☆ Save Variational Inference — Approximating Intractable Bayesian Posteriors

09/13/2026

Variational Inference is a way to perform Bayesian Inference when the posterior distribution is too expensive to compute exactly. Given observed data x and a latent variable z, we would ideally like to evaluate the posterior p(z|x). In many probabilistic models·however·this requires computing the evidence p(x), which may involve a large sum or an intractable integral over possible values of z. Variational Inference avoids evaluating the posterior directly. Instead, it introduces a tractable variational distribution q(z) and learns parameters that make q(z) approximate p(z|x) as closely as possible.

In simple terms: after observing x, we want to know which values of the hidden variable z are plausible. The exact posterior p(z|x) contains that information, but calculating it may be impractical. Variational Inference takes a different route: choose a distribution q(z) that is easy to work with, then optimize it so that it behaves like the posterior. The core idea is to turn a difficult inference problem into a tractable optimization problem.

Variational Inference learns a tractable distribution that approximates the posterior

How It Works

• Start with the posterior

  • Consider a probabilistic model with observed data x and an unobserved latent variable z. Once x is observed, the quantity of interest is the posterior p(z|x), which describes how plausible different values of z are given the data.

  • Bayes’ Rule gives the posterior as follows.

  • $$ p(z \mid x)=\frac{p(x \mid z)p(z)}{p(x)} $$

    The posterior distribution over latent variable z given observed data x

  • The hard part is the denominator, the evidence p(x). Computing it exactly means accounting for all possible values of z, usually through a sum or integral. For complex models, that calculation can become prohibitively expensive, which makes direct evaluation of p(z|x) difficult as well.

  • This is the problem Variational Inference is designed to address.

• Approximate p(z|x) with a tractable q(z)

  • When the posterior cannot be computed efficiently, we introduce another probability distribution that is easier to evaluate and optimize. This distribution is the variational distribution q(z).

  • $$ q(z) \approx p(z \mid x) $$

    The objective is to make q(z) a close approximation to the true posterior p(z|x).

  • q(z) is not treated as the correct answer from the beginning. Instead, we choose a tractable family of distributions and adjust its parameters so that the resulting q(z) better matches the posterior.

  • For example, if q(z) is defined by a mean and variance, those values become variational parameters. Variational Inference updates them to improve the approximation.

• Reframe inference as optimization

  • Once q(z) is introduced, the question changes. Instead of asking, “How do we compute p(z|x) exactly?”, we ask which variational parameters make q(z) approximate p(z|x) most effectively.
  • The problem is no longer to evaluate an intractable distribution directly, but to search for good values of a set of adjustable parameters.
  • That turns posterior inference into an optimization problem.
  • This reformulation is the key idea behind Variational Inference: even when the exact posterior is out of reach, a useful approximation can still be learned.

• Use KL Divergence to measure the approximation

  • To optimize q(z), we need a way to quantify how different it is from the posterior. Variational Inference typically uses KL Divergence for this purpose.

  • The target is to minimize the divergence between q(z) and p(z|x).

  • $$ \min_q D_{KL}\big(q(z)\parallel p(z \mid x)\big) $$

    A smaller KL Divergence means q(z) is closer to the posterior under this measure.

  • So a “good” q(z) is not chosen because it merely looks similar to the posterior. Its parameters are optimized against an explicit objective that measures the gap between the two distributions.

• Optimize the ELBO instead of the KL Divergence directly

  • There is one more complication. The KL Divergence above contains the true posterior p(z|x), which is exactly the quantity that was difficult to compute in the first place. So although minimizing KL Divergence describes the goal, it is not directly practical as the optimization objective.

  • Variational Inference resolves this by optimizing the Evidence Lower Bound, usually abbreviated as ELBO.

  • The log evidence·ELBO·and KL Divergence satisfy the following relationship.

  • $$ \log p(x)=\mathrm{ELBO}+D_{KL}\big(q(z)\parallel p(z \mid x)\big) $$

    For fixed observed data x, increasing the ELBO decreases the KL Divergence between q(z) and the posterior.

  • For fixed x, log p(x) does not change when the parameters of q(z) change. Maximizing the ELBO therefore pushes the KL Divergence lower and moves q(z) toward the posterior within the chosen variational family.

  • In practice, this means we can update the variational parameters by optimizing the ELBO without ever computing the exact posterior explicitly.

• Use the learned q(z) as an approximate posterior

  • As the variational parameters are updated, q(z) is optimized toward p(z|x) within the chosen variational family.
  • Once optimization is complete, the learned q(z) is used as an approximate posterior. Instead of working directly with the intractable p(z|x), inference about z can be performed using the tractable q(z).
  • The important limitation is that q(z) is still only an approximation. Variational Inference does not guarantee that it will exactly reproduce the true posterior.

• How it differs from sampling-based inference

  • Sampling-based inference is another way to work with a posterior that is difficult to evaluate directly. These methods generate many samples and use them to estimate properties of the posterior.
  • Variational Inference follows a different strategy. It defines a tractable q(z) and learns its parameters through optimization.
  • The main distinction is whether the posterior is estimated through sampling or approximated by optimizing a parameterized distribution.
  • Optimization can make Variational Inference efficient for large datasets and large models, but the quality of the result is constrained by how expressive the chosen variational family is.

Significance and Limitations

The key contribution of Variational Inference is that it turns intractable posterior inference into an optimization problem. In complex probabilistic models, computing the evidence can make the exact posterior p(z|x) impractical to evaluate. By introducing a tractable q(z) and optimizing the ELBO, Variational Inference provides a practical way to approximate that posterior and perform inference over latent variables without computing the exact distribution.

The tradeoff is approximation error. The learned q(z) is not the true posterior, and a limited variational family may fail to capture important structure in p(z|x) even when optimization works well. The optimization process is also not guaranteed to reach the best possible solution. In practice, Variational Inference therefore involves balancing computational efficiency, the expressive power of the variational family, and the stability of the optimization procedure.

Recommended prerequisite reading (3/5)

+2

• Variational Distribution — How Variational Inference Approximates the Posterior
• Intractable Posterior — Why Bayesian Inference Needs Approximate Inference
• Approximate Inference: Making Bayesian Inference Practical with Posterior Distribution Estimation

• Mode-Covering — An Approximation Property That Broadly Includes Patterns So the Data Does Not Miss Existing Modes
• Metropolis-Hastings Algorithm — How MCMC Samples from a Posterior

Recommended next reading (5/17)

+5

• Posterior Inference: Making Intractable Posteriors Usable
• Prior Distribution — Setting Probabilities Before Observing Data in Bayesian Inference
• Bayesian Network
• Stochastic Inference — Reasoning with Uncertainty in AI
• Normalizing Constant — The Key to Turning Relative Scores into a Probability Distribution

• Markov Chain Monte Carlo (MCMC) — Sampling the Posterior with a Markov Chain
• Gibbs Sampling: Sampling from a Posterior One Parameter at a Time
• Parametric Approximation — Modeling Data Distributions with Probabilistic Models
• Likelihood — A Criterion for Judging Which Model Best Explains the Observed Data
• Probability Normalization — How Softmax Converts Model Outputs into a Probability Distribution

• Multinoulli Distribution — Modeling One Choice Among Many Categories
• Valid Probability Distributions: Why Probabilities Must Sum to 1
• High-Probability Region — How Machine Learning Identifies Important Areas of a Data Distribution
• Probability Factorization — Computing Complex Joint Probabilities via a Chain of Conditional Probabilities
• Exponential Distribution — A Continuous Probability Distribution for Modeling Waiting Time Until an Event
• Probability Distribution Comparison — How to Measure and Interpret Differences Between Distributions
• Unnormalized Probability Models — Why Models Use Scores Before Normalization

Posts on the same topic (4/4)

• Markov Chain — How Transition Probabilities Drive State Evolution
• Stationary Distribution — How Long-Run Probabilities Emerge in a Markov Chain
• Joint Distribution Modeling — How AI Learns Relationships Between Variables
• Probabilistic Model — Modeling Uncertainty with Probability Distributions

Related concepts (2/2)

• Posterior Expectation — Estimating with the Mean of the Posterior Distribution,
• Posterior Distribution — Updating Probabilities with Observed Data in Bayesian Inference,

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🔖 Tags: Bayesian Inference · ELBO · KL Divergence · Latent Variable · posterior distribution · Variational Distribution · variational inference