# Bounding the Test Log-Likelihood of Generative Models

arXiv: https://arxiv.org/abs/1311.6184 · HTML: https://mlsanity.com/paper/1311.6184/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Nov 2013.

## Built on by (most cited first, 2 of 2)

- [A Trust Crisis In Simulation-Based Inference? Your Posterior Approximations Can Be Unfaithful](https://mlsanity.com/paper/2110.06581/) (2021)
- [Efficiently Access Diffusion Fisher: Within the Outer Product Span Space](https://mlsanity.com/paper/2505.23264/) (2025)

## Abstract (arXiv)

Several interesting generative learning algorithms involve a complex probability distribution over many random variables, involving intractable normalization constants or latent variable normalization. Some of them may even not have an analytic expression for the unnormalized probability function and no tractable approximation. This makes it difficult to estimate the quality of these models, once they have been trained, or to monitor their quality (e.g. for early stopping) while training. A previously proposed method is based on constructing a non-parametric density estimator of the model's probability function from samples generated by the model. We revisit this idea, propose a more efficient estimator, and prove that it provides a lower bound on the true test log-likelihood, and an unbiased estimator as the number of generated samples goes to infinity, although one that incorporates the effect of poor mixing. We further propose a biased variant of the estimator that can be used reliably with a finite number of samples for the purpose of model comparison.