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Model-Informed Flows for Bayesian Inference

Joohwan Ko, Justin Domke

TL;DR

This work addresses the challenge of posterior geometry in variational inference for complex hierarchical models by linking VIP with forward autoregressive flows. It proves that full-rank VIP transformations can be exactly represented by generalized forward autoregressive flows augmented with a translation term and prior-function inputs, motivating the Model-Informed Flow (MIF) architecture. Empirically, MIF delivers tighter posterior approximations and achieves state-of-the-art or competitive performance across hierarchical and non-hierarchical benchmarks, with insights from ablations and capacity studies. The approach provides a principled, architecture-driven path to integrating model structure into flexible variational families, advancing practical Bayesian inference for large, structured models.

Abstract

Variational inference often struggles with the posterior geometry exhibited by complex hierarchical Bayesian models. Recent advances in flow-based variational families and Variationally Inferred Parameters (VIP) each address aspects of this challenge, but their formal relationship is unexplored. Here, we prove that the combination of VIP and a full-rank Gaussian can be represented exactly as a forward autoregressive flow augmented with a translation term and input from the model's prior. Guided by this theoretical insight, we introduce the Model-Informed Flow (MIF) architecture, which adds the necessary translation mechanism, prior information, and hierarchical ordering. Empirically, MIF delivers tighter posterior approximations and matches or exceeds state-of-the-art performance across a suite of hierarchical and non-hierarchical benchmarks.

Model-Informed Flows for Bayesian Inference

TL;DR

This work addresses the challenge of posterior geometry in variational inference for complex hierarchical models by linking VIP with forward autoregressive flows. It proves that full-rank VIP transformations can be exactly represented by generalized forward autoregressive flows augmented with a translation term and prior-function inputs, motivating the Model-Informed Flow (MIF) architecture. Empirically, MIF delivers tighter posterior approximations and achieves state-of-the-art or competitive performance across hierarchical and non-hierarchical benchmarks, with insights from ablations and capacity studies. The approach provides a principled, architecture-driven path to integrating model structure into flexible variational families, advancing practical Bayesian inference for large, structured models.

Abstract

Variational inference often struggles with the posterior geometry exhibited by complex hierarchical Bayesian models. Recent advances in flow-based variational families and Variationally Inferred Parameters (VIP) each address aspects of this challenge, but their formal relationship is unexplored. Here, we prove that the combination of VIP and a full-rank Gaussian can be represented exactly as a forward autoregressive flow augmented with a translation term and input from the model's prior. Guided by this theoretical insight, we introduce the Model-Informed Flow (MIF) architecture, which adds the necessary translation mechanism, prior information, and hierarchical ordering. Empirically, MIF delivers tighter posterior approximations and matches or exceeds state-of-the-art performance across a suite of hierarchical and non-hierarchical benchmarks.

Paper Structure

This paper contains 38 sections, 5 theorems, 48 equations, 2 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Variational Inference
  4. Hierarchical Bayesian Models
  5. Example.
  6. Non-Centered Parameterization
  7. Variationally Inferred Parameters
  8. Equivalence of VIP in Model vs Variational Spaces
  9. Forward Autoregressive Flows
  10. Model-Informed Flows: Bridging VIP and Autoregressive Flows
  11. Model-Informed Flows
  12. Incorporating Prior Information through $f_i$ and $\log g_i$
  13. The Necessity of Conditioning on Previously Generated Latent Variables
  14. Related Work
  15. Experiments
  16. ...and 23 more sections

Key Result

Theorem 4

Let $T = T_{\mathrm{VIP}} \circ T_A$ where $T_{\mathrm{VIP}}$ is the VIP transformation (def:vip) and $T_A$ is the affine transformation from full-rank Gaussian. If $f_i$ and $\log g_i$ in the hierarchical Bayesian model (eq:hierarchical_bayesian_model) are arbitrary continuous functions, and the pa

Figures (2)

  • Figure 1: "Funnel"-type distributions commonly arise in hierarchical model. This figure shows the funnel distribution (gray contours) approximated with 5000 samples from three families (blue points): A Full-Rank Gaussian (FR) is very poor (KL-divergence of 1.86 nats). A standard Forward Autoregressive Flow (FAF) is much better (0.38 nats) but still imperfect. Our proposed model-informed flow (MIF) achieves a KL-divergence of effectively 0.
  • Figure 2: Best ELBO achieved by full MIF (blue) and the variant without latent‐variable conditioning (orange) as a function of MLP hidden‐unit count. Higher capacity allows the no‐latent variant to close the gap, showing that sufficiently expressive networks can implicitly learn dependencies otherwise provided by explicit latent inputs.

Theorems & Definitions (12)

  • Definition 1: VIP Transformation
  • Definition 2
  • Definition 3
  • Theorem 4
  • Definition 5
  • Corollary 5
  • Lemma 5
  • proof
  • Theorem 5
  • proof
  • ...and 2 more