Variational Inference in Machine Learning

Variational inference in machine learning is a family of probabilistic inference techniques that approximates complex probability distributions with simpler ones. Instead of computing an intractable posterior distribution directly, methods optimize a tractable approximation—often by minimizing a divergence measure such as the Kullback–Leibler divergence. Variational inference is widely used in Bayesian statistics, including latent-variable models and probabilistic graphical models.

Overview

In many Bayesian models, the posterior distribution (p(z \mid x)) over latent variables (z) given observed data (x) is difficult to compute because it involves an intractable integral (the evidence). Variational inference replaces this posterior with a tractable family of distributions (q(z)), chosen so that optimization is feasible. The goal is to find the member of the family that is closest to the true posterior under a chosen divergence, typically by minimizing (\mathrm{KL}(q(z),|,p(z \mid x))).

A central construct in this approach is the evidence lower bound (ELBO), which provides an objective that can be optimized without requiring the true marginal likelihood. Maximizing the ELBO is equivalent to minimizing the KL divergence between the variational approximation and the posterior (up to an additive constant). This perspective connects variational inference to broader concepts such as Bayesian inference and statistical learning.

Variational formulation and ELBO

Let a probabilistic model define a joint distribution (p(x,z)) and a variational distribution (q(z)). The marginal log-likelihood can be decomposed as: [ \log p(x) = \mathrm{ELBO}(q) + \mathrm{KL}(q(z),|,p(z\mid x)). ] Because the KL term is nonnegative, the ELBO serves as a lower bound on (\log p(x)). In practice, variational inference optimizes parameters of (q(z)) by maximizing the ELBO with respect to those parameters.

In models with conjugacy, coordinate ascent updates can yield closed-form solutions for parts of (q(z)), as in classical mean-field approaches. Mean-field variational inference assumes a factorized form (q(z)=\prod_i q_i(z_i)), which simplifies optimization but can underestimate posterior uncertainty. These trade-offs are important when applying variational inference to models such as probabilistic graphical models and hierarchical Bayesian structures built from Bayesian networks or related representations.

Optimization methods and gradient estimators

When the variational family or model structure prevents analytic updates, variational inference becomes an optimization problem. The ELBO may be differentiated with respect to variational parameters using gradient-based methods. For continuous latent variables, the reparameterization trick can reduce variance of Monte Carlo gradient estimates by expressing samples from (q(z)) as deterministic transformations of noise.

For discrete latent variables or situations where reparameterization is not straightforward, alternative estimators such as score-function methods are used. These ideas are closely related to Monte Carlo methods, since many variational objectives are estimated using sampling. Stochastic optimization can also support large-scale datasets, turning variational inference into a scalable learning procedure rather than a purely batch algorithm.

Variational inference in latent-variable models

Variational inference is frequently applied to latent-variable models where the posterior over latent states is complicated. In latent variable models, variational inference provides an approximate posterior that enables learning of model parameters via an ELBO objective. A common workflow is: (1) define an approximate distribution (q(z)), (2) compute or estimate the ELBO, and (3) optimize both variational and model parameters.

A prominent example is the family of methods using variational inference to train generative models. In variational autoencoders, an encoder network parameterizes (q(z\mid x)), while a decoder network parameterizes (p(x\mid z)). The model is trained by maximizing an ELBO-like objective, linking variational inference to deep learning and modern neural architectures.

Variational inference can also be used to approximate posteriors in mixture models, topic models, and other structured probabilistic settings. However, performance depends on how expressive the variational family is and on the choice of divergence direction, since minimizing (\mathrm{KL}(q|p)) tends to favor approximations that cover regions where (q) assigns probability and may ignore modes of the true posterior.

Limitations and practical considerations

A key limitation of many variational inference methods is the potential for biased uncertainty due to the mean-field assumption or insufficient variational expressiveness. The direction of the KL divergence can lead to mode-seeking behavior: the approximation may concentrate on one mode of the posterior rather than representing multiple modes. This issue is a general concern in approximate inference and can be relevant for complex models and multimodal posteriors.

Another practical consideration is computational cost. Variational methods often require evaluating expectations under (q(z)), which can involve Monte Carlo estimation and can be sensitive to estimator variance. Careful design of the variational family and the gradient estimator—such as use of the stochastic gradient descent family of optimizers—can improve stability.

Despite these limitations, variational inference remains a foundational approach because it transforms inference into optimization. Its ability to scale and integrate with neural models has helped popularize related objectives, including ELBO-based training and approximate Bayesian learning.

See also

• Markov chain Monte Carlo
• Expectation–maximization algorithm
• Mean-field theory
• Probabilistic programming
• Variational autoencoder

Categories: Machine learning, Bayesian statistics, Probabilistic inference, Variational methods, Deep learning