Likelihood-free Inference Research Articles

Efficient multifidelity likelihood-free Bayesian inference with adaptive computational resource allocation

Likelihood-free Bayesian inference algorithms are popular methods for inferring the parameters of complex stochastic models with intractable likelihoods. These algorithms characteristically rely heavily on repeated model simulations. However, whenever the computational cost of simulation is even moderately expensive, the significant burden incurred by likelihood-free algorithms leaves them infeasible for many practical applications. The multifidelity approach has been introduced in the context of approximate Bayesian computation to reduce the simulation burden of likelihood-free inference without loss of accuracy, by using the information provided by simulating computationally cheap, approximate models in place of the model of interest. In this work we demonstrate that multifidelity techniques can be applied in the general likelihood-free Bayesian inference setting. Analytical results on the optimal allocation of computational resources to simulations at different levels of fidelity are derived, and subsequently implemented practically. We provide an adaptive multifidelity likelihood-free inference algorithm that learns the relationships between models at different fidelities and adapts resource allocation accordingly, and demonstrate that this algorithm produces posterior estimates with near-optimal efficiency.

Treating detector systematics via a likelihood free inference method

Estimating the impact of systematic uncertainties in particle physics experiments is challenging, especially since the detector response is unknown analytically in most situations and needs to be estimated through Monte Carlo (MC) simulations. Typically, detector property variations are parameterized in ways that implicitly assume a specific physics model, which can introduce biases on quantities measured by an analysis. In this paper, we present a method to recover a model-independent, event-wise estimation of the detector response variation by applying a likelihood-free inference method to a set of MC simulations representing discrete detector realizations. The method provides a re-weighting scheme for every event, which can be used to apply the effects of detector property variations fully decoupled from the assumed physics model. Using a toy MC example inspired by fixed-baseline neutrino oscillation experiments, we demonstrate the performance of our method. We show that it fully decouples the modeling of the detector response from the physics parameters to be measured in a MC forward-folding analysis.

A Universal Equation to Predict Ωm from Halo and Galaxy Catalogs

We discover analytic equations that can infer the value of Ωm from the positions and velocity moduli of halo and galaxy catalogs. The equations are derived by combining a tailored graph neural network (GNN) architecture with symbolic regression. We first train the GNN on dark matter halos from Gadget N-body simulations to perform field-level likelihood-free inference, and show that our model can infer Ωm with ∼6% accuracy from halo catalogs of thousands of N-body simulations run with six different codes: Abacus, CUBEP3M, Gadget, Enzo, PKDGrav3, and Ramses. By applying symbolic regression to the different parts comprising the GNN, we derive equations that can predict Ωm from halo catalogs of simulations run with all of the above codes with accuracies similar to those of the GNN. We show that, by tuning a single free parameter, our equations can also infer the value of Ωm from galaxy catalogs of thousands of state-of-the-art hydrodynamic simulations of the CAMELS project, each with a different astrophysics model, run with five distinct codes that employ different subgrid physics: IllustrisTNG, SIMBA, Astrid, Magneticum, SWIFT-EAGLE. Furthermore, the equations also perform well when tested on galaxy catalogs from simulations covering a vast region in parameter space that samples variations in 5 cosmological and 23 astrophysical parameters. We speculate that the equations may reflect the existence of a fundamental physics relation between the phase-space distribution of generic tracers and Ωm, one that is not affected by galaxy formation physics down to scales as small as 10 h −1 kpc.

Machine-learning Cosmology from Void Properties

Cosmic voids are the largest and most underdense structures in the Universe. Their properties have been shown to encode precious information about the laws and constituents of the Universe. We show that machine-learning techniques can unlock the information in void features for cosmological parameter inference. We rely on thousands of void catalogs from the GIGANTES data set, where every catalog contains an average of 11,000 voids from a volume of . We focus on three properties of cosmic voids: ellipticity, density contrast, and radius. We train (1) fully connected neural networks on histograms from individual void properties and (2) deep sets from void catalogs to perform likelihood-free inference on the value of cosmological parameters. We find that our best models are able to constrain the value of Ω m , σ 8, and n s with mean relative errors of 10%, 4%, and 3%, respectively, without using any spatial information from the void catalogs. Our results provide an illustration for the use of machine learning to constrain cosmology with voids.

A recursive inference method based on invertible neural network for multi-level model updating using video monitoring data

Likelihood-free inference methods have been widely adopted, but they face significant challenges in updating multi-level computational models that have hierarchically embedded sub-models. This difficulty arises from the lack of direct observations of the quantities of interest of the sub-models. In addition, recent advancements in sensing and image processing technologies allow for the collection of a substantial amount of video monitoring data through non-contact sensing techniques. The implicit and very complicated relationship between the uncertain model parameters and video monitoring data adds an additional layer of challenge to the updating of multi-level models. This paper overcomes these challenges by proposing an innovative Recursive Inference method based on Invertible Neural Networks (RINN) for multi-level models. The proposed RINN framework first compresses the high-dimensional video monitoring data into low-dimension latent-space data using a convolutional autoencoder. Using synthetic video monitoring data generated by considering various uncertainty sources in the computational simulation models, a likelihood-free inference model is then trained through a conditional invertible neural network (cINN) and a summary neural network. This model efficiently approximates the posterior distributions of uncertain model parameters without evaluating the computationally intractable likelihood function, given any latent-space data obtained from the autoencoder. To facilitate continuous monitoring and model updating over an extended monitoring period, this paper further proposes a recursive model updating strategy that integrates the cINN-based likelihood-free inference with the particle filtering method. The updating of a degradation model of a miter gate application is employed as an example throughout this paper to explain and demonstrate the efficacy of the proposed RINN framework. The results of the case study show that the RINN is able to effectively reduce uncertainty in the degradation model parameters through strain video monitoring data of a miter gate, and thereby increase the confidence in the remaining useful life (RUL) estimation using the updated degradation model.

Exploring the polarization of axially symmetric supernovae with unsupervised deep learning

ABSTRACT The measurement of non-zero polarization can be used to infer the presence of departures from spherical symmetry in supernovae (SNe). The origin of the majority of the intrinsic polarization observed in SNe is in electron scattering, which induces a wavelength-independent continuum polarization that is generally observed to be low ($\lesssim\!\! 1{{\ \rm per\, cent}}$) for all SN types. The key indicator of asymmetry in SNe is the polarization observed across spectral lines, in particular the characteristic ‘inverse P Cygni’ profile. The results of a suite of 900 Monte Carlo radiative transfer simulations are presented here. These simulations cover a range of possible axisymmetric structures (including unipolar, bipolar, and equatorial enhancements) for the line-forming region of the Ca ii infrared triplet. Using a variational auto-encoder, seven key latent parameters are learned that describe the relationship between Stokes I and q, under the assumption of an axially symmetric line-forming region and resonant scattering. Likelihood-free inference techniques are used to invert the Stokes I and q line profiles, in the latent space, to derive the underlying geometries. For axially symmetric structures that yield an observable ‘dominant axis’ on the Stokes q–u plane, we propose the existence of a geometry ‘conjugate’ (which is indistinguishable under a rotation of π/2). Using this machine learning infrastructure, we attempt to identify possible geometries associated with spectropolarimetric observations of the Type Ib SN 2017gax.

Conditional density estimation using population Monte Carlo based approximate Bayesian computation

Most statistical methods require likelihood evaluation to draw a statistical inference. However, in some situations, likelihood evaluation becomes difficult analytically or computationally. Different likelihood-free methods are available that eliminate the need to compute the likelihood function. Approximate Bayesian Computation (ABC) is a framework that implements likelihood-free inference and replaces the likelihood evaluation with simulations by using forward modeling. The goal of ABC methods is to approximate the posterior distribution. However, posterior approximation via ABC methods is still considerably expensive for high dimensions. ABC requires many simulations that become computationally infeasible for complex models. Here, a technique is proposed that combines a somewhat more efficient form of ABC (Population Monte Carlo, PMC) with a Conditional Density Estimation (CDE) approach. The proposed framework provides an estimation of the posterior distribution which is referred to as PMC-CDE. A simulation study is performed that provides empirical evidence to show the efficiency of PMC-CDE in terms of integrated squared error loss. Furthermore, real-life datasets manifest the application of the proposed method.

The influence of inter-regional delays in generating large-scale brain networks of phase synchronization

Large-scale networks of phase synchronization are considered to regulate the communication between brain regions fundamental to cognitive function, but the mapping to their structural substrates, i.e., the structure-function relationship, remains poorly understood. Biophysical Network Models (BNMs) have demonstrated the influences of local oscillatory activity and inter-regional anatomical connections in generating alpha-band (8–12 Hz) networks of phase synchronization observed with Electroencephalography (EEG) and Magnetoencephalography (MEG). Yet, the influence of inter-regional conduction delays remains unknown. In this study, we compared a BNM with standard “distance-dependent delays”, which assumes constant conduction velocity, to BNMs with delays specified by two alternative methods accounting for spatially varying conduction velocities, “isochronous delays” and “mixed delays”. We followed the Approximate Bayesian Computation (ABC) workflow, i) specifying neurophysiologically informed prior distributions of BNM parameters, ii) verifying the suitability of the prior distributions with Prior Predictive Checks, iii) fitting each of the three BNMs to alpha-band MEG resting-state data (N = 75) with Bayesian optimization for Likelihood-Free Inference (BOLFI), and iv) choosing between the fitted BNMs with ABC model comparison on a separate MEG dataset (N = 30). Prior Predictive Checks revealed the range of dynamics generated by each of the BNMs to encompass those seen in the MEG data, suggesting the suitability of the prior distributions. Fitting the models to MEG data yielded reliable posterior distributions of the parameters of each of the BNMs. Finally, model comparison revealed the BNM with “distance-dependent delays”, as the most probable to describe the generation of alpha-band networks of phase synchronization seen in MEG. These findings suggest that distance-dependent delays might contribute to the neocortical architecture of human alpha-band networks of phase synchronization. Hence, our study illuminates the role of inter-regional delays in generating the large-scale networks of phase synchronization that might subserve the communication between regions vital to cognition.

Inference of the optical depth to reionization τ from Planck CMB maps with convolutional neural networks

The optical depth to reionization, τ, is the least constrained parameter of the cosmological Λ cold dark matter (ΛCDM) model. To date, its most precise value is inferred from large-scale polarized cosmic microwave background (CMB) power spectra from the High Frequency Instrument (HFI) aboard the Planck satellite. These maps are known to contain significant contamination by residual non-Gaussian systematic effects, which are hard to model analytically. Therefore, robust constraints on τ are currently obtained through an empirical cross-spectrum likelihood built from simulations. In this paper, we present a likelihood-free inference of τ from polarized Planck HFI maps which, for the first time, is fully based on neural networks (NNs). NNs have the advantage of not requiring an analytical description of the data and can be trained on state-of-the-art simulations, combining the information from multiple channels. By using Gaussian sky simulations and Planck SRoll2 simulations, including CMB, noise, and residual instrumental systematic effects, we trained, tested, and validated NN models considering different setups. We inferred the value of τ directly from Stokes Q and U maps at ∼4° pixel resolution, without computing angular power spectra. On Planck data, we obtained τNN = 0.0579 ± 0.0082, which is compatible with current EE cross-spectrum results but with a ∼30% larger uncertainty, which can be assigned to the inherent nonoptimality of our estimator and to the retraining procedure applied to avoid biases. While this paper does not improve on current cosmological constraints on τ, our analysis represents a first robust application of NN-based inference on real data, and highlights its potential as a promising tool for complementary analysis of near-future CMB experiments, also in view of the ongoing challenge to achieve the first detection of primordial gravitational waves.

Improving the Accuracy of Marginal Approximations in Likelihood-Free Inference via Localization

Likelihood-free methods are an essential tool for performing inference for implicit models which can be simulated from, but for which the corresponding likelihood is intractable. However, common likelihood-free methods do not scale well to a large number of model parameters. A promising approach to high-dimensional likelihood-free inference involves estimating low-dimensional marginal posteriors by conditioning only on summary statistics believed to be informative for the low-dimensional component, and then combining the low-dimensional approximations in some way. In this article, we demonstrate that such low-dimensional approximations can be surprisingly poor in practice for seemingly intuitive summary statistic choices. We describe an idealized low-dimensional summary statistic that is, in principle, suitable for marginal estimation. However, a direct approximation of the idealized choice is difficult in practice. We thus suggest an alternative approach to marginal estimation which is easier to implement and automate. Given an initial choice of low-dimensional summary statistic that might only be informative about a marginal posterior location, the new method improves performance by first crudely localizing the posterior approximation using all the summary statistics to ensure global identifiability, followed by a second step that hones in on an accurate low-dimensional approximation using the low-dimensional summary statistic. We show that the posterior this approach targets can be represented as a logarithmic pool of posterior distributions based on the low-dimensional and full summary statistics, respectively. The good performance of our method is illustrated in low to moderate dimensional examples. Computer code to implement the methods for the examples of this article is available at https://github.com/cdrovandi/ABC-marginal-approximations.

Robust Field-level Likelihood-free Inference with Galaxies

We train graph neural networks to perform field-level likelihood-free inference using galaxy catalogs from state-of-the-art hydrodynamic simulations of the CAMELS project. Our models are rotational, translational, and permutation invariant and do not impose any cut on scale. From galaxy catalogs that only contain 3D positions and radial velocities of ∼1000 galaxies in tiny volumes our models can infer the value of Ωm with approximately 12% precision. More importantly, by testing the models on galaxy catalogs from thousands of hydrodynamic simulations, each having a different efficiency of supernova and active galactic nucleus feedback, run with five different codes and subgrid models—IllustrisTNG, SIMBA, Astrid, Magneticum, SWIFT-EAGLE—we find that our models are robust to changes in astrophysics, subgrid physics, and subhalo/galaxy finder. Furthermore, we test our models on 1024 simulations that cover a vast region in parameter space—variations in five cosmological and 23 astrophysical parameters—finding that the model extrapolates really well. Our results indicate that the key to building a robust model is the use of both galaxy positions and velocities, suggesting that the network has likely learned an underlying physical relation that does not depend on galaxy formation and is valid on scales larger than ∼10 h −1 kpc.

Weighing the Milky Way and Andromeda galaxies with artificial intelligence

We present new constraints on the masses of the halos hosting the Milky Way and Andromeda galaxies derived using graph neural networks. Our models, trained on 2,000 state-of-the-art hydrodynamic simulations of the CAMELS project, only make use of the positions, velocities and stellar masses of the galaxies belonging to the halos, and are able to perform likelihood-free inference on halo masses while accounting for both cosmological and astrophysical uncertainties. Our constraints are in agreement with estimates from other traditional methods, within our derived posterior standard deviation.

Sequential Likelihood-Free Inference with Neural Proposal

Bayesian inference without the likelihood evaluation, or likelihood-free inference, has been a key research topic in simulation studies for gaining quantitatively validated simulation models on real-world datasets. As the likelihood evaluation is inaccessible, previous papers train the amortized neural network to estimate the ground-truth posterior for the simulation of interest. Training the network and accumulating the dataset alternatively in a sequential manner could save the total simulation budget by orders of magnitude. In the data accumulation phase, the new simulation inputs are chosen within a portion of the total simulation budget to accumulate upon the collected dataset so far. This newly accumulated data degenerates because the set of simulation inputs is hardly mixed, and this degenerated data collection process ruins the posterior inference. This paper introduces a new sampling approach, called Neural Proposal (NP), of the simulation input that resolves the biased data collection as it guarantees the i.i.d. sampling. The experiments show the improved performance of our sampler, especially for the simulations with multi-modal posteriors.

Distilling Importance Sampling for Likelihood Free Inference

Likelihood-free inference involves inferring parameter values given observed data and a simulator model. The simulator is computer code which takes parameters, performs stochastic calculations, and outputs simulated data. In this work, we view the simulator as a function whose inputs are (1) the parameters and (2) a vector of pseudo-random draws. We attempt to infer all these inputs conditional on the observations. This is challenging as the resulting posterior can be high dimensional and involves strong dependence. We approximate the posterior using normalizing flows, a flexible parametric family of densities. Training data is generated by likelihood-free importance sampling with a large bandwidth value ϵ , which makes the target similar to the prior. The training data is “distilled” by using it to train an updated normalizing flow. The process is iterated, using the updated flow as the importance sampling proposal, and slowly reducing ϵ so the target becomes closer to the posterior. Unlike most other likelihood-free methods, we avoid the need to reduce data to low-dimensional summary statistics, and hence can achieve more accurate results. We illustrate our method in two challenging examples, on queuing and epidemiology. Supplementary materials for this article are available online.

Sidestepping the inversion of the weak-lensing covariance matrix with Approximate Bayesian Computation

Weak gravitational lensing is one of the few direct methods to map the dark-matter distribution on large scales in the Universe, and to estimate cosmological parameters. We study a Bayesian inference problem where the data covariance C, estimated from a number ns of numerical simulations, is singular. In a cosmological context of large-scale structure observations, the creation of a large number of such N-body simulations is often prohibitively expensive. Inference based on a likelihood function often includes a precision matrix, Ψ=C−1. The covariance matrix corresponding to a p-dimensional data vector is singular for p≥ns, in which case the precision matrix is unavailable. We propose the likelihood-free inference method Approximate Bayesian Computation (ABC) as a solution that circumvents the inversion of the singular covariance matrix. We present examples of increasing degree of complexity, culminating in a realistic cosmological scenario of the determination of the weak-gravitational lensing power spectrum for the upcoming European Space Agency satellite Euclid. While we found the ABC parameter estimate variances to be mildly larger compared to likelihood-based approaches, which are restricted to settings with p<ns, we obtain unbiased parameter estimates with ABC even in extreme cases where p/ns≫1. The code has been made publicly available11https://github.com/emilleishida/CorrMatrix_ABC. to ensure the reproducibility of the results.

Calibrating Cosmological Simulations with Implicit Likelihood Inference Using Galaxy Growth Observables

In a novel approach employing implicit likelihood inference (ILI), also known as likelihood-free inference, we calibrate the parameters of cosmological hydrodynamic simulations against observations, which has previously been unfeasible due to the high computational cost of these simulations. For computational efficiency, we train neural networks as emulators on ∼1000 cosmological simulations from the CAMELS project to estimate simulated observables, taking as input the cosmological and astrophysical parameters, and use these emulators as surrogates for the cosmological simulations. Using the cosmic star formation rate density (SFRD) and, separately, the stellar mass functions (SMFs) at different redshifts, we perform ILI on selected cosmological and astrophysical parameters (Ω m , σ 8, stellar wind feedback, and kinetic black hole feedback) and obtain full six-dimensional posterior distributions. In the performance test, the ILI from the emulated SFRD (SMFs) can recover the target observables with a relative error of 0.17% (0.4%). We find that degeneracies exist between the parameters inferred from the emulated SFRD, confirmed with new full cosmological simulations. We also find that the SMFs can break the degeneracy in the SFRD, which indicates that the SMFs provide complementary constraints for the parameters. Further, we find that a parameter combination inferred from an observationally inferred SFRD reproduces the target observed SFRD very well, whereas, in the case of the SMFs, the inferred and observed SMFs show significant discrepancies that indicate potential limitations of the current galaxy formation modeling and calibration framework, and/or systematic differences and inconsistencies between observations of the SMFs.

Simulation-based inference of single-molecule force spectroscopy.

Single-molecule force spectroscopy (smFS) is a powerful approach to studying the microscopic mechanisms of molecular self-organization in biological cells and complex materials. However, extracting quantitative reduced models from these experiments is still challenging because coupling the molecule with the ever-present experimental measuring device introduces artifacts that blur the measurements. Performing statistical inference within the Bayesian framework is an attractive way to extract quantitative models from smFS experiments. But even minimalistic models lead to mathematically intractable likelihoods---ubiquitous in the computational sciences---that hinder standard approaches based on likelihood optimization. We develop a computational framework that performs Bayesian inference to extract reduced models based on diffusive dynamics on a free energy landscape. We build on a new generation of likelihood-free inference methods called simulation-based inference (SBI). By encoding a parametric mechanistic model into a simulator in combination with probabilistic deep learning, SBI enables us to directly estimate the Bayesian posterior, avoiding any likelihood calculation. Using synthetic data, we show how we can systematically disentangle the measurement of hidden molecular properties from experimental artifacts. By integrating physics-based parametric models with machine learning density estimation, SBI enables accurate Bayesian inference for models with an intractable likelihood. It paves the way for more complex and realistic models that would otherwise be ruled out due to their mathematical intractability. SBI is general, conceptually transparent, easy to use, and broadly applicable to other types of biophysical experiments.

Robust Field-level Inference of Cosmological Parameters with Dark Matter Halos

We train graph neural networks on halo catalogs from Gadget N-body simulations to perform field-level likelihood-free inference of cosmological parameters. The catalogs contain ≲5000 halos with masses ≳1010 h −1 M ⊙ in a periodic volume of ; every halo in the catalog is characterized by several properties such as position, mass, velocity, concentration, and maximum circular velocity. Our models, built to be permutationally, translationally, and rotationally invariant, do not impose a minimum scale on which to extract information and are able to infer the values of Ωm and σ 8 with a mean relative error of ∼6%, when using positions plus velocities and positions plus masses, respectively. More importantly, we find that our models are very robust: they can infer the value of Ωm and σ 8 when tested using halo catalogs from thousands of N-body simulations run with five different N-body codes: Abacus, CUBEP3M, Enzo, PKDGrav3, and Ramses. Surprisingly, the model trained to infer Ωm also works when tested on thousands of state-of-the-art CAMELS hydrodynamic simulations run with four different codes and subgrid physics implementations. Using halo properties such as concentration and maximum circular velocity allow our models to extract more information, at the expense of breaking the robustness of the models. This may happen because the different N-body codes are not converged on the relevant scales corresponding to these parameters.

Modularized Bayesian analyses and cutting feedback in likelihood-free inference

There has been much recent interest in modifying Bayesian inference for misspecified models so that it is useful for specific purposes. One popular modified Bayesian inference method is “cutting feedback” which can be used when the model consists of a number of coupled modules, with only some of the modules being misspecified. Cutting feedback methods represent the full posterior distribution in terms of conditional and sequential components, and then modify some terms in such a representation based on the modular structure for specification or computation of a modified posterior distribution. The main goal of this is to avoid contamination of inferences for parameters of interest by misspecified modules. Computation for cut posterior distributions is challenging, and here we consider cutting feedback for likelihood-free inference based on Gaussian mixture approximations to the joint distribution of parameters and data summary statistics. We exploit the fact that marginal and conditional distributions of a Gaussian mixture are Gaussian mixtures to give explicit approximations to marginal or conditional posterior distributions so that we can easily approximate cut posterior analyses. The mixture approach allows repeated approximation of posterior distributions for different data based on a single mixture fit. This is important for model checks which aid in the decision of whether to “cut”. A semi-modular approach to likelihood-free inference where feedback is partially cut is also developed. The benefits of the method are illustrated on two challenging examples, a collective cell spreading model and a continuous time model for asset returns with jumps.

Bayesian model comparison for simulation-based inference

Abstract Comparison of appropriate models to describe observational data is a fundamental task of science. The Bayesian model evidence, or marginal likelihood, is a computationally challenging, yet crucial, quantity to estimate to perform Bayesian model comparison. We introduce a methodology to compute the Bayesian model evidence in simulation-based inference (SBI) scenarios (often called likelihood-free inference). In particular, we leverage the recently proposed learned harmonic mean estimator and exploit the fact that it is decoupled from the method used to generate posterior samples, i.e. it requires posterior samples only, which may be generated by any approach. This flexibility, which is lacking in many alternative methods for computing the model evidence, allows us to develop SBI model comparison techniques for the three main neural density estimation approaches, including neural posterior estimation, neural likelihood estimation, and neural ratio estimation. We demonstrate and validate our SBI evidence calculation techniques on a range of inference problems, including a gravitational wave example. Moreover, we further validate the accuracy of the learned harmonic mean estimator, implemented in the harmonic software, in likelihood-based settings. These results highlight the potential of harmonic as a sampler-agnostic method to estimate the model evidence in both likelihood-based and simulation-based scenarios.