Unnormalized Variational Bayes

Abstract

We unify empirical Bayes and variational Bayes for approximating unnormalized densities. This framework, named unnormalized variational Bayes (UVB), is based on formulating a latent variable model for the random variable $Y=X+N(0,{\sigma }^2{I}_d)$ and using the evidence lower bound (ELBO), computed by a variational autoencoder, as a parametrization of the energy function of $Y$ which is then used to estimate $X$ with the empirical Bayes least-squares estimator. In this intriguing setup, the $\text{gradient}$ of the ELBO with respect to noisy inputs plays the central role in learning the energy function. Empirically, we demonstrate that UVB has a higher capacity to approximate energy functions than the parametrization with MLPs as done in neural empirical Bayes (DEEN). We especially showcase $\sigma =1$, where the differences between UVB and DEEN become visible and qualitative in the denoising experiments. For this high level of noise, the distribution of $Y$ is very smoothed and we demonstrate that one can traverse in a single run $-$ without a restart $-$ all MNIST classes in a variety of styles via walk-jump sampling with a fast-mixing Langevin MCMC sampler. We finish by probing the encoder/decoder of the trained models and confirm UVB $\ne$ VAE.

Submitted to Journal of Machine Learning Research

References in corpus (7)

• Adam: A Method for Stochastic Optimization
• Stochastic Backpropagation and Approximate Inference in Deep Generative Models
• Searching for Activation Functions
• Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning
• Deep Energy Estimator Networks
• On approximating $\nabla f$ with neural networks
• Learning and Inference in Imaginary Noise Models