Georges A. , Phys. Rev B 43, 3475, 1991.  Georges A. , J. Phys. A 24, 2173, 1991. , Learning in Graphical Models, Cambridge: MIT Press, 1998. , Cambridge: MIT Press, 1998. , Europhys. Lett. 44, 668, 1998. , Contribution to this volume, 2000. , Phys. Rev. 81, 988, 1951. , Special issue in honor of R.