Inference where we approximate the density of the posterior variationally. That is, we use cunning tricks to turn solve an inference problem by optimising over some parameter set, usually one that allows us to trade off difficulty for fidelity in some useful way.

This idea is not intrinsically Bayesian (i.e. the density we are approximating need not be a posterior density or the marginal likelihood of the evidence), but much of the hot literature on it is from Bayesians doing probabilistic deep learning, so for concreteness I will assume Bayesian uses here.

This is usually mentioned in contrast from the other main method of approximating such densities: sampling from them, usually using Markov Chain Monte Carlo. In practice the two are related (Salimans, Kingma, and Welling 2015) and nowadays frequently used together. (Rezende and Mohamed 2015; Caterini, Doucet, and Sejdinovic 2018)

See also mixture models, probabilistic deep learning, directed graphical models, reparameterization tricks.

## Introduction

The classic intro seems to be (Jordan et al. 1999), which considers diverse types of variational calculus applications and inference. Typical ML uses these days are more specific; an archetypal example would be the variational auto-encoder (Kingma and Welling 2014).

## Inference via KL divergence

Practically we often want a variational approximation on the marginal (log-)likelihood \(\log p_{\theta}(\mathbf{x})\) for some probabilistic model with observations \(\mathbf{x},\) unobserved latent factors \(\mathbf{x}\) and model parameters \(\mathbb{\theta}.\)

\[\begin{aligned} \log p_{\theta}(\mathbf{x}) &=\log \int p_{\theta}(\mathbf{x} | \mathbf{z}) p(\mathbf{z}) d \mathbf{z} \\ &=\log \int \frac{q_{\phi}(\mathbf{z} | \mathbf{x})}{q_{\phi}(\mathbf{z} | \mathbf{x})} p_{\theta}(\mathbf{x}|\mathbf{z}|) p(\mathbf{z}) d \mathbf{z} \\ &\geq-\mathbb{D}_{KL}\left[q_{\phi}(\mathbf{z} | \mathbf{x}) \| p(\mathbf{z})\right]+\mathbb{E}_{q}\left[\log p_{\theta}(\mathbf{x} | \mathbf{z})\right]\\ &=-\mathcal{F}(\mathbf{x}) \end{aligned}\]

\(\mathcal{F}\) is called the free energy.

## Mixture models

Mixture models are classic and for ages, seemed to be the default choice for variational approximation. I do not have much use for these.

## Reparameterization trick

See reparameterisation.

## Autoencoders

## Loss functions

In which probability metric should one approximate the target density? For tradition and convenience, we usually use KL-loss, but this is not ideal, and alternatives are hot topics.

Ingmar Schuster’s critique of black box loss raises some issues (Ranganath et al. 2016):

It’s called Operator VI as a fancy way to say that one is flexible in constructing how exactly the objective function uses \(\pi, q\) and test functions from some family \(\mathcal{F}\). I completely agree with the motivation: KL-Divergence in the form \(\int q(x) \log \frac{q(x)}{\pi(x)} \mathrm{d}x\) indeed underestimates the variance of \(\pi\) and approximates only one mode. Using KL the other way around, \(\int \pi(x) \log \frac{pi(x)}{q(x)} \mathrm{d}x\) takes all modes into account, but still tends to underestimate variance.

…the authors suggest an objective using what they call the Langevin-Stein Operator which does not make use of the proposal density \(q\) at all but uses test functions exclusively.

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