Evidence lower bound

Part of a series on
Bayesian statistics
Posterior = Likelihood × Prior ÷ Evidence
Background
Bayesian inference Bayesian probability Bayes' theorem Bernstein–von Mises theorem Coherence Cox's theorem Cromwell's rule Likelihood principle Principle of indifference Principle of maximum entropy
Model building
Conjugate prior Linear regression Empirical Bayes Hierarchical model
Posterior approximation
Markov chain Monte Carlo Laplace's approximation Integrated nested Laplace approximations Variational inference Approximate Bayesian computation
Estimators
Bayesian estimator Credible interval Maximum a posteriori estimation
Evidence approximation
Evidence lower bound Nested sampling
Model evaluation
Bayes factor Model averaging Posterior predictive
Mathematics portal
v t e

In variational Bayesian methods, the evidence lower bound (often abbreviated ELBO, also sometimes called the variational lower bound^[1] or negative variational free energy) is a useful lower bound on the log-likelihood of some observed data.

The ELBO is useful because it provides a guarantee on the worst-case for the log-likelihood of some distribution (e.g. ${\displaystyle p(X)}$) which models a set of data. The actual log-likelihood may be higher (indicating an even better fit to the distribution) because the ELBO includes a Kullback-Leibler divergence (KL divergence) term which decreases the ELBO due to an internal part of the model being inaccurate despite good fit of the model overall. Thus improving the ELBO score indicates either improving the likelihood of the model ${\displaystyle p(X)}$ or the fit of a component internal to the model, or both, and the ELBO score makes a good loss function, e.g., for training a deep neural network to improve both the model overall and the internal component. (The internal component is ${\displaystyle q_{\phi }(\cdot |x)}$, defined in detail later in this article.)