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A Frequentist Statistical Introduction to Variational Inference, Autoencoders, and Diffusion Models

Yen-Chi Chen

TL;DR

This paper addresses the gap between frequentist inference and modern generative models by providing a self-contained, frequentist treatment of Variational Inference ($VI$), Variational Autoencoders (VAEs), and Denoising Diffusion Models (DDMs). It shows VI as a scalable surrogate for intractable E-steps through the evidence lower bound ($ELBO$) and its Monte Carlo and amortized extensions, connecting EM, MCEM, and regularized Q-functions to deep learning. It frames VAEs and DDMS as deep latent variable models built on an amortized VI foundation, deriving practical objectives such as the noise-prediction formulation in diffusion models. It also discusses when VI should be viewed under a frequentist versus Bayesian lens and contrasts latent-variable generative power with scientific interpretability in latent-variable modeling.

Abstract

While Variational Inference (VI) is central to modern generative models like Variational Autoencoders (VAEs) and Denoising Diffusion Models (DDMs), its pedagogical treatment is split across disciplines. In statistics, VI is typically framed as a Bayesian method for posterior approximation. In machine learning, however, VAEs and DDMs are developed from a Frequentist viewpoint, where VI is used to approximate a maximum likelihood estimator. This creates a barrier for statisticians, as the principles behind VAEs and DDMs are hard to contextualize without a corresponding Frequentist introduction to VI. This paper provides that introduction: we explain the theory for VI, VAEs, and DDMs from a purely Frequentist perspective, starting with the classical Expectation-Maximization (EM) algorithm. We show how VI arises as a scalable solution for intractable E-steps and how VAEs and DDMs are natural, deep-learning-based extensions of this framework, thereby bridging the gap between classical statistical inference and modern generative AI.

A Frequentist Statistical Introduction to Variational Inference, Autoencoders, and Diffusion Models

TL;DR

This paper addresses the gap between frequentist inference and modern generative models by providing a self-contained, frequentist treatment of Variational Inference (), Variational Autoencoders (VAEs), and Denoising Diffusion Models (DDMs). It shows VI as a scalable surrogate for intractable E-steps through the evidence lower bound () and its Monte Carlo and amortized extensions, connecting EM, MCEM, and regularized Q-functions to deep learning. It frames VAEs and DDMS as deep latent variable models built on an amortized VI foundation, deriving practical objectives such as the noise-prediction formulation in diffusion models. It also discusses when VI should be viewed under a frequentist versus Bayesian lens and contrasts latent-variable generative power with scientific interpretability in latent-variable modeling.

Abstract

While Variational Inference (VI) is central to modern generative models like Variational Autoencoders (VAEs) and Denoising Diffusion Models (DDMs), its pedagogical treatment is split across disciplines. In statistics, VI is typically framed as a Bayesian method for posterior approximation. In machine learning, however, VAEs and DDMs are developed from a Frequentist viewpoint, where VI is used to approximate a maximum likelihood estimator. This creates a barrier for statisticians, as the principles behind VAEs and DDMs are hard to contextualize without a corresponding Frequentist introduction to VI. This paper provides that introduction: we explain the theory for VI, VAEs, and DDMs from a purely Frequentist perspective, starting with the classical Expectation-Maximization (EM) algorithm. We show how VI arises as a scalable solution for intractable E-steps and how VAEs and DDMs are natural, deep-learning-based extensions of this framework, thereby bridging the gap between classical statistical inference and modern generative AI.
Paper Structure (26 sections, 89 equations, 2 figures)

This paper contains 26 sections, 89 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Latent Variable Model
  3. EM algorithm
  4. MCEM: Monte Carlo EM
  5. Regularization form of the Q-function
  6. Example: limitation of the EM Algorithm
  7. Variational Approximation
  8. Gradient of the ELBO and the reparameterization trick
  9. Conditions for fast gradient ascent
  10. Amortized Variational Inference and the Variational Autoencoder
  11. Example: connecting amortized and non-amortized VI
  12. Gradient of the amortized ELBO
  13. Variational Autoencoder (VAE)
  14. Denoising Diffusion Model (DDM)
  15. A deep latent variable model
  16. ...and 11 more sections

Figures (2)

  • Figure 1: An illustration of the DDM framework. An observation $Y_0=X_i$ is a clean image (left). The variational distribution (the forward process) gradually adds Gaussian noise, moving from left to right. The generative model (the reverse process) learns to reverse this, starting from noise (right) and progressively denoising it to recover a clean image.
  • Figure 2: An illustration of the DDM training loop. For each observation (e.g., the cat and polar bear images), the forward process is used to generate a trajectory of noisy images. These trajectories are then used to compute the gradients and update the generative model's parameters in the reverse (denoising) process. This is repeated until convergence.