Jakob H. Macke

Maximilian Dax, Stephen R. Green, Jonathan Gair et al.

We combine amortized neural posterior estimation with importance sampling for fast and accurate gravitational-wave inference. We first generate a rapid proposal for the Bayesian posterior using neural networks, and then attach importance weights based on the underlying likelihood and prior. This provides (1) a corrected posterior free from network inaccuracies, (2) a performance diagnostic (the sample efficiency) for assessing the proposal and identifying failure cases, and (3) an unbiased estimate of the Bayesian evidence. By establishing this independent verification and correction mechanism we address some of the most frequent criticisms against deep learning for scientific inference. We carry out a large study analyzing 42 binary black hole mergers observed by LIGO and Virgo with the SEOBNRv4PHM and IMRPhenomXPHM waveform models. This shows a median sample efficiency of $\approx 10\%$ (two orders-of-magnitude better than standard samplers) as well as a ten-fold reduction in the statistical uncertainty in the log evidence. Given these advantages, we expect a significant impact on gravitational-wave inference, and for this approach to serve as a paradigm for harnessing deep learning methods in scientific applications.

Maximilian Dax, Stephen R. Green, Jonathan Gair et al.

Mergers of binary neutron stars (BNSs) emit signals in both the gravitational-wave (GW) and electromagnetic (EM) spectra. Famously, the 2017 multi-messenger observation of GW170817 led to scientific discoveries across cosmology, nuclear physics, and gravity. Central to these results were the sky localization and distance obtained from GW data, which, in the case of GW170817, helped to identify the associated EM transient, AT 2017gfo, 11 hours after the GW signal. Fast analysis of GW data is critical for directing time-sensitive EM observations; however, due to challenges arising from the length and complexity of signals, it is often necessary to make approximations that sacrifice accuracy. Here, we present a machine learning framework that performs complete BNS inference in just one second without making any such approximations. Our approach enhances multi-messenger observations by providing (i) accurate localization even before the merger; (ii) improved localization precision by $\sim30\%$ compared to approximate low-latency methods; and (iii) detailed information on luminosity distance, inclination, and masses, which can be used to prioritize expensive telescope time. Additionally, the flexibility and reduced cost of our method open new opportunities for equation-of-state studies. Finally, we demonstrate that our method scales to extremely long signals, up to an hour in length, thus serving as a blueprint for data analysis for next-generation ground- and space-based detectors.

Mila Gorecki, Jakob H. Macke, Michael Deistler · mila

Simulation-based inference (SBI) provides a powerful framework for inferring posterior distributions of stochastic simulators in a wide range of domains. In many settings, however, the posterior distribution is not the end goal itself -- rather, the derived parameter values and their uncertainties are used as a basis for deciding what actions to take. Unfortunately, because posterior distributions provided by SBI are (potentially crude) approximations of the true posterior, the resulting decisions can be suboptimal. Here, we address the question of how to perform Bayesian decision making on stochastic simulators, and how one can circumvent the need to compute an explicit approximation to the posterior. Our method trains a neural network on simulated data and can predict the expected cost given any data and action, and can, thus, be directly used to infer the action with lowest cost. We apply our method to several benchmark problems and demonstrate that it induces similar cost as the true posterior distribution. We then apply the method to infer optimal actions in a real-world simulator in the medical neurosciences, the Bayesian Virtual Epileptic Patient, and demonstrate that it allows to infer actions associated with low cost after few simulations.

Jonas Wildberger, Maximilian Dax, Stephen R. Green et al.

Deep learning techniques for gravitational-wave parameter estimation have emerged as a fast alternative to standard samplers $\unicode{x2013}$ producing results of comparable accuracy. These approaches (e.g., DINGO) enable amortized inference by training a normalizing flow to represent the Bayesian posterior conditional on observed data. By conditioning also on the noise power spectral density (PSD) they can even account for changing detector characteristics. However, training such networks requires knowing in advance the distribution of PSDs expected to be observed, and therefore can only take place once all data to be analyzed have been gathered. Here, we develop a probabilistic model to forecast future PSDs, greatly increasing the temporal scope of DINGO networks. Using PSDs from the second LIGO-Virgo observing run (O2) $\unicode{x2013}$ plus just a single PSD from the beginning of the third (O3) $\unicode{x2013}$ we show that we can train a DINGO network to perform accurate inference throughout O3 (on 37 real events). We therefore expect this approach to be a key component to enable the use of deep learning techniques for low-latency analyses of gravitational waves.

Manuel Glöckler, Michael Deistler, Jakob H. Macke

We present Sequential Neural Variational Inference (SNVI), an approach to perform Bayesian inference in models with intractable likelihoods. SNVI combines likelihood-estimation (or likelihood-ratio-estimation) with variational inference to achieve a scalable simulation-based inference approach. SNVI maintains the flexibility of likelihood(-ratio) estimation to allow arbitrary proposals for simulations, while simultaneously providing a functional estimate of the posterior distribution without requiring MCMC sampling. We present several variants of SNVI and demonstrate that they are substantially more computationally efficient than previous algorithms, without loss of accuracy on benchmark tasks. We apply SNVI to a neuroscience model of the pyloric network in the crab and demonstrate that it can infer the posterior distribution with one order of magnitude fewer simulations than previously reported. SNVI vastly reduces the computational cost of simulation-based inference while maintaining accuracy and flexibility, making it possible to tackle problems that were previously inaccessible.

Poornima Ramesh, Jan-Matthis Lueckmann, Jan Boelts et al.

Simulation-based inference (SBI) refers to statistical inference on stochastic models for which we can generate samples, but not compute likelihoods. Like SBI algorithms, generative adversarial networks (GANs) do not require explicit likelihoods. We study the relationship between SBI and GANs, and introduce GATSBI, an adversarial approach to SBI. GATSBI reformulates the variational objective in an adversarial setting to learn implicit posterior distributions. Inference with GATSBI is amortised across observations, works in high-dimensional posterior spaces and supports implicit priors. We evaluate GATSBI on two SBI benchmark problems and on two high-dimensional simulators. On a model for wave propagation on the surface of a shallow water body, we show that GATSBI can return well-calibrated posterior estimates even in high dimensions. On a model of camera optics, it infers a high-dimensional posterior given an implicit prior, and performs better than a state-of-the-art SBI approach. We also show how GATSBI can be extended to perform sequential posterior estimation to focus on individual observations. Overall, GATSBI opens up opportunities for leveraging advances in GANs to perform Bayesian inference on high-dimensional simulation-based models.

Jaivardhan Kapoor, Jakob H. Macke, Christian F. Baumgartner

Generative modeling of 3D brain MRIs presents difficulties in achieving high visual fidelity while ensuring sufficient coverage of the data distribution. In this work, we propose to address this challenge with composable, multiscale morphological transformations in a variational autoencoder (VAE) framework. These transformations are applied to a chosen reference brain image to generate MRI volumes, equipping the model with strong anatomical inductive biases. We structure the VAE latent space in a way such that the model covers the data distribution sufficiently well. We show substantial performance improvements in FID while retaining comparable, or superior, reconstruction quality compared to prior work based on VAEs and generative adversarial networks (GANs).

Annalena Kofler, Maximilian Dax, Stephen R. Green et al.

Gravitational-wave data analysis relies on accurate and efficient methods to extract physical information from noisy detector signals, yet the increasing rate and complexity of observations represent a growing challenge. Deep learning provides a powerful alternative to traditional inference, but existing neural models typically lack the flexibility to handle variations in data analysis settings. Such variations accommodate imperfect observations or are required for specialized tests, and could include changes in detector configurations, overall frequency ranges, or localized cuts. We introduce a flexible transformer-based architecture paired with a training strategy that enables adaptation to diverse analysis settings at inference time. Applied to parameter estimation, we demonstrate that a single flexible model -- called Dingo-T1 -- can (i) analyze 48 gravitational-wave events from the third LIGO-Virgo-KAGRA Observing Run under a wide range of analysis configurations, (ii) enable systematic studies of how detector and frequency configurations impact inferred posteriors, and (iii) perform inspiral-merger-ringdown consistency tests probing general relativity. Dingo-T1 also improves median sample efficiency on real events from a baseline of 1.4% to 4.2%. Our approach thus demonstrates flexible and scalable inference with a principled framework for handling missing or incomplete data -- key capabilities for current and next-generation observatories.

Roxana Zeraati, Anna Levina, Jakob H. Macke et al.

Neural activity fluctuates over a wide range of timescales within and across brain areas. Experimental observations suggest that diverse neural timescales reflect information in dynamic environments. However, how timescales are defined and measured from brain recordings vary across the literature. Moreover, these observations do not specify the mechanisms underlying timescale variations, nor whether specific timescales are necessary for neural computation and brain function. Here, we synthesize three directions where computational approaches can distill the broad set of empirical observations into quantitative and testable theories: We review (i) how different data analysis methods quantify timescales across distinct behavioral states and recording modalities, (ii) how biophysical models provide mechanistic explanations for the emergence of diverse timescales, and (iii) how task-performing networks and machine learning models uncover the functional relevance of neural timescales. This integrative computational perspective thus complements experimental investigations, providing a holistic view on how neural timescales reflect the relationship between brain structure, dynamics, and behavior.

Manuel Gloeckler, J. P. Manzano-Patrón, Stamatios N. Sotiropoulos et al.

Simulation plays a central role in scientific discovery. In many applications, the bottleneck is no longer running a simulator; it is choosing among large families of plausible simulators, each corresponding to different forward models/hypotheses consistent with observations. Over large model families, classical Bayesian workflows for model selection are impractical. Furthermore, amortized model selection methods typically hard-code a fixed model prior or complexity penalty at training time, requiring users to commit to a particular parsimony assumption before seeing the data. We introduce PRISM, a simulation-based encoder-decoder that infers a joint posterior over both discrete model structures and associated continuous parameters, while enabling test-time control of model complexity via a tunable model prior that the network is conditioned on. We show that PRISM scales to families with combinatorially many (up to billions) of model instantiations on a synthetic symbolic regression task. As a scientific application, we evaluate PRISM on biophysical modeling for diffusion MRI data, showing the ability to perform model selection across several multi-compartment models, on both synthetic and in vivo neuroimaging data.

Jan Boelts, Cornelius Schröder, Jonas Beck et al.

Neural Posterior Estimation (NPE) enables rapid parameter inference for complex simulators with intractable likelihoods. NPE trains an inference network to estimate a probability density over parameters given data, typically assumed to be \emph{continuous}. However, many scientific models involve parameter spaces that are \emph{mixed}, that is, they contain both discrete and continuous dimensions. We address this limitation by extending NPE to mixed parameter spaces through an inference network that jointly handles discrete and continuous parameters. The inference network factorizes the joint posterior into discrete and continuous components, combining an autoregressive classifier for the discrete parameters with a generative model for the continuous parameters, trained jointly under a single simulation-based objective. In addition, we propose a diagnostic tool to assess the calibration of the mixed posterior approximation. Across tractable toy examples and real-world scientific simulators, our joint inference approach yields accurate and calibrated posteriors. The inference framework is available in the \texttt{sbi} Python package.

Jan-Matthis Lueckmann, Jan Boelts, David S. Greenberg et al.

Recent advances in probabilistic modelling have led to a large number of simulation-based inference algorithms which do not require numerical evaluation of likelihoods. However, a public benchmark with appropriate performance metrics for such 'likelihood-free' algorithms has been lacking. This has made it difficult to compare algorithms and identify their strengths and weaknesses. We set out to fill this gap: We provide a benchmark with inference tasks and suitable performance metrics, with an initial selection of algorithms including recent approaches employing neural networks and classical Approximate Bayesian Computation methods. We found that the choice of performance metric is critical, that even state-of-the-art algorithms have substantial room for improvement, and that sequential estimation improves sample efficiency. Neural network-based approaches generally exhibit better performance, but there is no uniformly best algorithm. We provide practical advice and highlight the potential of the benchmark to diagnose problems and improve algorithms. The results can be explored interactively on a companion website. All code is open source, making it possible to contribute further benchmark tasks and inference algorithms.

Manuel Gloeckler, Michael Deistler, Christian Weilbach et al.

Amortized Bayesian inference trains neural networks to solve stochastic inference problems using model simulations, thereby making it possible to rapidly perform Bayesian inference for any newly observed data. However, current simulation-based amortized inference methods are simulation-hungry and inflexible: They require the specification of a fixed parametric prior, simulator, and inference tasks ahead of time. Here, we present a new amortized inference method -- the Simformer -- which overcomes these limitations. By training a probabilistic diffusion model with transformer architectures, the Simformer outperforms current state-of-the-art amortized inference approaches on benchmark tasks and is substantially more flexible: It can be applied to models with function-valued parameters, it can handle inference scenarios with missing or unstructured data, and it can sample arbitrary conditionals of the joint distribution of parameters and data, including both posterior and likelihood. We showcase the performance and flexibility of the Simformer on simulators from ecology, epidemiology, and neuroscience, and demonstrate that it opens up new possibilities and application domains for amortized Bayesian inference on simulation-based models.

Jan Boelts, Michael Deistler, Manuel Gloeckler et al.

Scientists and engineers use simulators to model empirically observed phenomena. However, tuning the parameters of a simulator to ensure its outputs match observed data presents a significant challenge. Simulation-based inference (SBI) addresses this by enabling Bayesian inference for simulators, identifying parameters that match observed data and align with prior knowledge. Unlike traditional Bayesian inference, SBI only needs access to simulations from the model and does not require evaluations of the likelihood function. In addition, SBI algorithms do not require gradients through the simulator, allow for massive parallelization of simulations, and can perform inference for different observations without further simulations or training, thereby amortizing inference. Over the past years, we have developed, maintained, and extended sbi, a PyTorch-based package that implements Bayesian SBI algorithms based on neural networks. The sbi toolkit implements a wide range of inference methods, neural network architectures, sampling methods, and diagnostic tools. In addition, it provides well-tested default settings, but also offers flexibility to fully customize every step of the simulation-based inference workflow. Taken together, the sbi toolkit enables scientists and engineers to apply state-of-the-art SBI methods to black-box simulators, opening up new possibilities for aligning simulations with empirically observed data.

Manuel Gloeckler, Shoji Toyota, Kenji Fukumizu et al.

Amortized simulation-based inference (SBI) methods train neural networks on simulated data to perform Bayesian inference. While this strategy avoids the need for tractable likelihoods, it often requires a large number of simulations and has been challenging to scale to time series data. Scientific simulators frequently emulate real-world dynamics through thousands of single-state transitions over time. We propose an SBI approach that can exploit such Markovian simulators by locally identifying parameters consistent with individual state transitions. We then compose these local results to obtain a posterior over parameters that align with the entire time series observation. We focus on applying this approach to neural posterior score estimation but also show how it can be applied, e.g., to neural likelihood (ratio) estimation. We demonstrate that our approach is more simulation-efficient than directly estimating the global posterior on several synthetic benchmark tasks and simulators used in ecology and epidemiology. Finally, we validate scalability and simulation efficiency of our approach by applying it to a high-dimensional Kolmogorov flow simulator with around one million data dimensions.

Julius Vetter, Guy Moss, Cornelius Schröder et al.

Scientific modeling applications often require estimating a distribution of parameters consistent with a dataset of observations - an inference task also known as source distribution estimation. This problem can be ill-posed, however, since many different source distributions might produce the same distribution of data-consistent simulations. To make a principled choice among many equally valid sources, we propose an approach which targets the maximum entropy distribution, i.e., prioritizes retaining as much uncertainty as possible. Our method is purely sample-based - leveraging the Sliced-Wasserstein distance to measure the discrepancy between the dataset and simulations - and thus suitable for simulators with intractable likelihoods. We benchmark our method on several tasks, and show that it can recover source distributions with substantially higher entropy than recent source estimation methods, without sacrificing the fidelity of the simulations. Finally, to demonstrate the utility of our approach, we infer source distributions for parameters of the Hodgkin-Huxley model from experimental datasets with thousands of single-neuron measurements. In summary, we propose a principled method for inferring source distributions of scientific simulator parameters while retaining as much uncertainty as possible.

Jonas Beck, Nathanael Bosch, Michael Deistler et al.

Ordinary differential equations (ODEs) are widely used to describe dynamical systems in science, but identifying parameters that explain experimental measurements is challenging. In particular, although ODEs are differentiable and would allow for gradient-based parameter optimization, the nonlinear dynamics of ODEs often lead to many local minima and extreme sensitivity to initial conditions. We therefore propose diffusion tempering, a novel regularization technique for probabilistic numerical methods which improves convergence of gradient-based parameter optimization in ODEs. By iteratively reducing a noise parameter of the probabilistic integrator, the proposed method converges more reliably to the true parameters. We demonstrate that our method is effective for dynamical systems of different complexity and show that it obtains reliable parameter estimates for a Hodgkin-Huxley model with a practically relevant number of parameters.

Michael Deistler, Jan Boelts, Peter Steinbach et al.

A central challenge in many areas of science and engineering is to identify model parameters that are consistent with prior knowledge and empirical data. Bayesian inference offers a principled framework for this task, but can be computationally prohibitive when models are defined by stochastic simulators. Simulation-based Inference (SBI) is a suite of methods developed to overcome this limitation, which has enabled scientific discoveries in fields such as particle physics, astrophysics, and neuroscience. The core idea of SBI is to train neural networks on data generated by a simulator, without requiring access to likelihood evaluations. Once trained, inference is amortized: The neural network can rapidly perform Bayesian inference on empirical observations without requiring additional training or simulations. In this tutorial, we provide a practical guide for practitioners aiming to apply SBI methods. We outline a structured SBI workflow and offer practical guidelines and diagnostic tools for every stage of the process -- from setting up the simulator and prior, choosing and training inference networks, to performing inference and validating the results. We illustrate these steps through examples from astrophysics, psychophysics, and neuroscience. This tutorial empowers researchers to apply state-of-the-art SBI methods, facilitating efficient parameter inference for scientific discovery.

Julius Vetter, Manuel Gloeckler, Daniel Gedon et al.

Simulation-based inference (SBI) offers a flexible and general approach to performing Bayesian inference: In SBI, a neural network is trained on synthetic data simulated from a model and used to rapidly infer posterior distributions for observed data. A key goal for SBI is to achieve accurate inference with as few simulations as possible, especially for expensive simulators. In this work, we address this challenge by repurposing recent probabilistic foundation models for tabular data: We show how tabular foundation models -- specifically TabPFN -- can be used as pre-trained autoregressive conditional density estimators for SBI. We propose Neural Posterior Estimation with Prior-data Fitted Networks (NPE-PFN) and show that it is competitive with current SBI approaches in terms of accuracy for both benchmark tasks and two complex scientific inverse problems. Crucially, it often substantially outperforms them in terms of simulation efficiency, sometimes requiring orders of magnitude fewer simulations. NPE-PFN eliminates the need for inference network selection, training, and hyperparameter tuning. We also show that it exhibits superior robustness to model misspecification and can be scaled to simulation budgets that exceed the context size limit of TabPFN. NPE-PFN provides a new direction for SBI, where training-free, general-purpose inference models offer efficient, easy-to-use, and flexible solutions for a wide range of stochastic inverse problems.

Stefan Wahl, Raphaela Schenk, Ali Farnoud et al.

Automated methods for discovering mechanistic simulator models from observational data offer a promising path toward accelerating scientific progress. Such methods often take the form of agentic-style iterative workflows that repeatedly propose and revise candidate models by imitating human discovery processes. However, existing LLM-based approaches typically implement such workflows via hand-crafted heuristic procedures, without an explicit probabilistic formulation. We recast model discovery as probabilistic inference, i.e., as sampling from an unknown distribution over mechanistic models capable of explaining the data. This perspective provides a unified way to reason about model proposal, refinement, and selection within a single inference framework. As a concrete instantiation of this view, we introduce ModelSMC, an algorithm based on Sequential Monte Carlo sampling. ModelSMC represents candidate models as particles which are iteratively proposed and refined by an LLM, and weighted using likelihood-based criteria. Experiments on real-world scientific systems illustrate that this formulation discovers models with interpretable mechanisms and improves posterior predictive checks. More broadly, this perspective provides a probabilistic lens for understanding and developing LLM-based approaches to model discovery.

Lisa Haxel, Jaivardhan Kapoor, Ulf Ziemann et al.

Brain-computer interfaces (BCIs) suffer from accuracy degradation as neural signals drift over time and vary across users, requiring frequent recalibration that limits practical deployment. We introduce EDAPT, a task- and model-agnostic framework that eliminates calibration through continual model adaptation. EDAPT first trains a baseline decoder using data from multiple users, then continually personalizes this model via supervised finetuning as the neural patterns evolve during use. We tested EDAPT across nine datasets covering three BCI tasks, and found that it consistently improved accuracy over conventional, static methods. These improvements primarily stem from combining population-level pretraining and online continual finetuning, with unsupervised domain adaptation providing further gains on some datasets. EDAPT runs efficiently, updating models within 200 milliseconds on consumer-grade hardware. Finally, decoding accuracy scales with total data budget rather than its allocation between subjects and trials. EDAPT provides a practical pathway toward calibration-free BCIs, reducing a major barrier to BCI deployment.

Guy Moss, Leah Sophie Muhle, Reinhard Drews et al.

Simulation-based inference (SBI) is an established approach for performing Bayesian inference on scientific simulators. SBI so far works best on low-dimensional parametric models. However, it is difficult to infer function-valued parameters, which frequently occur in disciplines that model spatiotemporal processes such as the climate and earth sciences. Here, we introduce an approach for efficient posterior estimation, using a Fourier Neural Operator (FNO) architecture with a flow matching objective. We show that our approach, FNOPE, can perform inference of function-valued parameters at a fraction of the simulation budget of state of the art methods. In addition, FNOPE supports posterior evaluation at arbitrary discretizations of the domain, as well as simultaneous estimation of vector-valued parameters. We demonstrate the effectiveness of our approach on several benchmark tasks and a challenging spatial inference task from glaciology. FNOPE extends the applicability of SBI methods to new scientific domains by enabling the inference of function-valued parameters.

Jaivardhan Kapoor, Auguste Schulz, Julius Vetter et al.

Modern datasets in neuroscience enable unprecedented inquiries into the relationship between complex behaviors and the activity of many simultaneously recorded neurons. While latent variable models can successfully extract low-dimensional embeddings from such recordings, using them to generate realistic spiking data, especially in a behavior-dependent manner, still poses a challenge. Here, we present Latent Diffusion for Neural Spiking data (LDNS), a diffusion-based generative model with a low-dimensional latent space: LDNS employs an autoencoder with structured state-space (S4) layers to project discrete high-dimensional spiking data into continuous time-aligned latents. On these inferred latents, we train expressive (conditional) diffusion models, enabling us to sample neural activity with realistic single-neuron and population spiking statistics. We validate LDNS on synthetic data, accurately recovering latent structure, firing rates, and spiking statistics. Next, we demonstrate its flexibility by generating variable-length data that mimics human cortical activity during attempted speech. We show how to equip LDNS with an expressive observation model that accounts for single-neuron dynamics not mediated by the latent state, further increasing the realism of generated samples. Finally, conditional LDNS trained on motor cortical activity during diverse reaching behaviors can generate realistic spiking data given reach direction or unseen reach trajectories. In summary, LDNS simultaneously enables inference of low-dimensional latents and realistic conditional generation of neural spiking datasets, opening up further possibilities for simulating experimentally testable hypotheses.

Maximilian Dax, Jonas Wildberger, Simon Buchholz et al.

Neural posterior estimation methods based on discrete normalizing flows have become established tools for simulation-based inference (SBI), but scaling them to high-dimensional problems can be challenging. Building on recent advances in generative modeling, we here present flow matching posterior estimation (FMPE), a technique for SBI using continuous normalizing flows. Like diffusion models, and in contrast to discrete flows, flow matching allows for unconstrained architectures, providing enhanced flexibility for complex data modalities. Flow matching, therefore, enables exact density evaluation, fast training, and seamless scalability to large architectures--making it ideal for SBI. We show that FMPE achieves competitive performance on an established SBI benchmark, and then demonstrate its improved scalability on a challenging scientific problem: for gravitational-wave inference, FMPE outperforms methods based on comparable discrete flows, reducing training time by 30% with substantially improved accuracy. Our work underscores the potential of FMPE to enhance performance in challenging inference scenarios, thereby paving the way for more advanced applications to scientific problems.

Richard Gao, Michael Deistler, Jakob H. Macke

Simulation-based inference (SBI) enables amortized Bayesian inference for simulators with implicit likelihoods. But when we are primarily interested in the quality of predictive simulations, or when the model cannot exactly reproduce the observed data (i.e., is misspecified), targeting the Bayesian posterior may be overly restrictive. Generalized Bayesian Inference (GBI) aims to robustify inference for (misspecified) simulator models, replacing the likelihood-function with a cost function that evaluates the goodness of parameters relative to data. However, GBI methods generally require running multiple simulations to estimate the cost function at each parameter value during inference, making the approach computationally infeasible for even moderately complex simulators. Here, we propose amortized cost estimation (ACE) for GBI to address this challenge: We train a neural network to approximate the cost function, which we define as the expected distance between simulations produced by a parameter and observed data. The trained network can then be used with MCMC to infer GBI posteriors for any observation without running additional simulations. We show that, on several benchmark tasks, ACE accurately predicts cost and provides predictive simulations that are closer to synthetic observations than other SBI methods, especially for misspecified simulators. Finally, we apply ACE to infer parameters of the Hodgkin-Huxley model given real intracellular recordings from the Allen Cell Types Database. ACE identifies better data-matching parameters while being an order of magnitude more simulation-efficient than a standard SBI method. In summary, ACE combines the strengths of SBI methods and GBI to perform robust and simulation-amortized inference for scientific simulators.

Cornelius Schröder, Jakob H. Macke

Many scientific models are composed of multiple discrete components, and scientists often make heuristic decisions about which components to include. Bayesian inference provides a mathematical framework for systematically selecting model components, but defining prior distributions over model components and developing associated inference schemes has been challenging. We approach this problem in a simulation-based inference framework: We define model priors over candidate components and, from model simulations, train neural networks to infer joint probability distributions over both model components and associated parameters. Our method, simulation-based model inference (SBMI), represents distributions over model components as a conditional mixture of multivariate binary distributions in the Grassmann formalism. SBMI can be applied to any compositional stochastic simulator without requiring likelihood evaluations. We evaluate SBMI on a simple time series model and on two scientific models from neuroscience, and show that it can discover multiple data-consistent model configurations, and that it reveals non-identifiable model components and parameters. SBMI provides a powerful tool for data-driven scientific inquiry which will allow scientists to identify essential model components and make uncertainty-informed modelling decisions.

Manuel Glöckler, Michael Deistler, Jakob H. Macke

Bayesian inference usually requires running potentially costly inference procedures separately for every new observation. In contrast, the idea of amortized Bayesian inference is to initially invest computational cost in training an inference network on simulated data, which can subsequently be used to rapidly perform inference (i.e., to return estimates of posterior distributions) for new observations. This approach has been applied to many real-world models in the sciences and engineering, but it is unclear how robust the approach is to adversarial perturbations in the observed data. Here, we study the adversarial robustness of amortized Bayesian inference, focusing on simulation-based estimation of multi-dimensional posterior distributions. We show that almost unrecognizable, targeted perturbations of the observations can lead to drastic changes in the predicted posterior and highly unrealistic posterior predictive samples, across several benchmark tasks and a real-world example from neuroscience. We propose a computationally efficient regularization scheme based on penalizing the Fisher information of the conditional density estimator, and show how it improves the adversarial robustness of amortized Bayesian inference.

Maximilian Dax, Stephen R. Green, Jonathan Gair et al.

Simulation-based inference with conditional neural density estimators is a powerful approach to solving inverse problems in science. However, these methods typically treat the underlying forward model as a black box, with no way to exploit geometric properties such as equivariances. Equivariances are common in scientific models, however integrating them directly into expressive inference networks (such as normalizing flows) is not straightforward. We here describe an alternative method to incorporate equivariances under joint transformations of parameters and data. Our method -- called group equivariant neural posterior estimation (GNPE) -- is based on self-consistently standardizing the "pose" of the data while estimating the posterior over parameters. It is architecture-independent, and applies both to exact and approximate equivariances. As a real-world application, we use GNPE for amortized inference of astrophysical binary black hole systems from gravitational-wave observations. We show that GNPE achieves state-of-the-art accuracy while reducing inference times by three orders of magnitude.

Maximilian Dax, Stephen R. Green, Jonathan Gair et al.

We demonstrate unprecedented accuracy for rapid gravitational-wave parameter estimation with deep learning. Using neural networks as surrogates for Bayesian posterior distributions, we analyze eight gravitational-wave events from the first LIGO-Virgo Gravitational-Wave Transient Catalog and find very close quantitative agreement with standard inference codes, but with inference times reduced from O(day) to a minute per event. Our networks are trained using simulated data, including an estimate of the detector-noise characteristics near the event. This encodes the signal and noise models within millions of neural-network parameters, and enables inference for any observed data consistent with the training distribution, accounting for noise nonstationarity from event to event. Our algorithm -- called "DINGO" -- sets a new standard in fast-and-accurate inference of physical parameters of detected gravitational-wave events, which should enable real-time data analysis without sacrificing accuracy.

Alvaro Tejero-Cantero, Jan Boelts, Michael Deistler et al.

Scientists and engineers employ stochastic numerical simulators to model empirically observed phenomena. In contrast to purely statistical models, simulators express scientific principles that provide powerful inductive biases, improve generalization to new data or scenarios and allow for fewer, more interpretable and domain-relevant parameters. Despite these advantages, tuning a simulator's parameters so that its outputs match data is challenging. Simulation-based inference (SBI) seeks to identify parameter sets that a) are compatible with prior knowledge and b) match empirical observations. Importantly, SBI does not seek to recover a single 'best' data-compatible parameter set, but rather to identify all high probability regions of parameter space that explain observed data, and thereby to quantify parameter uncertainty. In Bayesian terminology, SBI aims to retrieve the posterior distribution over the parameters of interest. In contrast to conventional Bayesian inference, SBI is also applicable when one can run model simulations, but no formula or algorithm exists for evaluating the probability of data given parameters, i.e. the likelihood. We present $\texttt{sbi}$, a PyTorch-based package that implements SBI algorithms based on neural networks. $\texttt{sbi}$ facilitates inference on black-box simulators for practising scientists and engineers by providing a unified interface to state-of-the-art algorithms together with documentation and tutorials.

Alexandre René, André Longtin, Jakob H. Macke

To understand how rich dynamics emerge in neural populations, we require models exhibiting a wide range of activity patterns while remaining interpretable in terms of connectivity and single-neuron dynamics. However, it has been challenging to fit such mechanistic spiking networks at the single neuron scale to empirical population data. To close this gap, we propose to fit such data at a meso scale, using a mechanistic but low-dimensional and hence statistically tractable model. The mesoscopic representation is obtained by approximating a population of neurons as multiple homogeneous `pools' of neurons, and modelling the dynamics of the aggregate population activity within each pool. We derive the likelihood of both single-neuron and connectivity parameters given this activity, which can then be used to either optimize parameters by gradient ascent on the log-likelihood, or to perform Bayesian inference using Markov Chain Monte Carlo (MCMC) sampling. We illustrate this approach using a model of generalized integrate-and-fire neurons for which mesoscopic dynamics have been previously derived, and show that both single-neuron and connectivity parameters can be recovered from simulated data. In particular, our inference method extracts posterior correlations between model parameters, which define parameter subsets able to reproduce the data. We compute the Bayesian posterior for combinations of parameters using MCMC sampling and investigate how the approximations inherent to a mesoscopic population model impact the accuracy of the inferred single-neuron parameters.

Artur Speiser, Lucas-Raphael Müller, Ulf Matti et al.

Single-molecule localization fluorescence microscopy constructs super-resolution images by sequential imaging and computational localization of sparsely activated fluorophores. Accurate and efficient fluorophore localization algorithms are key to the success of this computational microscopy method. We present a novel localization algorithm based on deep learning which significantly improves upon the state of the art. Our contributions are a novel network architecture for simultaneous detection and localization, and new loss function which phrases detection and localization as a Bayesian inference problem, and thus allows the network to provide uncertainty-estimates. In contrast to standard methods which independently process imaging frames, our network architecture uses temporal context from multiple sequentially imaged frames to detect and localize molecules. We demonstrate the power of our method across a variety of datasets, imaging modalities, signal to noise ratios, and fluorophore densities. While existing localization algorithms can achieve optimal localization accuracy at low fluorophore densities, they are confounded by high densities. Our method is the first deep-learning based approach which achieves state-of-the-art on the SMLM2016 challenge. It achieves the best scores on 12 out of 12 data-sets when comparing both detection accuracy and precision, and excels at high densities. Finally, we investigate how unsupervised learning can be used to make the network robust against mismatch between simulated and real data. The lessons learned here are more generally relevant for the training of deep networks to solve challenging Bayesian inverse problems on spatially extended domains in biology and physics.

Alessio Ansuini, Alessandro Laio, Jakob H. Macke et al.

Deep neural networks progressively transform their inputs across multiple processing layers. What are the geometrical properties of the representations learned by these networks? Here we study the intrinsic dimensionality (ID) of data-representations, i.e. the minimal number of parameters needed to describe a representation. We find that, in a trained network, the ID is orders of magnitude smaller than the number of units in each layer. Across layers, the ID first increases and then progressively decreases in the final layers. Remarkably, the ID of the last hidden layer predicts classification accuracy on the test set. These results can neither be found by linear dimensionality estimates (e.g., with principal component analysis), nor in representations that had been artificially linearized. They are neither found in untrained networks, nor in networks that are trained on randomized labels. This suggests that neural networks that can generalize are those that transform the data into low-dimensional, but not necessarily flat manifolds.

David S. Greenberg, Marcel Nonnenmacher, Jakob H. Macke

How can one perform Bayesian inference on stochastic simulators with intractable likelihoods? A recent approach is to learn the posterior from adaptively proposed simulations using neural network-based conditional density estimators. However, existing methods are limited to a narrow range of proposal distributions or require importance weighting that can limit performance in practice. Here we present automatic posterior transformation (APT), a new sequential neural posterior estimation method for simulation-based inference. APT can modify the posterior estimate using arbitrary, dynamically updated proposals, and is compatible with powerful flow-based density estimators. It is more flexible, scalable and efficient than previous simulation-based inference techniques. APT can operate directly on high-dimensional time series and image data, opening up new applications for likelihood-free inference.

David G. T. Barrett, Ari S. Morcos, Jakob H. Macke

Deep neural networks (DNNs) transform stimuli across multiple processing stages to produce representations that can be used to solve complex tasks, such as object recognition in images. However, a full understanding of how they achieve this remains elusive. The complexity of biological neural networks substantially exceeds the complexity of DNNs, making it even more challenging to understand the representations that they learn. Thus, both machine learning and computational neuroscience are faced with a shared challenge: how can we analyze their representations in order to understand how they solve complex tasks? We review how data-analysis concepts and techniques developed by computational neuroscientists can be useful for analyzing representations in DNNs, and in turn, how recently developed techniques for analysis of DNNs can be useful for understanding representations in biological neural networks. We explore opportunities for synergy between the two fields, such as the use of DNNs as in-silico model systems for neuroscience, and how this synergy can lead to new hypotheses about the operating principles of biological neural networks.

Jan-Matthis Lueckmann, Giacomo Bassetto, Theofanis Karaletsos et al.

Approximate Bayesian Computation (ABC) provides methods for Bayesian inference in simulation-based stochastic models which do not permit tractable likelihoods. We present a new ABC method which uses probabilistic neural emulator networks to learn synthetic likelihoods on simulated data -- both local emulators which approximate the likelihood for specific observed data, as well as global ones which are applicable to a range of data. Simulations are chosen adaptively using an acquisition function which takes into account uncertainty about either the posterior distribution of interest, or the parameters of the emulator. Our approach does not rely on user-defined rejection thresholds or distance functions. We illustrate inference with emulator networks on synthetic examples and on a biophysical neuron model, and show that emulators allow accurate and efficient inference even on high-dimensional problems which are challenging for conventional ABC approaches.

Jan-Matthis Lueckmann, Pedro J. Goncalves, Giacomo Bassetto et al.

Mechanistic models of single-neuron dynamics have been extensively studied in computational neuroscience. However, identifying which models can quantitatively reproduce empirically measured data has been challenging. We propose to overcome this limitation by using likelihood-free inference approaches (also known as Approximate Bayesian Computation, ABC) to perform full Bayesian inference on single-neuron models. Our approach builds on recent advances in ABC by learning a neural network which maps features of the observed data to the posterior distribution over parameters. We learn a Bayesian mixture-density network approximating the posterior over multiple rounds of adaptively chosen simulations. Furthermore, we propose an efficient approach for handling missing features and parameter settings for which the simulator fails, as well as a strategy for automatically learning relevant features using recurrent neural networks. On synthetic data, our approach efficiently estimates posterior distributions and recovers ground-truth parameters. On in-vitro recordings of membrane voltages, we recover multivariate posteriors over biophysical parameters, which yield model-predicted voltage traces that accurately match empirical data. Our approach will enable neuroscientists to perform Bayesian inference on complex neuron models without having to design model-specific algorithms, closing the gap between mechanistic and statistical approaches to single-neuron modelling.

Marcel Nonnenmacher, Srinivas C. Turaga, Jakob H. Macke

A powerful approach for understanding neural population dynamics is to extract low-dimensional trajectories from population recordings using dimensionality reduction methods. Current approaches for dimensionality reduction on neural data are limited to single population recordings, and can not identify dynamics embedded across multiple measurements. We propose an approach for extracting low-dimensional dynamics from multiple, sequential recordings. Our algorithm scales to data comprising millions of observed dimensions, making it possible to access dynamics distributed across large populations or multiple brain areas. Building on subspace-identification approaches for dynamical systems, we perform parameter estimation by minimizing a moment-matching objective using a scalable stochastic gradient descent algorithm: The model is optimized to predict temporal covariations across neurons and across time. We show how this approach naturally handles missing data and multiple partial recordings, and can identify dynamics and predict correlations even in the presence of severe subsampling and small overlap between recordings. We demonstrate the effectiveness of the approach both on simulated data and a whole-brain larval zebrafish imaging dataset.

Artur Speiser, Jinyao Yan, Evan Archer et al.

Calcium imaging permits optical measurement of neural activity. Since intracellular calcium concentration is an indirect measurement of neural activity, computational tools are necessary to infer the true underlying spiking activity from fluorescence measurements. Bayesian model inversion can be used to solve this problem, but typically requires either computationally expensive MCMC sampling, or faster but approximate maximum-a-posteriori optimization. Here, we introduce a flexible algorithmic framework for fast, efficient and accurate extraction of neural spikes from imaging data. Using the framework of variational autoencoders, we propose to amortize inference by training a deep neural network to perform model inversion efficiently. The recognition network is trained to produce samples from the posterior distribution over spike trains. Once trained, performing inference amounts to a fast single forward pass through the network, without the need for iterative optimization or sampling. We show that amortization can be applied flexibly to a wide range of nonlinear generative models and significantly improves upon the state of the art in computation time, while achieving competitive accuracy. Our framework is also able to represent posterior distributions over spike-trains. We demonstrate the generality of our method by proposing the first probabilistic approach for separating backpropagating action potentials from putative synaptic inputs in calcium imaging of dendritic spines.

Mijung Park, Jakob H. Macke

Neural population activity often exhibits rich variability and temporal structure. This variability is thought to arise from single-neuron stochasticity, neural dynamics on short time-scales, as well as from modulations of neural firing properties on long time-scales, often referred to as "non-stationarity". To better understand the nature of co-variability in neural circuits and their impact on cortical information processing, we need statistical models that are able to capture multiple sources of variability on different time-scales. Here, we introduce a hierarchical statistical model of neural population activity which models both neural population dynamics as well as inter-trial modulations in firing rates. In addition, we extend the model to allow us to capture non-stationarities in the population dynamics itself (i.e., correlations across neurons). We develop variational inference methods for learning model parameters, and demonstrate that the method can recover non-stationarities in both average firing rates and correlation structure. Applied to neural population recordings from anesthetized macaque primary visual cortex, our models provide a better account of the structure of neural firing than stationary dynamics models.