By Walter E Thirring

Mathematical Physics, Nat. Sciences, Physics, arithmetic

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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. [9] Georges A. , Phys. Rev B 43, 3475, 1991. [10] 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.