Are variational parameters always means and variances?

No. Their form depends on the chosen distribution family.

Does a negative log scale mean a negative standard deviation?

No. Exponentiating any finite real log scale produces a positive standard deviation.

Can different parameter values describe the same distribution?

Depending on the parameterization, yes; numerical coordinates must be interpreted through the distribution they define.

AI Glossary / Graphs and symbolic AI / Advanced

Variational parameters

Definition

Variational parameters are adjustable quantities that specify an approximate distribution used in variational inference. Optimizing them selects a distribution that approximates a target, commonly a Bayesian posterior. For example, the mean and scale of an approximate normal distribution can be variational parameters.

What it means

These parameters describe an approximation to uncertainty. They should not automatically be read as the unknown physical quantities being studied. If a model contains an uncertain sensor offset, the approximation may have one parameter for its estimated center and another for its spread. The offset, the estimated center, and the uncertainty around it play different roles.

How it works

Write the approximate distribution as q(z;phi), with phi collecting its adjustable values. Optimize an objective such as the evidence lower bound with respect to phi. Parameterization must preserve valid distributions: scales must be positive and categorical probabilities must sum to one. An unconstrained log scale can represent a positive scale through exponentiation. A factorized Gaussian uses separate location and scale parameters, while a full-covariance Gaussian additionally represents dependence. These choices determine what the family can express; the optimizer then selects particular values.

Three practical examples

A fictional calibration model treats an instrument's offset as uncertain. Its approximate normal posterior uses a location and a scale. Reporting only the location would hide uncertainty that matters when deciding whether calibration is necessary.
A fictional document classifier represents uncertain category membership with three probabilities. It can use unconstrained scores transformed into probabilities, rather than allowing an optimizer to produce negative probabilities. The scores themselves are not percentages and must not be displayed as such.
A fictional demand model contains two uncertain effects. Separate normal factors can describe their individual locations and spreads, but they cannot encode correlation between those effects. Adding covariance parameters changes the representation, rather than merely running the existing optimizer for longer.

Think of it this way

Think of a camera's focus and zoom controls. They adjust how a scene is represented without being the objects in the scene. Likewise, variational parameters adjust a probability distribution representing uncertainty. The analogy is limited: optimizing them follows a defined mathematical objective, not a person's visual preference.

Worked example

Suppose a deliberately tractable target posterior is normal with mean 2 and variance 9. Approximate it with q = Normal(m,s squared), where m is any real number and s is positive. Here the unknown variable is z, while m and s are the two variational parameters.
For this example, the divergence from q to the target is one half of [(s squared + (m-2) squared)/9 - 1 + ln(9/s squared)]. Natural logarithms are used.
At m=0 and s=1, the divergence is approximately 0.876390. Moving the location to m=2 while keeping s=1 reduces it to approximately 0.654168. The center is now correct, but the approximation remains too narrow. Setting m=2 and s=3 makes the divergence zero because the two distributions coincide.
We can alternatively optimize m and rho, with s=exp(rho). The exact solution becomes m=2 and rho=ln(3), approximately 1.098612. The spread is still 3, not 1.098612. Setting rho=0 gives s=1, not zero uncertainty.
If the implementation fixes s=1 and optimizes only m, no amount of further location adjustment can recover variance 9. The family must permit a different scale. This illustrates why the list of adjustable parameters matters independently of whether optimization succeeds.

Illustrative case study

In a fictional calibration review, two prototype reports both display an estimated offset of 2 units. One prototype fixed its approximate standard deviation at 1; the other fitted it and recovered 3 in the worked example. The identical central estimates initially make their outputs look interchangeable.
The reviewer asks both prototypes to display a central interval using the normal multiplier 1.96. The first reports approximately [0.04, 3.96]; the second reports approximately [-3.88, 7.88]. Their interval widths are 3.92 and 11.76 units, respectively. These are approximately 95% intervals for the stated normal distributions, not guarantees about the instrument's real offset.
The team finds that the narrow interval is a consequence of fixing a variational parameter. It changes the configuration and checks the recovered distribution against the known target. It also labels log scale separately from standard deviation so that an internal value of 1.098612 is not accidentally published as the physical spread. This scenario is illustrative rather than a measured deployment result.

Common misconceptions and mistakes

A fitted variational mean does not remove uncertainty about the latent quantity. In the example, mean 2 coexists with standard deviation 3.
A variance and a standard deviation are different: the target's values are 9 and 3. Their logarithms are also different.
A parameter's numerical value depends on the representation. The scale 3 and log scale ln(3) describe the same distribution.
Not every numerical setting is a variational parameter. A chosen iteration limit controls the optimization procedure; it does not directly specify q. Similarly, changing a prior changes the model rather than simply adjusting its posterior approximation.

Business use

Ask reports to distinguish estimated locations, uncertainty measures, internal transformed parameters, and optimization settings. In a small validation model, recover known answers and check units after transformations. For the calibration example, matching the center alone would miss a threefold error in interval width. Keep a record of which distribution parameters are fixed and which are fitted, so a restrictive configuration cannot silently masquerade as strong evidence.

Check your understanding

What are the variational parameters in q=Normal(m,s squared)?

Answer: The location m and positive standard deviation s.

What does rho=0 imply when s=exp(rho)?

Answer: A standard deviation of 1.

Why does optimizing only m fail in the restricted example?

Answer: The fixed scale prevents q from matching the target's variance.

Related terms

Variational posterior
Latent variable
Mean field approximation
Covariance
Optimization
Variational inference
variational family
evidence lower bound
parameterization

Learn next

Differentiate the worked divergence with respect to m and s to recover the optimum analytically. Then add a second uncertain variable and count the extra parameters needed to represent covariance, distinguishing representational flexibility from optimization reliability.

Sources

Kucukelbir and colleagues, Automatic Differentiation Variational Inference, Section 2.4 on variational approximations and their parameters: https://jmlr.org/papers/volume18/16-107/16-107.pdf The normal target, numerical calculations, calibration scenario, and reporting comparisons are original illustrations.

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