Welcome to sbi!

Welcome to sbi!#

sbi is a Python package for simulation-based inference, designed to meet the needs of both researchers and practitioners. Whether you need fine-grained control or an easy-to-use interface, sbi has you covered.

With sbi, you can perform parameter inference using Bayesian inference: Given a simulator that models a real-world process, SBI estimates the full posterior distribution over the simulator’s parameters based on observed data. This distribution indicates the most likely parameter values while additionally quantifying uncertainty and revealing potential interactions between parameters.

sbi provides access to simulation-based inference methods via a user-friendly interface:

import torch
from sbi.inference import NPE

# define shifted Gaussian simulator.
def simulator(θ): return θ + torch.randn_like(θ)
# draw parameters from Gaussian prior.
θ = torch.randn(1000, 2)
# simulate data
x = simulator(θ)

# choose sbi method and train
inference = NPE()
inference.append_simulations(θ, x).train()

# do inference given observed data
x_o = torch.ones(2)
posterior = inference.build_posterior()
samples = posterior.sample((1000,), x=x_o)

Overview#

To get started, install the sbi package with:

python -m pip install sbi

for more advanced install options, see our Install Guide.

Then, check out our material:

🧠 Tutorials

Step-by-step introductions.

Tutorials
🛠 How-To Guides

Practical recipes for common tasks.

How-to guide
📚 API Reference

Full documentation of modules and functions.

API reference

Motivation and approach#

Many areas of science and engineering make extensive use of complex, stochastic, numerical simulations to describe the structure and dynamics of the processes being investigated.

A key challenge in simulation-based science is constraining these simulation models’ parameters, which are interpretable quantities, with observational data. Bayesian inference provides a general and powerful framework to invert the simulators, i.e. describe the parameters that are consistent both with empirical data and prior knowledge.

In the case of simulators, a key quantity required for statistical inference, the likelihood of observed data given parameters, \(\mathcal{L}(\theta) = p(x_o|\theta)\), is typically intractable, rendering conventional statistical approaches inapplicable.

sbi implements powerful machine-learning methods that address this problem. Roughly, these algorithms can be categorized as:

  • Neural Posterior Estimation (amortized NPE and sequential SNPE),

  • Neural Likelihood Estimation ((S)NLE), and

  • Neural Ratio Estimation ((S)NRE).

Depending on the characteristics of the problem, e.g. the dimensionalities of the parameter space and the observation space, one of the methods will be more suitable.

Conceptual diagram for simulation-based inference. It depicts a model, prior distribution, simulated data, neural density estimator, and posterior distribution. The process includes a feedback loop using the estimated posterior distribution to adaptively generate additional informative simulations. The diagram also illustrates consistent and inconsistent samples.

Goal: Algorithmically identify mechanistic models that are consistent with data.

Each of the methods above needs three inputs: A candidate mechanistic model, prior knowledge or constraints on model parameters, and observational data (or summary statistics thereof).

The methods then proceed by

  1. sampling parameters from the prior followed by simulating synthetic data from these parameters,

  2. learning the (probabilistic) association between data (or data features) and underlying parameters, i.e. to learn statistical inference from simulated data. How this association is learned differs between the above methods, but all use deep neural networks.

  3. This learned neural network is then applied to empirical data to derive the full space of parameters consistent with the data and the prior, i.e. the posterior distribution. The posterior assigns high probability to parameters that are consistent with both the data and the prior, and low probability to inconsistent parameters. While NPE directly learns the posterior distribution, NLE and NRE need an extra MCMC sampling step to construct a posterior.

  4. If needed, an initial estimate of the posterior can be used to adaptively generate additional informative simulations.

Cranmer, Brehmer, Louppe (2020)

Implemented algorithms#

sbi implements a variety of amortized and sequential SBI methods.

Amortized methods return a posterior that can be applied to many different observations without retraining (e.g., NPE), whereas sequential methods focus the inference on one particular observation to be more simulation-efficient (e.g., SNPE).

Below, we list all implemented methods and their corresponding publications. For usage in sbi, see the Inference API reference and the tutorial on implemented methods.

Posterior estimation ((S)NPE)#

  • Fast ε-free Inference of Simulation Models with Bayesian Conditional Density Estimation by Papamakarios & Murray (NeurIPS 2016) PDF

  • Flexible statistical inference for mechanistic models of neural dynamics by Lueckmann, Goncalves, Bassetto, Öcal, Nonnenmacher & Macke (NeurIPS 2017) PDF

  • Automatic posterior transformation for likelihood-free inference by Greenberg, Nonnenmacher & Macke (ICML 2019) PDF

  • BayesFlow: Learning complex stochastic models with invertible neural networks by Radev, S. T., Mertens, U. K., Voss, A., Ardizzone, L., & Köthe, U. (IEEE transactions on neural networks and learning systems 2020) Paper

  • Truncated proposals for scalable and hassle-free simulation-based inference by Deistler, Goncalves & Macke (NeurIPS 2022) Paper

  • Flow matching for scalable simulation-based inference by Dax, M., Wildberger, J., Buchholz, S., Green, S. R., Macke, J. H., & Schölkopf, B. (NeurIPS, 2023) Paper

  • Compositional Score Modeling for Simulation-Based Inference by Geffner, T., Papamakarios, G., & Mnih, A. (ICML 2023) Paper

Likelihood-estimation ((S)NLE)#

  • Sequential neural likelihood: Fast likelihood-free inference with autoregressive flows by Papamakarios, Sterratt & Murray (AISTATS 2019) PDF

  • Variational methods for simulation-based inference by Glöckler, Deistler, Macke (ICLR 2022) Paper

  • Flexible and efficient simulation-based inference for models of decision-making by Boelts, Lueckmann, Gao, Macke (Elife 2022) Paper

Likelihood-ratio-estimation ((S)NRE)#

  • Likelihood-free MCMC with Amortized Approximate Likelihood Ratios by Hermans, Begy & Louppe (ICML 2020) PDF

  • On Contrastive Learning for Likelihood-free Inference by Durkan, Murray & Papamakarios (ICML 2020) PDF

  • Towards Reliable Simulation-Based Inference with Balanced Neural Ratio Estimation by Delaunoy, Hermans, Rozet, Wehenkel & Louppe (NeurIPS 2022) PDF

  • Contrastive Neural Ratio Estimation by Benjamin Kurt Miller, Christoph Weniger & Patrick Forré (NeurIPS 2022) PDF

Diagnostics#

  • Simulation-based calibration by Talts, Betancourt, Simpson, Vehtari, Gelman (arXiv 2018) Paper

  • Expected coverage (sample-based) as computed in Deistler, Goncalves, & Macke (NeurIPS 2022) Paper and in Rozet & Louppe Paper

  • Local C2ST by Linhart, Gramfort & Rodrigues (NeurIPS 2023) Paper

  • TARP by Lemos, Coogan, Hezaveh & Perreault-Levasseur (ICML 2023) Paper

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