Variational inference

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Last modified: 2026-11-12 01:08:55 (PST)

The posterior of an unobserved quantity is often too hard to compute. Variational inference replaces it with a simpler stand-in that is close to it, where closeness is measured by the KL divergence.

1 The evidence lower bound

Definition 1 (Evidence lower bound (ELBO)) Let \(x\) be observed data, and let \(z\) be an unobserved discrete random variable, with joint probability mass function \({\mathbb{p}\mathopen{}\left(x, z\right)\mathclose{}}\). Let \(q\) be a probability mass function for \(z\). Require that \(q\) gives probability zero to every value \(z\) where the posterior \({\mathbb{p}\mathopen{}\left(z \mid x\right)\mathclose{}}\) is zero. The evidence lower bound (ELBO) of \(q\) is the average, over \(z\) drawn from \(q\), of the difference between the logarithm of \({\mathbb{p}\mathopen{}\left(x, z\right)\mathclose{}}\) and the logarithm of \(q(z)\):

\[\mathrm{ELBO}(q) = {\mathbb{E}_{q}\mathopen{}\left[{\operatorname{log}\mathopen{}\left\{{\mathbb{p}\mathopen{}\left(x, z\right)\mathclose{}}\right\}\mathclose{}} - {\operatorname{log}\mathopen{}\left\{q(z)\right\}\mathclose{}}\right]\mathclose{}}.\]

Written as a sum over the values \(z\) with \(q(z) > 0\), the average is

\[\mathrm{ELBO}(q) = \sum_z q(z) \mathopen{}\left({\operatorname{log}\mathopen{}\left\{{\mathbb{p}\mathopen{}\left(x, z\right)\mathclose{}}\right\}\mathclose{}} - {\operatorname{log}\mathopen{}\left\{q(z)\right\}\mathclose{}}\right)\mathclose{}.\]

Theorem 1 (The log evidence is the ELBO plus a divergence) Let \({\mathbb{p}\mathopen{}\left(x, z\right)\mathclose{}}\) and \(q\) be as in Definition 1. Let \({\mathbb{p}\mathopen{}\left(x\right)\mathclose{}} = \sum_z {\mathbb{p}\mathopen{}\left(x, z\right)\mathclose{}}\) be the evidence of the data, also called its marginal likelihood. The evidence is nonzero, because a data value \(x\) was observed. Then the logarithm of the evidence is the ELBO of \(q\) plus the Kullback–Leibler divergence of \(q\) from the posterior:

\[{\operatorname{log}\mathopen{}\left\{{\mathbb{p}\mathopen{}\left(x\right)\mathclose{}}\right\}\mathclose{}} = \mathrm{ELBO}(q) + {{D_{\mathrm{KL}}}\mathopen{}\left(q \,\Vert\, {\mathbb{p}\mathopen{}\left(\cdot \mid x\right)\mathclose{}}\right)\mathclose{}}.\]

Proof

Proof. Let \(z\) be a value with \(q(z) > 0\). Then \({\mathbb{p}\mathopen{}\left(z \mid x\right)\mathclose{}} \ne 0\), and \({\mathbb{p}\mathopen{}\left(x, z\right)\mathclose{}} = {\mathbb{p}\mathopen{}\left(z \mid x\right)\mathclose{}} \, {\mathbb{p}\mathopen{}\left(x\right)\mathclose{}}\). Apply \(\operatorname{log}\) to both sides and rearrange:

\[{\operatorname{log}\mathopen{}\left\{{\mathbb{p}\mathopen{}\left(x\right)\mathclose{}}\right\}\mathclose{}} = {\operatorname{log}\mathopen{}\left\{{\mathbb{p}\mathopen{}\left(x, z\right)\mathclose{}}\right\}\mathclose{}} - {\operatorname{log}\mathopen{}\left\{{\mathbb{p}\mathopen{}\left(z \mid x\right)\mathclose{}}\right\}\mathclose{}}.\]

Add and subtract \({\operatorname{log}\mathopen{}\left\{q(z)\right\}\mathclose{}}\) on the right:

\[{\operatorname{log}\mathopen{}\left\{{\mathbb{p}\mathopen{}\left(x\right)\mathclose{}}\right\}\mathclose{}} = \mathopen{}\left({\operatorname{log}\mathopen{}\left\{{\mathbb{p}\mathopen{}\left(x, z\right)\mathclose{}}\right\}\mathclose{}} - {\operatorname{log}\mathopen{}\left\{q(z)\right\}\mathclose{}}\right)\mathclose{} + \mathopen{}\left({\operatorname{log}\mathopen{}\left\{q(z)\right\}\mathclose{}} - {\operatorname{log}\mathopen{}\left\{{\mathbb{p}\mathopen{}\left(z \mid x\right)\mathclose{}}\right\}\mathclose{}}\right)\mathclose{}.\]

Multiply both sides by \(q(z)\), then sum over the values \(z\) with \(q(z) > 0\). The left side sums to \({\operatorname{log}\mathopen{}\left\{{\mathbb{p}\mathopen{}\left(x\right)\mathclose{}}\right\}\mathclose{}}\), because \(q\) sums to \(1\):

\[{\operatorname{log}\mathopen{}\left\{{\mathbb{p}\mathopen{}\left(x\right)\mathclose{}}\right\}\mathclose{}} = \sum_z q(z) \mathopen{}\left({\operatorname{log}\mathopen{}\left\{{\mathbb{p}\mathopen{}\left(x, z\right)\mathclose{}}\right\}\mathclose{}} - {\operatorname{log}\mathopen{}\left\{q(z)\right\}\mathclose{}}\right)\mathclose{} + \sum_z q(z) \mathopen{}\left({\operatorname{log}\mathopen{}\left\{q(z)\right\}\mathclose{}} - {\operatorname{log}\mathopen{}\left\{{\mathbb{p}\mathopen{}\left(z \mid x\right)\mathclose{}}\right\}\mathclose{}}\right)\mathclose{}.\]

The first sum is \(\mathrm{ELBO}(q)\) by its definition. The second sum is \({{D_{\mathrm{KL}}}\mathopen{}\left(q \,\Vert\, {\mathbb{p}\mathopen{}\left(\cdot \mid x\right)\mathclose{}}\right)\mathclose{}}\) by the definition of the divergence.

Example 1 (The ELBO and the divergence for a three-valued latent variable)

Let \(z\) take the values \(1, 2, 3\) with prior probabilities \(0.5, 0.3, 0.2\). A data value \(x\) is observed, and its probabilities given \(z\) are \(0.1, 0.4, 0.8\). The R chunk below finds the posterior, then computes the ELBO and the divergence for three choices of \(q\), with natural logarithms.

Show code

prior <- c(0.5, 0.3, 0.2)
likelihood <- c(0.1, 0.4, 0.8)
joint <- prior * likelihood
evidence <- sum(joint)
posterior <- joint / evidence
log_evidence <- log(evidence)

elbo_and_divergence <- function(q) {
  log_ratio <- log(joint) - log(q)
  elbo <- sum(q * log_ratio)
  log_gap <- log(q) - log(posterior)
  divergence <- sum(q * log_gap)
  c(elbo = elbo, divergence = divergence, total = elbo + divergence)
}

uniform <- rep(1 / 3, 3)
shifted <- c(0.2, 0.3, 0.5)
rows <- rbind(
  uniform = elbo_and_divergence(uniform),
  shifted = elbo_and_divergence(shifted),
  posterior = elbo_and_divergence(posterior)
)
round(rows, 4)
#>              elbo divergence   total
#> uniform   -1.2176     0.1089 -1.1087
#> shifted   -1.1219     0.0132 -1.1087
#> posterior -1.1087     0.0000 -1.1087

The evidence is \(0.33\), so its logarithm is \(-1.1087\). In every row the ELBO plus the divergence equals this value, as Theorem 1 says. The ELBO is largest, and the divergence is \(0\), for \(q\) equal to the posterior.

Remark 1 (Variational inference maximizes the ELBO). The logarithm of the evidence \({\operatorname{log}\mathopen{}\left\{{\mathbb{p}\mathopen{}\left(x\right)\mathclose{}}\right\}\mathclose{}}\) does not depend on \(q\). By Theorem 1, choosing \(q\) to maximize the ELBO is therefore the same as choosing \(q\) to minimize its Kullback–Leibler divergence from the posterior. The ELBO never exceeds the logarithm of the evidence, because the divergence is at least \(0\). Variational inference searches a restricted family of choices for \(q\) this way (Blei et al. 2017).

References

Blei, David M., Alp Kucukelbir, and Jon D. McAuliffe. 2017. “Variational Inference: A Review for Statisticians.” Journal of the American Statistical Association 112 (518): 859–77. https://doi.org/10.1080/01621459.2017.1285773.