Variational Inference and Pathfinder

Summary

Variational inference replaces integration with optimization: find the closest member of a tractable family $\mathcal{Q}$ to the target. Two design choices — the family and the divergence — determine everything, and both are hard. The book’s honest assessment is that VI’s accuracy is difficult to trust in high dimensions, and even a global minimizer remains misspecified. But Pathfinder is singled out as genuinely useful precisely because it does not try to be accurate: it uses L-BFGS to walk toward the mode and picks the best normal approximation along the optimization path, giving fast initialization for HMC and, run in parallel from many starts, a way to find modes and weight their relative mass.

Overview

The variational problem (Eq. 13.1, Ch. 13.2, p. 241)

$$q^{\ast }=\underset{q\in \mathcal{Q}}{\mathrm{argmin}}D(q\Vert p)$$

“The two main design choices of VI are the family $\mathcal{Q}$ of approximations and the divergence $D$ … These choices need to balance inferential accuracy and computational cost, and finding a good balance is challenging.”

What VI is for: “VIs are useful as rough posterior approximations and can be used to supply starting values for Hamiltonian Monte Carlo and other simulation-based algorithms. Variational approximations can also be useful as quick inference in early phases of workflow, in the end if the specific variational approximation and inference has been carefully tested to reach the desired inference accuracy for the quantities of interest, or for approximate computation for large models such as neural nets with highly multimodal posteriors that cannot be covered well by existing simulation algorithms” (Carpenter 2026).

The family $\mathcal{Q}$

Tractable families and the richness tradeoff

“By tractable, we mean that certain operations can be easily computed: for example, calculating expectation values and drawing samples. The most commonly used family of approximations is the family of normals”, minimized over the mean and covariance matrix by deterministic or stochastic optimization.

Richer families: “a nonlinear neural network transformation of the normal, also known as a transformation flow (Agrawal, Sheldon, and Domke 2021) or a distribution constructed using a basis expansion” (Cai, Modi, Margossian, et al. 2024).

The tradeoff: “the richer the family of approximations, the more challenging the optimization in (13.1) becomes.”

The divergence $D$

Reverse KL, and why it is the default (Eq. 13.2)

$$\text{KL}(q\Vert p)=\int (\log q(z)-\log p(z))q(z)dz$$

“which can be minimized with respect to $q$, even if $p$ is only known up to a normalizing constant” — the property that makes it usable at all.

The contrast with modal methods: “In contrast to methods described in Section 13.1, VI does not explicitly fit to the mode; rather, it minimizes an average difference in log density.”

Alternatives, with their tradeoffs:

• Score-based divergences — “which compare gradients of log densities rather than log densities, possess attractive optimization properties” (Cai, Modi, Pillaud-Vivien, et al. 2024; Cai, Modi, Margossian, et al. 2024).
• $\alpha$-divergences — “can yield better solutions than minimizers of $\text{KL}(q\Vert p)$ for certain inferential tasks but can be much harder to minimize” (Dhaka, Catalina, Welandawe, et al. 2021; Geffner and Domke 2021; Daudel, Douc, and Roueff 2023).

And the deeper problem: “which divergence works best depends on the inferential task at hand. Different tasks can sometimes compete, with an accurate estimate of one quantity implying a poor estimate of another, a tradeoff known to arise for different measures of uncertainty” (Margossian, Pillaud-Vivien, and Saul 2025).

Main Content

Two practical challenges

Optimization error and approximation error (Ch. 13.2, p. 241)

“First, we may not be able to exactly solve the optimization problem and find a minimizer of $D(q\Vert p)$.

Second, even if we do obtain a global minimizer of $D(q\Vert p)$, the approximation remains misspecified, meaning $q^{\ast }\ne p$, especially if $\mathcal{Q}$ is a constrained family of approximations, such as the family of normals.

A misspecified approximation can, in some limited settings, still recover statistical properties of $p$ such as its mean (MacKay 2003; Margossian and Saul 2025), even while misestimating other quantities such as its variance. In many cases, it is unclear how well $q^{\ast }$ approximates $p$” (Yao, Vehtari, Simpson, et al. 2018b).

The exercise set of Approximate Algorithms and Approximate Models makes the variance claim exact: mean-field VI on a multivariate normal recovers conditional variances exactly but underestimates marginal variances and entropy.

Is VI really "faster than MCMC"?

“VI is often presented as a fast alternative to MCMC. Arguably, the optimization problem for VI can be relatively straightforward to solve when the variational family $\mathcal{Q}$ is constrained, compared to achieving convergence with MCMC.

This comparison is not entirely fair, since MCMC can achieve higher precision than VI when both algorithms converge (Figure 13.1). Moreover, it is challenging to get similar accuracy and trust in convergence for VI, compared to MCMC, in high dimensions” (Dhaka, Catalina, Andersen, et al. 2020; Dhaka, Catalina, Welandawe, et al. 2021).

Where it genuinely wins: “VI is used to fit deep learning models to big data, a setting where MCMC struggles to handle challenging posteriors and where VI’s fast approximation provides useful, if imperfect, inference, as measured by the performance of the trained model.”

ADVI

Automatic differentiation variational inference (Kucukelbir et al. 2017)

Stan’s implementation. “ADVI minimizes reverse Kullback-Leibler divergences (13.2) over the family of normals, either with a diagonal covariance matrix (‘mean-field’ assumption) or a dense covariance matrix.”

The reported successful use case, with its conditions: “Some modelers have reported they have successfully used Stan’s ADVI when there has been a need for repeated inference with similar big data sets, and where ADVI’s expected accuracy for the particular class of models has been first assessed using simulation experiments.”

Note the two conditions: repeated inference on similar data (so the assessment amortizes) and accuracy pre-assessed by simulation (so the approximation is not taken on faith). Both are necessary.

Pathfinder

What Pathfinder does (Zhang, Carpenter, et al. 2022)

“Pathfinder is a variational algorithm that uses the fast deterministic quasi-Newton L-BFGS optimization algorithm and chooses the best approximate normal distribution along the optimization path according to the stochastic Kullback-Leibler divergence estimate.

The goal is not to get the best possible variational fit but rather to get a fast and useful approximation.”

Where the randomness comes from: “Individual Pathfinder paths depend on the initial values, and the use of stochastic Kullback-Leibler estimates also induces randomness in the normal approximations along the path.”

Multi-Pathfinder: “Running many Pathfinders in parallel is fast, and multi-Pathfinder uses many normal approximations as a mixture importance sampling proposal distribution from which we can obtain approximate posterior draws.”

The three uses of Pathfinder, all endorsed in this book

1. Sanity check. “Quickly checking that the model code produced posterior is somewhat sensible.”
2. Initialization. “Obtaining Markov chain initial values that are closer to the typical set.” See Initial Values, Adaptation, and Warmup, where slow transit to the typical set is one of the three named initialization failures.
3. Mode finding and weighting. “If the posterior is multimodal, Pathfinder can be used to find many modes and to assess the relative mass in each mode using importance sampling.”

Use 3 is developed in Failure 6 — Multimodality: “Instead of running a large number of costly Markov chains, we can run the much faster Pathfinder algorithm from different initial values to help find more modes, and then start fewer Markov chains based on the found modes. Pathfinder can also discard modes that have negligible posterior mass so that further computation is focused where it matters.”

§27.4 demonstrates “the use of Pathfinder to obtain quick initial results for testing and to initialize slower MCMC sampling” — Model Building - Time-Series Decomposition for Birthdays.

Why Pathfinder is the VI the book actually recommends

The key design decision is stated plainly: “The goal is not to get the best possible variational fit.”

That sidesteps both of the warnings above. Because Pathfinder is used to initialize and to locate rather than to infer, the misspecification of the normal family and the difficulty of trusting VI convergence in high dimensions stop mattering — the MCMC that follows does the inference. Contrast ADVI, whose recommended use requires a prior simulation study precisely because it is being asked to produce the answer.

Connections

• Pathfinder-as-initializer is the practical resolution of “use reasonable starting points” — it automates what the book otherwise asks you to do by hand.
• The optimization/mode contrast with Approximations Based on Joint and Conditional Posterior Modes is substantive: Laplace fits at the mode with local curvature; VI minimizes an average discrepancy and can end up elsewhere.
• Unlike the Laplace approximation, VI has no equally clean self-check; the diagnostics that exist are cited in Approximate inference is itself iterative.

See Also

• Approximate Algorithms and Approximate Models — the chapter frame, and the exercises quantifying VI’s variance underestimation
• Initial Values, Adaptation, and Warmup — the problem Pathfinder solves best
• Failure Modes and Steps Forward — Pathfinder for multimodality
• Simulation-Based and Amortized Inference — the other optimization-and-neural-network family
• Diagnosing Variational Inference (PSIS k-hat and VSBC) — how to check whether a variational fit can be trusted
• Automatic Differentiation Variational Inference (ADVI) — the ADVI algorithm in detail
• Normalizing Flows as Conditional Density Estimators — same flow architecture, opposite KL direction
• Simulation-Based Calibration - Overview — SBC/VSBC is how an approximate (variational) posterior is validated