# Leveraging Nested MLMC for Sequential Neural Posterior Estimation with Intractable Likelihoods

arXiv: https://arxiv.org/abs/2401.16776 · HTML: https://mlsanity.com/paper/2401.16776/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 60 references that are in the map.
- First posted on arXiv 30 Jan 2024.

## Builds on (most cited first, 26 of 26)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [Revisiting Classifier Two-Sample Tests](https://mlsanity.com/paper/1610.06545/) (2016)
- [Sequential Neural Likelihood: Fast Likelihood-free Inference with Autoregressive Flows](https://mlsanity.com/paper/1805.07226/) (2018)
- [Automatic Posterior Transformation for Likelihood-Free Inference](https://mlsanity.com/paper/1905.07488/) (2019)
- [Flexible statistical inference for mechanistic models of neural dynamics](https://mlsanity.com/paper/1711.01861/) (2017)
- [Benchmarking Simulation-Based Inference](https://mlsanity.com/paper/2101.04653/) (2021)
- [Variational Bayesian Optimal Experimental Design](https://mlsanity.com/paper/1903.05480/) (2019)
- [On the Convergence of SGD with Biased Gradients](https://mlsanity.com/paper/2008.00051/) (2020)
- [Likelihood-free inference by ratio estimation](https://mlsanity.com/paper/1611.10242/) (2016)
- [Bayesian Optimization for Likelihood-Free Inference of Simulator-Based Statistical Models](https://mlsanity.com/paper/1501.03291/) (2015)
- [Mining gold from implicit models to improve likelihood-free inference](https://mlsanity.com/paper/1805.12244/) (2018)
- [Bayesian Experimental Design for Implicit Models by Mutual Information Neural Estimation](https://mlsanity.com/paper/2002.08129/) (2020)
- [Sequential Neural Score Estimation: Likelihood-Free Inference with Conditional Score Based Diffusion Models](https://mlsanity.com/paper/2210.04872/) (2022)
- [Truncated proposals for scalable and hassle-free simulation-based inference](https://mlsanity.com/paper/2210.04815/) (2022)
- [Bayesian Synthetic Likelihood](https://mlsanity.com/paper/2305.05120/) (2023)
- [A Trust Crisis In Simulation-Based Inference? Your Posterior Approximations Can Be Unfaithful](https://mlsanity.com/paper/2110.06581/) (2021)
- [Unbiased MLMC stochastic gradient-based optimization of Bayesian experimental designs](https://mlsanity.com/paper/2005.08414/) (2020)
- [On Russian Roulette Estimates for Bayesian Inference with Doubly-Intractable Likelihoods](https://mlsanity.com/paper/1306.4032/) (2013)
- [SUMO: Unbiased Estimation of Log Marginal Probability for Latent Variable Models](https://mlsanity.com/paper/2004.00353/) (2020)
- [Multilevel Monte Carlo estimation of expected information gains](https://mlsanity.com/paper/1811.07546/) (2018)
- [Variational Bayes with Intractable Likelihood](https://mlsanity.com/paper/1503.08621/) (2015)
- [Unbiased Markov chain Monte Carlo with couplings](https://mlsanity.com/paper/1708.03625/) (2017)
- [Validation of Approximate Likelihood and Emulator Models for Computationally Intensive Simulations](https://mlsanity.com/paper/1905.11505/) (2019)

## Abstract (arXiv)

There is a growing interest in studying sequential neural posterior estimation (SNPE) techniques due to their advantages for simulation-based models with intractable likelihoods. The methods aim to learn the posterior from adaptively proposed simulations using neural network-based conditional density estimators. As an SNPE technique, the automatic posterior transformation (APT) method proposed by Greenberg et al. (2019) performs well and scales to high-dimensional data. However, the APT method requires computing the expectation of the logarithm of an intractable normalizing constant, i.e., a nested expectation. Although atomic proposals were used to render an analytical normalizing constant, it remains challenging to analyze the convergence of learning. In this paper, we reformulate APT as a nested estimation problem. Building on this, we construct several multilevel Monte Carlo (MLMC) estimators for the loss function and its gradients to accommodate different scenarios, including two unbiased estimators, and a biased estimator that trades a small bias for reduced variance and controlled runtime and memory usage. We also provide convergence results of stochastic gradient descent to quantify the interaction of the bias and variance of the gradient estimator. Numerical experiments for approximating complex posteriors with multimodality in moderate dimensions are provided to examine the effectiveness of the proposed methods.