☆ Save Variational Distribution — How Variational Inference Approximates the Posterior

09/13/2026

A variational distribution is a tractable probability distribution used in Variational Inference to approximate a posterior that is difficult to compute directly. In Bayesian Inference, after observing data x, we want the posterior p(z|x) to describe which values of the latent variable z are plausible. In complex probabilistic models, however, evaluating that posterior may require computing the evidence p(x), which can involve an intractable sum or integral. Variational Inference avoids this direct calculation by introducing a manageable distribution q(z) and learning parameters that make it approximate the true posterior as closely as possible.

In simple terms: the distribution we ultimately care about is p(z|x), but it may be too expensive to evaluate exactly. Rather than substituting an arbitrary distribution, Variational Inference chooses a tractable q(z) and adjusts its parameters so that it becomes a better approximation to the posterior. A variational distribution is therefore a probability distribution learned to stand in for an otherwise intractable posterior, allowing a difficult inference problem to be handled through optimization.

How the variational distribution q(z) is optimized to approximate the posterior

How It Works

Begin with the posterior inference problem
- In Bayesian Inference, once we observe x, the goal is to determine which values of the latent variable z are plausible through the posterior distribution p(z|x).
- Computing that posterior requires the prior distribution and likelihood, as well as the evidence p(x).
- As a model becomes more complex, evaluating the evidence may require summing or integrating over many possible states of z. That calculation can become intractable.
- When exact posterior computation is impractical, we need a tractable distribution that can approximate it instead.
Use q(z) as the variational approximation
- The distribution introduced to approximate p(z|x) is the variational distribution q(z).
- $$ q(z) \approx p(z \mid x) $$
  
  The goal is to make the variational distribution q(z) a close approximation to the true posterior p(z|x).
- q(z) is not assumed to match the posterior from the beginning. We first choose a form that is computationally manageable, then learn its parameters so that the resulting distribution moves closer to p(z|x).
- Its role is to provide a tractable approximation that can be optimized even when the posterior itself cannot be evaluated directly.
Learn q(z) through its variational parameters
- q(z) is not a fixed distribution that remains unchanged once selected. Its shape is controlled by adjustable values known as variational parameters.
- For example, if the chosen distribution is parameterized by a mean and variance, those values can serve as its variational parameters.
- Variational Inference repeatedly updates these parameters so that q(z) becomes a better approximation to the true posterior p(z|x).
- This reframes the problem from “How do we compute the posterior exactly?” to “Which parameter values produce a better q(z)?” The inference problem therefore becomes an optimization problem.
Measure approximation quality with KL divergence
- To optimize q(z), we need a criterion that tells us how far the current approximation is from the target posterior p(z|x).
- Kullback-Leibler divergence, or KL divergence, measures the discrepancy between the two probability distributions. Variational Inference seeks parameter values that make this divergence as small as possible.
- $$ q^*(z)=\arg\min_{q(z)}D_{KL}\big(q(z)\parallel p(z \mid x)\big) $$
  
  Here, q(z) is the variational distribution, p(z|x) is the target posterior, and q*(z) is the approximation that minimizes KL divergence.
- Suppose two versions of q(z), produced by different parameter settings, have KL divergence values of 0.8 and 0.2. Under this criterion, the version with 0.2 is the better approximation to the posterior. Learning q(z) therefore means adjusting its parameters in a direction that reduces KL divergence.
Optimize the ELBO in practice
- There is one complication: KL divergence contains the true posterior p(z|x), which is precisely the quantity that is difficult to compute. Directly evaluating and minimizing that divergence is therefore impractical.
- Variational Inference avoids this problem by using the Evidence Lower Bound, or ELBO, as the optimization objective.
- $$ \log p(x)=\mathrm{ELBO}+D_{KL}\big(q(z)\parallel p(z \mid x)\big) $$
  
  For fixed observed data x, increasing the ELBO decreases the KL divergence between q(z) and the posterior.
- For fixed x, the log evidence log p(x) does not change as the parameters of q(z) are updated. Because the two terms on the right must still add up to the same value, increasing the ELBO reduces the KL divergence.
- In practice, this means the variational parameters can be learned by maximizing a tractable objective instead of explicitly computing the posterior and comparing q(z) with it.
Use the learned q(z) as an approximate posterior
- As ELBO optimization proceeds, q(z) is adjusted toward p(z|x) within the chosen variational family.
- Once optimization is complete, the learned q(z) serves as an approximate posterior and can be used to perform inference about the latent variable z.
- During optimization, q(z) is the distribution being adjusted toward the posterior. After optimization, it becomes the tractable distribution used in place of the posterior that was too difficult to compute directly.
- Because q(z) is restricted to the chosen variational family·however·it is not guaranteed to reproduce the true posterior exactly.
Use variational distributions in latent variable models
- Variational distributions are especially useful when uncertainty about an unobserved latent variable needs to be represented as a probability distribution rather than a single value.
- In a latent variable model, the latent space cannot be observed directly from the data. q(z) approximates the posterior over that hidden space, making it possible to infer the underlying latent structure.
- The Variational Autoencoder (VAE) is a well-known model that uses variational distributions to learn distributions over latent representations.
- This allows latent representations to be modeled probabilistically rather than reduced to a single fixed value.

Significance and Limitations

The main value of a variational distribution is that it provides a tractable q(z) for working with a posterior that cannot be computed directly. By optimizing the variational parameters and maximizing the ELBO, q(z) can be moved toward the posterior without explicitly evaluating p(z|x). This makes Bayesian Inference practical in complex machine learning systems and allows latent variable models to infer hidden structure probabilistically.

The limitation is that q(z) approximates the posterior only within the chosen variational family, the set of distributions available as candidates for the approximation. A restrictive family, such as a mean-field approximation, may be unable to represent important dependencies in the true posterior, leaving approximation error even after optimization. The final quality of q(z) also depends on how well the optimization process succeeds. In practice, Variational Inference therefore requires a balance between computational efficiency and approximation accuracy.

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