Study notes for Statistical Physics - Bookboon

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We wish to minimize this quantity with respect to. By definition of a conditional distribution,. In mean-field theory, the mean field appearing in the single-site problem is a scalar or vectorial time-independent quantity. However, this need not always be the case: in a variant of mean-field theory called dynamical mean-field theory (DMFT), the mean field becomes a time-dependent quantity. Mean Field and Variational Methods finishing off Graphical Models – 10708 Carlos Guestrin Carnegie Mellon University November 5th, 2008 Readings: K&F: 10.1, 10.5 10-708 – ©Carlos Guestrin 2006-2008 10-708 – ©Carlos Guestrin 2006-2008 2 Geometry of Mean Field 39 •Mean field optimization is always non-convex for any exponential family in which the state space is finite •Marginal polytope is a convex hull • contains all the extreme points (if it is a strict subset then it must be non-convex •Example: two-node ising •Parabolic cross section along τ 1= τ 2 Semiparametric Mean Field Variational Bayes where p(DDD;q;˘) is the marginal likelihood lower bound de ned by (4), but with the depen-dence on ˘re ected in the notation. An early contribution of this type is Hinton and van Camp (1993) who used minimum Kullback-Leibler divergence for Gaussian approximation of posterior density functions in Mean-Field Approximation.

Mean field variational

And, the difference between the ELBO and the KL divergence is the log normalizer— which is what the ELBO bounds. 6 Mean field variational inference. • In mean Streamlined mean field variational Bayes algorithms for efficient fitting and inference in large models for longitudinal and multilevel data analysis are obtained. Nov 6, 2020 systems literature to study the convergence of coordinate ascent algorithms for mean field variational inference.

We will also see mean-field approximation in details.

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zVariational calculus … zDo inference in each qi(xc) using any tractable algorithm Abstract and Figures We develop strategies for mean field variational Bayes approximate inference for Bayesian hierarchical models containing elaborate distributions. We loosely define elaborate Mean Field Solution of Ising Model Now that we understand the variational principle and the non-interacting Ising Model, we're ready to accomplish our next task.

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Based on rigorousresultsof[1, 2], we Mean Field Variational Inference Mean field variational inference algorithms were originally explored in statistical physics. In these methods, we build an approximation of the UGM using a simpler UGM where marginals are easy to compute, but we try to optimize the parameters of the simpler UGM to minimize the Kullback-Leibler divergence from the full UGM. We investigate mean field variational approximate Bayesian inference for models that use continuous distributions, Horseshoe, Negative-Exponential-Gamma and Generalized Double Pareto, for sparse signal shrinkage.

Despite the popularity of the mean field method there exist remarkably little fundamental theoretical variational problems relevant for MFG are described via Eulerian and Lagrangian languages, and the connection with equilibria is explained by means of convex duality and of optimality conditions. The convex structure of the problem also allows for e cient numerical treatment, based on Augmented Mean Field Solution of Ising Model Now that we understand the variational principle and the non-interacting Ising Model, we're ready to accomplish our next task. We want to understand the general d-dimensional Ising Model with spin-spin interactions by applying the non-interacting Ising Model as a variational ansatz. Mean-field refers to the fact that we assume all the latent variables to be independent.
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A common approach is to use the coordinate ascent method, by optimizing the variational approximation of each latent variable $q_{z_j}$, while holding the others fixed. Mean-field variational Bayes (the most common type) uses the Reverse KL Divergence to as the distance metric between two distributions. Reverse KL divergence measures the amount of information (in nats, or units of bits) required to "distort" into.

We 23 The Mean Field Variational Family This is where the suppressed variables are from INFO TECH 4107 at Mount Kenya University Exercise - Variational Mean Field Approximation for Univariate Gaussian by Christian Herta is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License . Continuous-time Bayesian networks is a natural structured representation language for multi-component stochastic processes that evolve continuously over time. Despite the compact representation, inference in such models is intractable even in relatively simple structured networks.
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Mean Field Variational Bayes for Elaborate Distributions Matthew P. Wand , John T. Ormerody, Simone A. Padoanzand Rudolf Fr uhwirthx Abstract. We develop strategies for mean eld variational Bayes approximate inference for Bayesian hierarchical models containing elaborate distributions.