Source-linked AI summary
Mining gold from implicit models to improve likelihood-free inference
Johann Brehmer, Gilles Louppe, Juan Pavez, Kyle Cranmer
TL;DR
Implicit simulators can generate data while leaving the likelihood needed for inference intractable and expensive to estimate. The paper extracts joint likelihood-ratio and score information from simulator trajectories, uses it to train neural surrogates and local summaries, and reports more sample-efficient, higher-precision inference than established methods. These gains are strongest for small parameter changes, while local approximations weaken farther from their reference point.
Problem
Simulators often define implicit models with intractable likelihoods, making likelihood-free inference computationally demanding when simulator samples are expensive.
Method
The paper extracts joint likelihood ratios and joint scores from latent simulator trajectories, then uses augmented losses and local score-based summaries to train neural inference surrogates.
Results
The experiments show more precise likelihood-ratio estimation and parameter measurements with less training data than established methods.
Takeaways & Limitations
Simulator-derived augmented information can improve the sample efficiency and quality of likelihood-free inference when it is available for extraction.
Takeaways & Limitations
SALLY and SALLINO work best near the reference point; their local approximation can deteriorate farther away, reducing sensitivity and weakening bounds.
Abstract
from arXiv · showhide
Simulators often provide the best description of real-world phenomena. However, they also lead to challenging inverse problems because the density they implicitly define is often intractable. We present a new suite of simulation-based inference techniques that go beyond the traditional Approximate Bayesian Computation approach, which struggles in a high-dimensional setting, and extend methods that use surrogate models based on neural networks. We show that additional information, such as the joint likelihood ratio and the joint score, can often be extracted from simulators and used to augment the training data for these surrogate models. Finally, we demonstrate that these new techniques are more sample efficient and provide higher-fidelity inference than traditional methods.
1 Introduction
Scientific simulators can generate observations from mechanistic models, but their implicit likelihoods are often intractable, making parameter inference difficult and sample efficiency important. The paper introduces techniques that use additional simulator information to improve neural likelihood and likelihood-ratio surrogates.
- Simulators generate observations from mechanistic stochastic processes but typically do not provide a tractable likelihood p(x|θ).
- Intractable implicit models create inverse problems for inferring parameters θ from observed data across fields including particle physics, epidemiology, and population genetics.
- ABC and neural density estimation provide likelihood-free inference approaches, but computationally expensive simulators make sample efficiency important.
- The proposed techniques improve training efficiency for neural surrogates estimating likelihoods or likelihood ratios.
- The methods extract additional information characterizing the simulator’s latent process and use augmented-data loss functions to improve inference quality.
2 Related work
Likelihood-free inference includes direct simulator comparison methods and surrogate-model methods. This work extends surrogate training by extracting augmented information from simulators and using it to learn likelihoods, ratios, and locally optimal summaries more efficiently.
- Likelihood-free methods either compare observations directly with simulator outputs or train tractable surrogate models for inference.
- Likelihood-ratio methods: The likelihood ratio trick estimates r(x|θ0, θ1) by classifying equal-sized samples drawn from p(x|θ0) and p(x|θ1).
- Likelihood-ratio methods: The optimal classifier decision function can be inverted to recover the likelihood ratio, while calibration can restore the relation when the classifier is only monotonic in that ratio.
- Neural density estimation: Neural density estimation trains neural surrogates for likelihoods or posteriors, including normalizing flows and autoregressive models.
- Novel contributions: The paper’s central contribution is extracting simulator information beyond observable samples and designing loss functions that use it to learn likelihood surrogates more efficiently.
- Novel contributions: The augmented data also defines locally optimal summary statistics usable with density estimation or Approximate Bayesian Computation.
- Novel contributions: The work generalizes earlier particle-physics applications to broader scientific simulators under weaker assumptions and introduces SCANDAL with experimental results.
3 Extracting more information from the simulator
The simulator’s latent execution trace exposes tractable conditional factors even when marginalizing over the latent space makes p(x|θ) intractable. The paper uses this structure to extract joint scores and likelihood ratios that augment simulator samples for inference.
- A simulator generates latent states z_i from conditional densities and then produces an observation x from a final conditional density.
- The marginal likelihood becomes intractable when the latent space is enormous, with real-world trajectories sometimes containing millions of latent variables.
- Although inference often uses only observations x, simulators commonly expose latent variables z and tractable conditional distributions for each stochastic step and final output.
- For each simulated sample, the joint score can be calculated from the simulator’s latent process.
- The joint score is accumulated from parameter gradients of latent-step log probabilities as the simulation follows its realized trajectory.
- The joint likelihood ratio compares how likely the same latent execution trace is under two parameter settings.
- Generalized Galton board: In the generalized Galton board, different latent paths can produce the same observed bin, requiring an explicit sum over paths as the lattice grows.
- Generalized Galton board: The toy simulator contains non-differentiable control flow, but its joint score can still be computed by accumulating factors during execution.
4 Learning from augmented data
The paper shows how simulator-derived joint likelihood ratios and joint scores can be transformed into observable-space quantities for training likelihood and likelihood-ratio surrogates. These augmented-data methods also support local score-based inference, with strong advantages for nearby parameter changes but reduced usefulness farther away.
- Key idea: Joint likelihood ratios and joint scores can be regressed on observables to recover the intractable likelihood ratio and score.The relevant minimum-mean-squared-error functionals are minimized by r(x|θ0, θ1) and t(x|θ0), respectively.
- Surrogate learning: The resulting augmented-data techniques train surrogate models for likelihoods or likelihood ratios, including parameterized estimators across θ.The parameter values can be supplied as additional inputs so one model represents dependence across the parameter space.
- Likelihood-ratio methods: RASCAL improves likelihood-ratio regression by combining ratio loss with a score loss, while CASCAL applies the same strategy to classifier-based likelihood-ratio estimation.For RASCAL, the predicted score is obtained by differentiating the log likelihood-ratio surrogate; CASCAL differentiates the calibrated classifier-derived ratio.
- Likelihood estimation: SCANDAL augments differentiable conditional neural density estimators with score information and estimates the likelihood itself rather than only its ratio.Depending on the architecture, the resulting surrogate can also provide a generative model.
- Local methods: A local expansion around θref makes the score vector sufficient near the reference point and allows score-based summary statistics to be extracted from a single simulator setting.SALLINO further projects the score onto the parameter-pair direction, enabling univariate density estimation rather than multivariate density estimation.
- Scope and limitations: Augmented data is especially useful for small parameter changes, whereas non-overlapping joint distributions and invalid local approximations limit performance for distant parameter points.For large parameter separations, the joint likelihood ratio can have high variance and the score may no longer characterize likelihood ratios.
5 Experiments
The experiments evaluate augmented-data inference techniques across Galton board, Lotka–Volterra, and particle-physics simulators. Across these settings, joint score and ratio information generally improves sample efficiency, while local methods can plateau away from a reference point.
- Generalized Galton board: SCANDAL and RASCAL are dramatically more sample efficient than neural density estimation and the likelihood ratio trick on the generalized Galton board.ROLR improves upon neural density estimation and reaches the same asymptotic error as SCANDAL more slowly.
- Lotka-Volterra model: The Lotka–Volterra study evaluates likelihood and ratio estimation using augmented observations that include the joint score and joint likelihood ratio.The evaluation focuses on nearby parameter points because densities often do not overlap across a wide prior.
- Lotka-Volterra model: Density and likelihood-ratio estimation exhibit a trade-off: MAF performs well for densities, whereas SCANDAL is more sample efficient for likelihood ratios.The score term’s variance degrades SCANDAL’s density-estimation performance, especially for larger α.
- Particle physics: All new inference techniques outperform the traditional histogram method when the training samples are sufficiently large.The comparison uses mean squared error of approximate log likelihood ratios in the particle-physics experiment.
- Particle physics: RASCAL reduces the training data required for good performance by two orders of magnitude relative to CARL.RASCAL uses both joint ratio and joint score information, while CARL uses only simulator samples.
6 Conclusions
The paper introduces simulation-based inference methods that exploit likelihood-ratio and score information extracted from stochastic simulators. Across three experiments, these methods estimate likelihood ratios precisely and enable more precise parameter measurements with less training data than established methods.
- Conclusions: The methods estimate likelihoods or likelihood ratios from simulator-generated data when the likelihood is only implicitly defined.They extend established approaches that use only simulator samples.
- Conclusions: Joint likelihood ratios and joint scores conditioned on simulator latent variables can often be extracted and used as additional training information.These quantities characterize trajectories through the data-generation process.
- Conclusions: The additional information defines loss functionals minimized by the likelihood or likelihood ratio, improving neural-network training efficiency and inference quality.A separate local approach uses the score as a sufficient statistic near a reference parameter value.
- Conclusions: Across three experiments, the techniques precisely estimate likelihood ratios and enable parameter measurements with higher precision and less training data than established methods.The augmented data can also improve neural density and likelihood-ratio estimators or define locally optimal summary statistics for ABC.
- Conclusions: The work motivates tools that automatically generate joint scores and ratios from simulator code, with a proof-of-principle implementation based on PYRO.The proposed direction builds on probabilistic programming and automatic differentiation.