By Manfred Opper, David Saad

ISBN-10: 0262150549

ISBN-13: 9780262150545

A massive challenge in sleek probabilistic modeling is the massive computational complexity inquisitive about general calculations with multivariate chance distributions whilst the variety of random variables is big. simply because distinct computations are infeasible in such instances and Monte Carlo sampling concepts may possibly succeed in their limits, there's a desire for tactics that let for effective approximate computations. one of many easiest approximations relies at the suggest box technique, which has an extended historical past in statistical physics. the strategy is well-known, really within the starting to be box of graphical models.Researchers from disciplines resembling statistical physics, desktop technological know-how, and mathematical data are learning how you can enhance this and comparable tools and are exploring novel program components. major ways contain the variational technique, which fits past factorizable distributions to accomplish systematic advancements; the faucet (Thouless-Anderson-Palmer) strategy, which contains correlations through together with potent response phrases within the suggest box thought; and the extra normal tools of graphical models.Bringing jointly rules and methods from those varied 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 acquired, and demonstrates their software to numerous components of probabilistic modeling.

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**Additional info for Advanced Mean Field Methods: Theory and Practice**

**Sample text**

There are in fact many different approximations that one could make to a Gibbs free energy, and much of the rest of this chapter is devoted to surveying them. Mean Field Theory: The Variational Approach 5 One very popular way to construct an approximate Gibbs free energy involves a variational argument. The derivation given here will be from a physicist's perspec tive; for an introduction to variational methods from a different point of view, see [12]. Assume that we have some system which can be in, say, K different states.

We minimize F(Q) E[Q] - S[Q] in two steps: In the first step, we perform a constrained minimization in the family of all distributions Qm which satisfy = = (S)Q m (46) = wherem is fixed. We define the Gibb '8 Free Energy as the constrained minimum G(m) = min{E[Q] - S[Q] I (S)Q Q m} . (47) = In the second step, we minimize G with respect to the vector m. Since the full minimizer of F[Q] equals the true distribution P, the minimizer of G(m) coincides with the vector of true expectations (Si) . Constrained optimization problems like (47) can be transformed into uncon strained ones by introducing appropriate Lagrange multipliers hi where we have to minimize (48) and the hi's must be chosen such that (46) holds.

This is also the case in the presence of evidence. When evidence is present, we separate the total set of variables in evidence variables e and the remaining hidden variables s: x (s, e). The conditional distribution becomes = (22) When couplings from hidden nodes to hidden or evidence nodes are zero, Wij 0, all i j E s, the conditional probability reduces to a factorized distribution = p(sle) -+ q(s) = II a (si}{) (23) In this case, the " evidence" factorizes as well: (24) As outlined in section 4, we define the mean field approximation to p(sle) as the factorized distribution q(s) that has identical marginals: p(sile) q(Si).

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

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