[Paper Review] Compositional Score Modeling for Simulation-based Inference

Tomas Geffner, George Papamakarios+1 more|arXiv (Cornell University)|Sep 28, 2022

Model Reduction and Neural Networks4 citations

TL;DR

This paper proposes Factorized Neural Posterior Score Estimation (F-NPSE) and its progressive variant (PF-NPSE), a simulation-based inference method that models scores of posterior distributions induced by individual observations and combines them to sample from the joint posterior of multiple observations. The approach enables sample-efficient, scalable inference without relying on MCMC or variational inference, achieving accurate posterior approximation with minimal simulator calls per training case.

ABSTRACT

Neural Posterior Estimation methods for simulation-based inference can be ill-suited for dealing with posterior distributions obtained by conditioning on multiple observations, as they tend to require a large number of simulator calls to learn accurate approximations. In contrast, Neural Likelihood Estimation methods can handle multiple observations at inference time after learning from individual observations, but they rely on standard inference methods, such as MCMC or variational inference, which come with certain performance drawbacks. We introduce a new method based on conditional score modeling that enjoys the benefits of both approaches. We model the scores of the (diffused) posterior distributions induced by individual observations, and introduce a way of combining the learned scores to approximately sample from the target posterior distribution. Our approach is sample-efficient, can naturally aggregate multiple observations at inference time, and avoids the drawbacks of standard inference methods.

Motivation & Objective

• To address the inefficiency of Neural Posterior Estimation (NPE) when conditioning on multiple observations, which requires many simulator calls per training case.
• To overcome the limitations of Neural Likelihood Estimation (NLE), which depends on standard inference methods like MCMC or variational inference that introduce approximation errors and fail on multimodal posteriors.
• To develop a method that combines the sample efficiency of NPE with the multi-observation capability of NLE, while avoiding reliance on standard inference algorithms.
• To enable posterior approximation for arbitrary numbers of i.i.d. observations at inference time with low simulation cost during training.

Proposed method

• The method models the score (gradient of log-density) of the posterior distribution induced by each individual observation using a conditional score network.
• It introduces a novel forward process that diffuses the posterior of each observation separately, enabling compositional aggregation of scores at inference time.
• At inference, the scores from individual observations are combined additively to approximate the score of the joint posterior distribution.
• Sampling from the joint posterior is performed using annealed Langevin dynamics, which leverages the learned score network to generate samples in a score-based generative process.
• An alternative, hyperparameter-free sampling method is also proposed, which runs in O(T) time and avoids step-size tuning.
• The approach reparameterizes the model to ensure posterior estimates remain within bounded support, with kernel density estimation used for visualization.

Experimental results

Research questions

• RQ1 Can we construct a simulation-based inference method that efficiently handles multiple observations at inference time while requiring only a single simulator call per training case?
• RQ2 Can we avoid the performance pitfalls of standard inference methods like MCMC and variational inference in posterior approximation for complex, multimodal posteriors?
• RQ3 Does compositional score modeling—aggregating scores from individual observations—yield accurate joint posterior approximations without retraining?
• RQ4 How does the trade-off between sample efficiency and error accumulation affect performance when varying the number of observations used in the score aggregation process?

Key findings

• F-NPSE and PF-NPSE achieve accurate posterior approximation for multiple observations using only a single simulator call per training case, significantly improving sample efficiency over NPE.
• The best performance is typically achieved with intermediate values of m (e.g., 3 or 6), indicating that both F-NPSE (m=1) and NPSE (m=n) are suboptimal, highlighting a trade-off between error accumulation and efficiency.
• The method is robust to hyperparameter choices: conservative parameterization of the score network and variations in Langevin sampler step count (5–10 steps per noise level) have minimal impact on performance.
• On the Weinberg simulator benchmark, F-NPSE successfully concentrates the posterior around the true parameter value as the number of observations increases, demonstrating correct learning and aggregation.
• An alternative sampling method without hyperparameters is proposed and shown to be effective, though it sometimes underperforms annealed Langevin dynamics, suggesting room for further optimization.