Package: bayesflow
BayesFlowWelcome to our BayesFlow library for efficient simulation-based Bayesian workflows! Our library enables users to create specialized neural networks foramortized Bayesian inference, which repay users with rapid statistical inference after a potentially longer simulation-based training phase.For starters, check out some of our walk-through notebooks:Quickstart amortized posterior estimationTackling strange bimodal distributionsDetecting model misspecification in posterior inferencePrincipled Bayesian workflow for cognitive modelsPosterior estimation for ODEsPosterior estimation for SIR-like modelsModel comparison for cognitive modelsHierarchical model comparison for cognitive modelsProject DocumentationThe project documentation is available athttps://bayesflow.orgInstallationSeeINSTALL.rstfor installation instructions.Conceptual OverviewA cornerstone idea of amortized Bayesian inference is to employ generative
neural networks for parameter estimation, model comparison, and model validation
when working with intractable simulators whose behavior as a whole is too
complex to be described analytically. The figure below presents a higher-level
overview of neurally bootstrapped Bayesian inference.Getting Started: Parameter EstimationThe core functionality of BayesFlow is amortized Bayesian posterior estimation, as described in our paper:Radev, S. T., Mertens, U. K., Voss, A., Ardizzone, L., & Köthe, U. (2020).
BayesFlow: Learning complex stochastic models with invertible neural networks.IEEE Transactions on Neural Networks and Learning Systems, available
for free at:https://arxiv.org/abs/2003.06281.However, since then, we have substantially extended the BayesFlow library such that
it is now much more general and cleaner than what we describe in the above paper.Minimal ExampleimportnumpyasnpimportbayesflowasbfTo introduce you to the basic workflow of the library, let's consider
a simple 2D Gaussian model, from which we want to obtain
posterior inference.  We assume a Gaussian simulator (likelihood)
and a Gaussian prior for the means of the two components,
which are our only model parameters in this example:defsimulator(theta,n_obs=50,scale=1.0):returnnp.random.default_rng().normal(loc=theta,scale=scale,size=(n_obs,theta.shape[0]))defprior(D=2,mu=0.,sigma=1.0):returnnp.random.default_rng().normal(loc=mu,scale=sigma,size=D)Then, we connect thepriorwith thesimulatorusing aGenerativeModelwrapper:generative_model=bf.simulation.GenerativeModel(prior,simulator)Next, we create our BayesFlow setup consisting of a summary and an inference network:summary_net=bf.networks.DeepSet()inference_net=bf.networks.InvertibleNetwork(num_params=2)amortized_posterior=bf.amortizers.AmortizedPosterior(inference_net,summary_net)Finally, we connect the networks with the generative model via aTrainerinstance:trainer=bf.trainers.Trainer(amortizer=amortized_posterior,generative_model=generative_model,memory=True)We are now ready to train an amortized posterior approximator. For instance,
to run online training, we simply call:losses=trainer.train_online(epochs=10,iterations_per_epoch=500,batch_size=32)Before inference, we can use simulation-based calibration (SBC,https://arxiv.org/abs/1804.06788) to check the computational faithfulness of
the model-amortizer combination:fig=trainer.diagnose_sbc_histograms()The histograms are roughly uniform and lie within the expected range for
well-calibrated inference algorithms as indicated by the shaded gray areas.
Accordingly, our amortizer seems to have converged to the intended target.Amortized inference on new (real or simulated) data is then easy and fast.
For example, we can simulate 200 new data sets and generate 500 posterior draws
per data set:new_sims=trainer.configurator(generative_model(200))posterior_draws=amortized_posterior.sample(new_sims,n_samples=500)We can then quickly inspect the how well the model can recover its parameters
across the simulated data sets.fig=bf.diagnostics.plot_recovery(posterior_draws,new_sims['parameters'])For any individual data set, we can also compare the parameters' posteriors with
their corresponding priors:fig=bf.diagnostics.plot_posterior_2d(posterior_draws[0],prior=generative_model.prior)We see clearly how the posterior shrinks relative to the prior for both
model parameters as a result of conditioning on the data.References and Further ReadingRadev, S. T., Mertens, U. K., Voss, A., Ardizzone, L., & Köthe, U. (2020).
BayesFlow: Learning complex stochastic models with invertible neural networks.IEEE Transactions on Neural Networks and Learning Systems, 33(4), 1452-1466.Radev, S. T., Graw, F., Chen, S., Mutters, N. T., Eichel, V. M., Bärnighausen, T., & Köthe, U. (2021).
OutbreakFlow: Model-based Bayesian inference of disease outbreak dynamics with invertible neural networks and its application to the COVID-19 pandemics in Germany.PLoS computational biology, 17(10), e1009472.Bieringer, S., Butter, A., Heimel, T., Höche, S., Köthe, U., Plehn, T., & Radev, S. T. (2021).
Measuring QCD splittings with invertible networks.SciPost Physics, 10(6), 126.von Krause, M., Radev, S. T., & Voss, A. (2022).
Mental speed is high until age 60 as revealed by analysis of over a million participants.Nature Human Behaviour, 6(5), 700-708.Model MisspecificationWhat if we are dealing with misspecified models? That is, how faithful is our
amortized inference if the generative model is a poor representation of reality?
A modified loss function optimizes the learned summary statistics towards a unit
Gaussian and reliably detects model misspecification during inference time.In order to use this method, you should only provide thesummary_loss_funargument
to theAmortizedPosteriorinstance:amortized_posterior=bf.amortizers.AmortizedPosterior(inference_net,summary_net,summary_loss_fun='MMD')The amortizer knows how to combine its losses and you can inspect the summary space for outliers during inference.References and Further ReadingSchmitt, M., Bürkner P. C., Köthe U., & Radev S. T. (2022). Detecting Model
Misspecification in Amortized Bayesian Inference with Neural Networks.ArXiv
preprint, available for free at:https://arxiv.org/abs/2112.08866Model ComparisonBayesFlow can not only be used for parameter estimation, but also to perform approximate Bayesian model comparison via posterior model probabilities or Bayes factors.
Let's extend the minimal example from before with a second model $M_2$ that we want to compare with our original model $M_1$:defsimulator(theta,n_obs=50,scale=1.0):returnnp.random.default_rng().normal(loc=theta,scale=scale,size=(n_obs,theta.shape[0]))defprior_m1(D=2,mu=0.,sigma=1.0):returnnp.random.default_rng().normal(loc=mu,scale=sigma,size=D)defprior_m2(D=2,mu=2.,sigma=1.0):returnnp.random.default_rng().normal(loc=mu,scale=sigma,size=D)For the purpose of this illustration, the two toy models only differ with respect to their prior specification ($M_1: \mu = 0, M_2: \mu = 2$). We create both models as before and use aMultiGenerativeModelwrapper to combine them in ameta_model:model_m1=bf.simulation.GenerativeModel(prior_m1,simulator,simulator_is_batched=False)model_m2=bf.simulation.GenerativeModel(prior_m2,simulator,simulator_is_batched=False)meta_model=bf.simulation.MultiGenerativeModel([model_m1,model_m2])Next, we construct our neural network with aPMPNetworkfor approximating posterior model probabilities:summary_net=bf.networks.DeepSet()probability_net=bf.networks.PMPNetwork(num_models=2)amortized_bmc=bf.amortizers.AmortizedModelComparison(probability_net,summary_net)We combine all previous steps with aTrainerinstance and train the neural approximator:trainer=bf.trainers.Trainer(amortizer=amortized_bmc,generative_model=meta_model)losses=trainer.train_online(epochs=3,iterations_per_epoch=100,batch_size=32)Let's simulate data sets from our models to check our networks' performance:sims=trainer.configurator(meta_model(5000))When feeding the data to our trained network, we almost immediately obtain posterior model probabilities for each of the 5000 data sets:model_probs=amortized_bmc.posterior_probs(sims)How good are these predicted probabilities in the closed world? We can have a look at the calibration:cal_curves=bf.diagnostics.plot_calibration_curves(sims["model_indices"],model_probs)Our approximator shows excellent calibration, with the calibration curve being closely aligned to the diagonal, an expected calibration error (ECE) near 0 and most predicted probabilities being certain of the model underlying a data set. We can further assess patterns of misclassification with a confusion matrix:conf_matrix=bf.diagnostics.plot_confusion_matrix(sims["model_indices"],model_probs)For the vast majority of simulated data sets, the "true" data-generating model is correctly identified. With these diagnostic results backing us up, we can proceed and apply our trained network to empirical data.BayesFlow is also able to conduct model comparison for hierarchical models. See thistutorial notebookfor an introduction to the associated workflow.References and Further ReadingRadev S. T., D’Alessandro M., Mertens U. K., Voss A., Köthe U., & Bürkner P.
C. (2021). Amortized Bayesian Model Comparison with Evidental Deep Learning.IEEE Transactions on Neural Networks and Learning Systems.
doi:10.1109/TNNLS.2021.3124052 available for free at:https://arxiv.org/abs/2004.10629Schmitt, M., Radev, S. T., & Bürkner, P. C. (2022). Meta-Uncertainty in
Bayesian Model Comparison. InInternational Conference on Artificial Intelligence
and Statistics, 11-29, PMLR, available for free at:https://arxiv.org/abs/2210.07278Elsemüller, L., Schnuerch, M., Bürkner, P. C., & Radev, S. T. (2023). A Deep
Learning Method for Comparing Bayesian Hierarchical Models.ArXiv preprint,
available for free at:https://arxiv.org/abs/2301.11873Likelihood EmulationIn order to learn the exchangeable (i.e., permutation invariant) likelihood from the minimal example instead of the posterior, you may use theAmortizedLikelihoodwrapper:likelihood_net=bf.networks.InvertibleNetwork(num_params=2)amortized_likelihood=bf.amortizers.AmortizedLikelihood(likelihood_net)This wrapper can interact with aTrainerinstance in the same way as theAmortizedPosterior. Finally, you can also learn the likelihood and the posteriorsimultaneouslyby using theAmortizedPosteriorLikelihoodwrapper and choosing your preferred training scheme:joint_amortizer=bf.amortizers.AmortizedPosteriorLikelihood(amortized_posterior,amortized_likelihood)Learning both densities enables us to approximate marginal likelihoods or perform approximate leave-one-out cross-validation (LOO-CV) for prior or posterior predictive model comparison, respectively.References and Further ReadingRadev, S. T., Schmitt, M., Pratz, V., Picchini, U., Köthe, U., & Bürkner, P.-C. (2023).
JANA: Jointly amortized neural approximation of complex Bayesian models.Proceedings of the Thirty-Ninth Conference on Uncertainty in Artificial Intelligence, 216, 1695-1706.
(arXiv)(PMLR)SupportThis work is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy -– EXC-2181 - 390900948 (the Heidelberg Cluster of Excellence STRUCTURES) and -- EXC-2075 - 390740016 (the Stuttgart Cluster of Excellence SimTech), the Informatics for Life initiative funded by the Klaus Tschira Foundation, and Google Cloud through the Academic Research Grants program.Citing BayesFlowYou can cite BayesFlow along the lines of:We approximated the posterior with neural posterior estimation and learned summary statistics (NPE; Radev et al., 2020), as implemented in the BayesFlow software for amortized Bayesian workflows (Radev et al., 2023b).We approximated the likelihood with neural likelihood estimation (NLE; Papamakarios et al., 2019) without hand-cafted summary statistics, as implemented in the BayesFlow software for amortized Bayesian workflows (Radev et al., 2023b).We performed simultaneous posterior and likelihood estimation with jointly amortized neural approximation (JANA; Radev et al., 2023a), as implemented in the BayesFlow software for amortized Bayesian workflows (Radev et al., 2023b).Radev, S. T., Schmitt, M., Schumacher, L., Elsemüller, L., Pratz, V., Schälte, Y., Köthe, U., & Bürkner, P.-C. (2023). BayesFlow: Amortized Bayesian workflows with neural networks.arXiv:2306.16015. (arXiv)Radev, S. T., Mertens, U. K., Voss, A., Ardizzone, L., Köthe, U. (2020). BayesFlow: Learning complex stochastic models with invertible neural networks.IEEE Transactions on Neural Networks and Learning Systems, 33(4), 1452-1466. (arXiv)(IEEE TNNLS)Radev, S. T., Schmitt, M., Pratz, V., Picchini, U., Köthe, U., & Bürkner, P.-C. (2023). JANA: Jointly amortized neural approximation of complex Bayesian models.Proceedings of the Thirty-Ninth Conference on Uncertainty in Artificial Intelligence, 216, 1695-1706. (arXiv)(PMLR)BibTeX:@misc{radev2023bayesflow,
 title = {{BayesFlow}: Amortized Bayesian workflows with neural networks},
 author = {Stefan T Radev and Marvin Schmitt and Lukas Schumacher and Lasse Elsem\"{u}ller and Valentin Pratz and Yannik Sch\"{a}lte and Ullrich K\"{o}the and Paul-Christian B\"{u}rkner},
 year = {2023},
 publisher= {arXiv},
 url={https://arxiv.org/abs/2306.16015}
}

@article{radev2020bayesflow,
  title={{BayesFlow}: Learning complex stochastic models with invertible neural networks},
  author={Radev, Stefan T. and Mertens, Ulf K. and Voss, Andreas and Ardizzone, Lynton and K{\"o}the, Ullrich},
  journal={IEEE transactions on neural networks and learning systems},
  volume={33},
  number={4},
  pages={1452--1466},
  year={2020},
  publisher={IEEE}
}

@inproceedings{pmlr-v216-radev23a,
  title = 	 {{JANA}: Jointly amortized neural approximation of complex {B}ayesian models},
  author =       {Radev, Stefan T. and Schmitt, Marvin and Pratz, Valentin and Picchini, Umberto and K\"othe, Ullrich and B\"urkner, Paul-Christian},
  booktitle = 	 {Proceedings of the Thirty-Ninth Conference on Uncertainty in Artificial Intelligence},
  pages = 	 {1695--1706},
  year = 	 {2023},
  volume = 	 {216},
  series = 	 {Proceedings of Machine Learning Research},
  publisher =    {PMLR}
}