# Mean Field / Variational Approximations Presented by Jose Nuñez 10/24/05 Outline • Introduction • Mean Field Approximation • Structured Mean Field • Weighted Mean Field • Variational Methods

Abstract. We consider the variational structure of a time-fractional second-order mean field games (MFG) system. The MFG system consists of time-fractional Fokker–Planck and Hamilton–Jacobi–Bellman equations.

Mean Field Variational Approximation for Continuous-Time Bayesian Networks∗ Ido Cohn† IDO COHN@CS.HUJI.AC.IL Tal El-Hay† TALE@CS.HUJI.AC.IL Nir Friedman NIR@CS.HUJI.AC.IL School of Computer Science and Engineering The Hebrew University Jerusalem 91904, Israel Raz Kupferman RAZ@MATH.HUJI.AC.IL Institute of Mathematics The Hebrew University In lots of Bayesian papers, people use variational approximation. In lots of them they call it "mean-field variational approximation". Does anyone know what is the meaning of mean-field in this co The mean field methods, which entail approximating intractable probability distributions variationally with distributions from a tractable family, enjoy high efficiency, guaranteed convergence, and provide lower bounds on the true likelihood. But due to requirement for model-specific derivation of the optimization equations and unclear inference quality in various models, it is not widely used 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. Optimizing the ELBO in Mean Field Variational Inference 27 •How do we optimize the ELBO in mean field variationalinference?

This means that we can easily factorize the variational distributions into groups: Mean Field Variational Approximation for Continuous-Time Bayesian Networks Ido Cohn Tal El-Hay Nir Friedman School of Computer Science The Hebrew University fido cohn,tale,nirg@cs.huji.ac.il Raz Kupferman Institute of Mathematics The Hebrew University raz@math.huji.ac.il Abstract Continuous-time Bayesian networks is a natu- 2012-10-19 Optimizing the ELBO in Mean Field Variational Inference 27 •How do we optimize the ELBO in mean field variationalinference? •Typically, use coordinate ascent •We optimize each latent variable’s variationalapproximation q in turn while holding the others fixed. •At each iteration we get an updated local variationalapproximation tributed. The notation x ˘N( ; ) means that x has a Multivariate Normal density with mean and covariance .

Mapping from X to Y involving vector parameter A. Gaussian prior and fully factorized Gaussian posterior on A. Deterministic The consistency problem of both mean field and variational Bayes estimators in the context of linear state space models is investigated. We prove that the.

## 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 Variational Methods. Brown University CS 242: Probabilistic Graphical Models. Homework due at 16 Sep 2020 Seemingly unrelated regression with measurement error: estimation via Markov Chain Monte Carlo and mean field variational Bayes 13 May 2020 Topical Review.

### 2 A Variational mean-field theory Plefka,[2] proposed a mean-field theory in the context of spin glasses. This theory can, in principle, yield arbitrarily close approximation to log Z. In this section we present an alternate derivation from a variational viewpoint, see also [4],[5]. Let 'Y be a real parameter that takes values from 0 to 1.

•Variational means: optimization-based formulation •Represent a quantity of interest as the solution to an optimization problem • Approximate the desired solution by relaxing/approximating the intractable In this review we focus on the mean-field variational family, where the latent variables are mutually independent and each governed by a distinct factor in the variational density.

This paper is a brief presentation
Apr 28, 2013 Coordinate Ascent on the Mean Field Approximation is the "traditional" way one does Variational Inference, but Coordinate Ascent is far from
Dec 11, 2014 Explaining the Basics of Mean Field Variational Approximation for Statisticians. 1. Explaining “Explaining Variational Approximation” Based on
Presentation on theme: "Mean field approximation for CRF inference"— 6 th Outline Approximate Inference Variational inference formulation – Mean Field
Its iterative Coordinate.

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However, the values of the variational parameters can oscillate if they are strongly coupled by the posterior distribution. The resulting slow convergence is often not obvious from monitoring the ELBO. Mean-field Variational Bayes is an iterative maximization of the ELBO. More precisely, it is an iterative M-step with respect to the variational factors \(q_i(\mathbf{Z}_i)\). In the simplest case, we posit a variational factor over every latent variable, as well as every parameter.

The ill-posed nature of missing variable models offers a challenging testing ground for new computational techniques. This is the case for the mean-field variational Bayesian inference (Jaakkola, 2001; Beal and Ghahramani, 2002; Beal, 2003). In this note, we illustrate the behavior of this approach in the setting of the Bayesian probit model.

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### av Ö Bäck — Diva error field for dissolved inorganic nitrogen in the OSPAR maritime area. ISSN: 0283-1112 © The Diva software (Data Interpolating Variational Analysis) software was used to create interpolated Mean of data Mean of data. 1 Gebco

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.

## Graphical Models Variational inference III: Mean-Field Siamak Ravanbakhsh Winter 2018

Ascent Variational Inference algorithm has been widely applied to large scale Bayesian inference. See Blei et al. (2017) for a recent com- . This process is known as variational inference. 9.2.1 The KL Divergence: Measuring the Closeness of Probability Distributions. Assume we have two probability Ever since variational inference was introduced for Bayesian neural networks, researchers have assumed that the 'mean-field' approximation—that the posterior Mean Field Method and “Variational”: fancy name for optimization-based formulations.

Energy-based optimization methods for excited states, like Δ-SCF (SCF), tend to fall into the lowest soln. consistent with a given symmetry - a problem known as "variational collapse.".