Guide to neural SBI methods

This page maps each method in neuralsbi to the paper it comes from and the function that runs it. For a worked example, read vignette("neuralsbi") instead.

Key papers

  • Cranmer et al. (2020) survey the field and set the NPE/NLE/NRE naming used here.
  • Lueckmann et al. (2021) compare these methods on a shared set of tasks.
  • Deistler et al. (2025) give the applied workflow, from simulator and prior through training to diagnostics.

Neural posterior estimation (NPE)

  • Papamakarios and Murray (2016) introduced neural posterior estimation, training a conditional density estimator on prior draws so the fitted network is the posterior.
  • npe() implements the single-round amortized case. Conditioning on a new observation costs one forward pass through posterior(), then sample().

Sequential neural posterior estimation

  • Greenberg et al. (2019) introduced automatic posterior transformation, which reweights the loss when draws come from a proposal rather than the prior.
  • Deistler et al. (2022) introduced truncated proposals, which restrict the prior to the current posterior’s highest-probability region and leave the plain NPE loss valid.
  • npe_sequential() implements truncated proposals. The result is not amortized and holds only near the observation it was trained on.

Neural likelihood estimation (NLE)

  • Papamakarios et al. (2018) introduced sequential neural likelihood, learning the surrogate likelihood \(q(x \mid \theta)\) and recovering the posterior through Bayes’ rule.
  • nle() implements the single-round case, so repeated independent observations are a sum over one trained density. Evaluate it with log_lik() and sample it through posterior().
  • Neal (2003) introduced slice sampling, which posterior() uses by default. stan_code() writes the estimator out as Stan code instead.

Neural likelihood-ratio estimation (NRE)

  • Hermans et al. (2019) introduced amortized approximate likelihood-ratio estimation, training a binary classifier for \(r(\theta, x) = p(x \mid \theta) / p(x)\) rather than either density.
  • Durkan et al. (2020) introduced the atomic contrastive objective, which scores the true parameter against contrasting draws from the same minibatch.
  • nre() implements the atomic objective with 10 atoms per simulation by default. Evaluate it with log_ratio() and sample it through posterior().

Density estimators

Select one through density_estimator= in npe() or nle(). A closed-form "linear_gaussian" option is also available: it needs no torch, is exact for linear-Gaussian models, and serves as a test oracle.

Mixture density network (MDN)

  • Bishop introduced mixture density networks in the 1994 technical report Mixture Density Networks, which has no DOI.
  • Papamakarios and Murray (2016) used an MDN as the density estimator for NPE.
  • npe(density_estimator = "mdn") maps x to the weights, means, and full covariances of a Gaussian mixture over \(\theta\). n_components and hidden size it.

Masked autoregressive flow (MAF)

  • Germain et al. (2015) introduced the MADE masking scheme for autoregressive density estimation.
  • Papamakarios et al. (2017) introduced masked autoregressive flows, stacking affine autoregressive transforms to a standard-normal base.
  • npe(density_estimator = "maf") is the default. n_transforms sets the depth.

Neural spline flow (NSF)

  • Durkan et al. (2019) introduced neural spline flows, replacing the affine transform with a monotonic rational-quadratic spline for sharply non-Gaussian posteriors.
  • npe(density_estimator = "nsf") implements the autoregressive form. n_bins and tail_bound control the spline.

Diagnostics

Simulation-based calibration (SBC)

  • Talts et al. (2018) introduced simulation-based calibration, which ranks the true parameter among posterior draws over many prior draws. Calibrated posteriors give uniform ranks.
  • sbc() computes the ranks, one parameter at a time, and plot_sbc() displays them.

Expected coverage

  • Hermans et al. (2021) showed that published SBI posteriors are often overconfident, and made coverage the check that exposes it.
  • expected_coverage() turns the SBC ranks into nominal against empirical coverage at each credible level, and plot_coverage() displays them. Calibrated posteriors lie on the diagonal.

TARP

  • Lemos et al. (2023) introduced tests of accuracy with random points, which measure coverage of distance-based credible regions around random reference points. This is a joint test, so it catches a posterior with calibrated marginals but wrong correlations.
  • tarp() runs the test and plot_tarp() displays it.

Classifier two-sample test (C2ST)

  • Lopez-Paz and Oquab (2016) introduced the classifier two-sample test, which trains a classifier to tell two sets of draws apart. Accuracy near 0.5 means they are indistinguishable.
  • c2st() runs the procedure Lueckmann et al. (2021) use in sbibm, so its numbers are comparable with published benchmark results: a two-hidden-layer ReLU network of 10 * d units per layer, 5-fold cross-validated accuracy, both sample sets z-scored by the moments of the first. The network trains on torch, so this classifier needs torch installed. classifier = "logistic" swaps in cross-validated logistic regression, which needs no torch and is faster, but is linear: it sees a shift in location and is close to blind to a difference in spread or dependence.

References

Cranmer, Kyle, Johann Brehmer, and Gilles Louppe. 2020. “The Frontier of Simulation-Based Inference.” Proceedings of the National Academy of Sciences 117 (48): 30055–62. https://doi.org/10.1073/pnas.1912789117.
Deistler, Michael, Jan Boelts, Peter Steinbach, et al. 2025. Simulation-Based Inference: A Practical Guide. https://doi.org/10.48550/arXiv.2508.12939.
Deistler, Michael, Pedro J. Goncalves, and Jakob H. Macke. 2022. Truncated Proposals for Scalable and Hassle-Free Simulation-Based Inference. https://doi.org/10.48550/arXiv.2210.04815.
Durkan, Conor, Artur Bekasov, Iain Murray, and George Papamakarios. 2019. Neural Spline Flows. https://doi.org/10.48550/arXiv.1906.04032.
Durkan, Conor, Iain Murray, and George Papamakarios. 2020. On Contrastive Learning for Likelihood-Free Inference. https://doi.org/10.48550/arXiv.2002.03712.
Germain, Mathieu, Karol Gregor, Iain Murray, and Hugo Larochelle. 2015. MADE: Masked Autoencoder for Distribution Estimation. https://doi.org/10.48550/arXiv.1502.03509.
Greenberg, David S., Marcel Nonnenmacher, and Jakob H. Macke. 2019. Automatic Posterior Transformation for Likelihood-Free Inference. https://doi.org/10.48550/arXiv.1905.07488.
Hermans, Joeri, Volodimir Begy, and Gilles Louppe. 2019. Likelihood-Free MCMC with Amortized Approximate Ratio Estimators. https://doi.org/10.48550/arXiv.1903.04057.
Hermans, Joeri, Arnaud Delaunoy, François Rozet, Antoine Wehenkel, Volodimir Begy, and Gilles Louppe. 2021. A Trust Crisis in Simulation-Based Inference? Your Posterior Approximations Can Be Unfaithful. https://doi.org/10.48550/arXiv.2110.06581.
Lemos, Pablo, Adam Coogan, Yashar Hezaveh, and Laurence Perreault-Levasseur. 2023. Sampling-Based Accuracy Testing of Posterior Estimators for General Inference. https://doi.org/10.48550/arXiv.2302.03026.
Lopez-Paz, David, and Maxime Oquab. 2016. Revisiting Classifier Two-Sample Tests. https://doi.org/10.48550/arXiv.1610.06545.
Lueckmann, Jan-Matthis, Jan Boelts, David S. Greenberg, Pedro J. Gonçalves, and Jakob H. Macke. 2021. Benchmarking Simulation-Based Inference. https://doi.org/10.48550/arXiv.2101.04653.
Neal, Radford M. 2003. “Slice Sampling.” The Annals of Statistics 31 (3): 705–67. https://doi.org/10.1214/aos/1056562461.
Papamakarios, George, and Iain Murray. 2016. Fast \(\epsilon\)-Free Inference of Simulation Models with Bayesian Conditional Density Estimation. https://doi.org/10.48550/arXiv.1605.06376.
Papamakarios, George, Theo Pavlakou, and Iain Murray. 2017. Masked Autoregressive Flow for Density Estimation. https://doi.org/10.48550/arXiv.1705.07057.
Papamakarios, George, David C. Sterratt, and Iain Murray. 2018. Sequential Neural Likelihood: Fast Likelihood-Free Inference with Autoregressive Flows. https://doi.org/10.48550/arXiv.1805.07226.
Talts, Sean, Michael Betancourt, Daniel Simpson, Aki Vehtari, and Andrew Gelman. 2018. Validating Bayesian Inference Algorithms with Simulation-Based Calibration. https://doi.org/10.48550/arXiv.1804.06788.