# The Thermodynamic Variational Objective

arXiv: https://arxiv.org/abs/1907.00031 · HTML: https://mlsanity.com/paper/1907.00031/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Jun 2019.

## Built on by (most cited first, 26 of 26)

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- [Information-Theoretic Diffusion](https://mlsanity.com/paper/2302.03792/) (2023)
- [Differentiable Annealed Importance Sampling and the Perils of Gradient Noise](https://mlsanity.com/paper/2107.10211/) (2021)
- [Nested Variational Inference](https://mlsanity.com/paper/2106.11302/) (2021)
- [Surrogate Likelihoods for Variational Annealed Importance Sampling](https://mlsanity.com/paper/2112.12194/) (2021)
- [Neural Thermodynamic Integration: Free Energies from Energy-based Diffusion Models](https://mlsanity.com/paper/2406.02313/) (2024)
- [Provable benefits of annealing for estimating normalizing constants: Importance Sampling, Noise-Contrastive Estimation, and beyond](https://mlsanity.com/paper/2310.03902/) (2023)
- [q-Paths: Generalizing the Geometric Annealing Path using Power Means](https://mlsanity.com/paper/2107.00745/) (2021)
- [Multiple Importance Sampling ELBO and Deep Ensembles of Variational Approximations](https://mlsanity.com/paper/2202.10951/) (2022)
- [Adaptive Annealed Importance Sampling with Constant Rate Progress](https://mlsanity.com/paper/2306.15283/) (2023)
- [Learning Proposals for Probabilistic Programs with Inference Combinators](https://mlsanity.com/paper/2103.00668/) (2021)
- [Bounding Evidence and Estimating Log-Likelihood in VAE](https://mlsanity.com/paper/2206.09453/) (2022)
- [Unsupervised representation learning with recognition-parametrised probabilistic models](https://mlsanity.com/paper/2209.05661/) (2022)
- [Variational Open-Domain Question Answering](https://mlsanity.com/paper/2210.06345/) (2022)
- [Fast Offline Policy Optimization for Large Scale Recommendation](https://mlsanity.com/paper/2208.05327/) (2022)
- [Variational Inference with Holder Bounds](https://mlsanity.com/paper/2111.02947/) (2021)
- [Reparameterized Variational Rejection Sampling](https://mlsanity.com/paper/2309.14612/) (2023)
- [Differentiable Annealed Importance Sampling Minimizes The Symmetrized Kullback-Leibler Divergence Between Initial and Target Distribution](https://mlsanity.com/paper/2405.14840/) (2024)
- [TUBE: Tangent Upper Bound on Evidence for Discrete Diffusion Language Models](https://mlsanity.com/paper/2605.24292/) (2026)
- [Estimating optimal PAC-Bayes bounds with Hamiltonian Monte Carlo](https://mlsanity.com/paper/2310.20053/) (2023)
- [Globally Convergent Variational Inference](https://mlsanity.com/paper/2501.08201/) (2025)
- [Learning Nonparametric High-Dimensional Generative Models: The Empirical-Beta-Copula Autoencoder](https://mlsanity.com/paper/2309.09916/) (2023)
- [Towards Model-Agnostic Posterior Approximation for Fast and Accurate Variational Autoencoders](https://mlsanity.com/paper/2403.08941/) (2024)
- [Tight Bounds for Jensen's Gap with Applications to Variational Inference](https://mlsanity.com/paper/2502.03988/) (2025)

## Abstract (arXiv)

We introduce the thermodynamic variational objective (TVO) for learning in both continuous and discrete deep generative models. The TVO arises from a key connection between variational inference and thermodynamic integration that results in a tighter lower bound to the log marginal likelihood than the standard variational variational evidence lower bound (ELBO) while remaining as broadly applicable. We provide a computationally efficient gradient estimator for the TVO that applies to continuous, discrete, and non-reparameterizable distributions and show that the objective functions used in variational inference, variational autoencoders, wake sleep, and inference compilation are all special cases of the TVO. We use the TVO to learn both discrete and continuous deep generative models and empirically demonstrate state of the art model and inference network learning.