## What is mean field inference?

In the mean-field approximation (a common type of variational Bayes), we assume that the unknown variables can be partitioned so that each partition is independent of the others. The mean-field approximation partitions the unknown variables and assumes each partition is independent (a simplifying assumption).

### What is meant by mean field approximation?

The mean field approximation is a technique that can be used to calculate approximate partition functions for systems composed of interacting particles. The problem with calculating such partition functions exactly comes when one attempts to enumerate all the possible microstates and calculate their energy.

Why is it called variational inference?

The term variational is used because you pick the best q in Q — the term derives from the “calculus of variations,” which deals with optimization problems that pick the best function (in this case, a distribution q).

Where is variational inference used?

In modern machine learning, variational (Bayesian) inference, which we will refer to here as variational Bayes, is most often used to infer the conditional distribution over the latent variables given the observations (and parameters). This is also known as the posterior distribution over the latent variables.

## How is Elbo calculated?

Variational Inference: Computation of ELBO and CAVI algorithm

1. p(x,c,μ)=p(μ)∏ni=1p(ci)p(xi|ci,μ)
2. q(μ,c)=∏Kk=1q(μk;mk,s2k)∏ni=1q(ci;φi)
3. ELBO(m,s2,φ)=∑Kk=1E[logp(μk);mk,s2k]++∑ni=1(E[logp(ci);φi]+E[logp(xi|ci,μ);m,s2,φi])+−∑ni=1E[logq(ci;φi)]−∑Kk=1E[logq(μk;mk,s2k)]
4. φik∝exp{E[μk;mk,s2k]xi−E[μ2k;mk,s2k]/2}

### What is mean-field model?

In physics and probability theory, mean-field theory (aka MFT or rarely self-consistent field theory) studies the behavior of high-dimensional random (stochastic) models by studying a simpler model that approximates the original by averaging over degrees of freedom (the number of values in the final calculation of a …

What are mean field models?

Why do we need variational inference?

Variational Bayesian methods are primarily used for two purposes: To provide an analytical approximation to the posterior probability of the unobserved variables, in order to do statistical inference over these variables.

## What does ELBO mean?

In statistics, the evidence lower bound (ELBO, also variational lower bound or negative variational free energy) is a quantity which is often optimized in Variational Bayesian methods.

### What is mean by a field?

noun. an expanse of open or cleared ground, especially a piece of land suitable or used for pasture or tillage. Sports. a piece of ground devoted to sports or contests; playing field.

What is mean field control?

Mean-field game theory is the study of strategic decision making by small interacting agents in very large populations. A related concept to that of mean-field games is “mean-field-type control”. In this case a social planner controls a distribution of states and chooses a control strategy.

Where does the mean field approximation come from?

This assumption is known as the mean-field approximation. In practice, this methods originates from mean-field theory. From wikipedia: “Mean-field theory […] studies the behavior of high-dimensional random (stochastic) models by studying a simpler model that approximates the original by averaging over degrees of freedom.

## Why is mean field approximation used in variational inference?

Simply because for many cases, we cannot directly compute the posterior distribution, i.e. the posterior is on an intractable form — often involving integrals — which cannot be (easily) computed. This post focuses on the simplest approach to Variational Inference based on mean-field approximation.

### What do you need to know about mean field theory?

From wikipedia: “Mean-field theory […] studies the behavior of high-dimensional random (stochastic) models by studying a simpler model that approximates the original by averaging over degrees of freedom. […]

Why are combinatorial problems arise in mean field theory?

Often combinatorial problems arise that make things like computing the partition function of a system difficult. MFT is an approximation method that often makes the original solvable and open to calculation.