Hierarchical Bayesian Inference Research Articles

Inferring the spin distribution of binary black holes using deep learning

Abstract The spin characteristics of black holes offer valuable insights into the evolutionary pathways of their progenitor stars, crucial for understanding the broader population properties of black holes. Traditional Hierarchical Bayesian inference techniques employed to discern these properties often entail substantial time investments, and consensus regarding the spin distribution of Binary Black Hole (BBH) systems remains elusive. In this study, leveraging observations from GWTC-3, we adopt a machine learning approach to infer the spin distribution of black holes within BBH systems. Specifically, we develop a Deep Neural Network (DNN) and train it using data generated from a Beta distribution. Our training strategy, involving the segregation of data into 10 bins, not only expedites model training but also enhances the DNN's versatility and adaptability to accommodate the burgeoning volume of gravitational wave observations. Utilizing Monte Carlo-bootstrap (MC-bootstrap) to generate observation-simulated samples, we derive spin distribution parameters: $\alpha=1.3^{+0.25}_{-0.18},\beta=1.70^{+0.24}_{-0.29}$ for the larger BH sample and $\alpha=1.37^{+0.31}_{-0.20},\beta=1.63^{+0.30}_{-0.20}$ for the smaller BH sample. Within our constraints, the distributions of component spin magnitudes suggest the likelihood of both black holes in the BBH merger possessing non-zero spin.

Mathematical modelling, selection and hierarchical inference to determine the minimal dose in IFNα therapy against myeloproliferative neoplasms.

Myeloproliferative neoplasms (MPN) are blood cancers that appear after acquiring a driver mutation in a hematopoietic stem cell. These hematological malignancies result in the overproduction of mature blood cells and, if not treated, induce a risk of cardiovascular events and thrombosis. Pegylated IFN$\alpha $ is commonly used to treat MPN, but no clear guidelines exist concerning the dose prescribed to patients. We applied a model selection procedure and ran a hierarchical Bayesian inference method to decipher how dose variations impact the response to the therapy. We inferred that IFN$\alpha $ acts on mutated stem cells by inducing their differentiation into progenitor cells; the higher the dose, the higher the effect. We found that the treatment can induce long-term remission when a sufficient (patient-dependent) dose is reached. We determined this minimal dose for individuals in a cohort of patients and estimated the most suitable starting dose to give to a new patient to increase the chances of being cured.

Pain perception as hierarchical Bayesian inference: A test case for the theory of constructed emotion.

An intriguing perspective about human emotion, the theory of constructed emotion considers emotions as generative models according to the Bayesian brain hypothesis. This theory brings fresh insight to existing findings, but its complexity renders it challenging to test experimentally. We argue that laboratory studies of pain could support the theory because although some may not consider pain to be a genuine emotion, the theory must at minimum be able to explain pain perception and its dysfunction in pathology. We review emerging evidence that bear on this question. We cover behavioral and neural laboratory findings, computational models, placebo hyperalgesia, and chronic pain. We conclude that there is substantial evidence for a predictive processing account of painful experience, paving the way for a better understanding of neuronal and computational mechanisms of other emotions.

Eccentricity-induced systematic error on parametrized tests of general relativity: Hierarchical Bayesian inference applied to a binary black hole population

Eccentricity-induced systematic error on parametrized tests of general relativity: Hierarchical Bayesian inference applied to a binary black hole population

A comprehensive multi-fidelity surrogate framework based on Gaussian process for datasets with heterogeneous responses

In recent decades, the multi-fidelity (MF) surrogate framework has seen widespread application across various scenarios. This framework significantly enhances the efficiency of surrogate modeling by integrating data of varying fidelity levels. However, research on the MF surrogate has predominantly focused on regression settings, aiming at approximating continuous responses, with limited attention to classification problems involving discrete responses. Moreover, no existing studies have addressed a heterogeneous MF dataset, where the data types (continuous or discrete) vary between different fidelities. This paper introduces a comprehensive MF surrogate framework based on Gaussian process, designed to handle a wide range of MF dataset types. Our framework employs a hierarchical Kriging scheme and Bayesian inference, adaptable to any MF dataset type. Given the challenges posed by discrete data in analytical inference, we incorporate the Laplace approximation for the latent function associated with discrete data in our framework. Additionally, we introduce a local scaling factor specifically for low-correlation classification problems in MF settings. The superiority of our method over conventional ones is demonstrated through both synthetic and engineering examples. Notably, in an engineering example featuring a heterogeneous MF dataset, our proposed MF surrogate model achieves an error reduction exceeding 50%.

Linking fast and slow: The case for generative models.

A pervasive challenge in neuroscience is testing whether neuronal connectivity changes over time due to specific causes, such as stimuli, events, or clinical interventions. Recent hardware innovations and falling data storage costs enable longer, more naturalistic neuronal recordings. The implicit opportunity for understanding the self-organised brain calls for new analysis methods that link temporal scales: from the order of milliseconds over which neuronal dynamics evolve, to the order of minutes, days, or even years over which experimental observations unfold. This review article demonstrates how hierarchical generative models and Bayesian inference help to characterise neuronal activity across different time scales. Crucially, these methods go beyond describing statistical associations among observations and enable inference about underlying mechanisms. We offer an overview of fundamental concepts in state-space modeling and suggest a taxonomy for these methods. Additionally, we introduce key mathematical principles that underscore a separation of temporal scales, such as the slaving principle, and review Bayesian methods that are being used to test hypotheses about the brain with multiscale data. We hope that this review will serve as a useful primer for experimental and computational neuroscientists on the state of the art and current directions of travel in the complex systems modelling literature.

Discussion on "Bayesian meta-analysis of penetrance for cancer risk" by Thanthirige Lakshika M. Ruberu, Danielle Braun, Giovanni Parmigiani, and Swati Biswas.

Ruberu etal. (2023) introduce an elegant approach to fit a complicated meta-analysis problem with diverse reporting modalities into the framework of hierarchical Bayesian inference. We discuss issues related to some of the involved parametric model assumptions.

Ensuring Consistency between Noise and Detection in Hierarchical Bayesian Inference

Hierarchical Bayesian inference can simultaneously account for both measurement uncertainty and selection effects within astronomical catalogs. In particular, the imposed hierarchy encodes beliefs about the interdependence of the physical processes that generate the observed data. We show that several proposed approximations within the literature actually correspond to inferences that are incompatible with any physical detection process. This generically leads to biases and is associated with the assumption that detectability is independent of the observed data given the true source parameters. We show several examples of how this error can affect astrophysical inferences based on catalogs of coalescing binaries observed through gravitational waves, including misestimating the redshift evolution of the merger rate as well as incorrectly inferring that general relativity is the correct theory of gravity when it is not. In general, one cannot directly fit for the “detected distribution” and “divide out” the selection effects in post-processing. Similarly, when comparing theoretical predictions to observations, it is better to simulate detected data (including both measurement noise and selection effects) rather than comparing estimates of the detected distributions of event parameters (which include only selection effects). While the biases introduced by model misspecification from incorrect assumptions may be smaller than statistical uncertainty for moderate catalog sizes (O(100) events), they will nevertheless pose a significant barrier to precision measurements of astrophysical populations.

Binary Black Hole Spins: Model Selection with GWTC-3

The origin of the spins of stellar-mass black holes is still controversial, and angular momentum transport inside massive stars is one of the main sources of uncertainty. Here, we apply hierarchical Bayesian inference to derive constraints on spin models from the 59 most confident binary black hole merger events in the third gravitational-wave transient catalogue (GWTC-3). We consider up to five parameters: chirp mass, mass ratio, redshift, effective spin, and precessing spin. For the model selection, we use a set of binary population synthesis simulations spanning drastically different assumptions for black hole spins and natal kicks. In particular, our spin models range from the maximal to minimal efficiency of angular momentum transport in stars. We find that if we include the precessing spin parameter into our analysis, models predicting only vanishingly small spins are in tension with GWTC-3 data. On the other hand, models in which most spins are vanishingly small but that also include a subpopulation of tidally spun-up black holes are a good match to the data. Our results show that the precessing spin parameter has a crucial impact on model selection.

Hierarchical Bayesian Inference of Globular Cluster Properties

Abstract We present a hierarchical Bayesian inference approach to estimating the structural properties and the phase space center of a globular cluster (GC) given the spatial and kinematic information of its stars based on lowered isothermal cluster models. As a first step towards more realistic modelling of GCs, we built a differentiable, accurate emulator of the lowered isothermal distribution function using interpolation. The reliable gradient information provided by the emulator allows the use of Hamiltonian Monte Carlo methods to sample large Bayesian models with hundreds of parameters, thereby enabling inference on hierarchical models. We explore the use of hierarchical Bayesian modelling to address several issues encountered in observations of GC including an unknown GC center, incomplete data, and measurement errors. Our approach not only avoids the common technique of radial binning but also incorporates the aforementioned uncertainties in a robust and statistically consistent way. Through demonstrating the reliability of our hierarchical Bayesian model on simulations, our work lays out the foundation for more realistic and complex modelling of real GC data.

Growing pains: understanding the impact of likelihood uncertainty on hierarchical Bayesian inference for gravitational-wave astronomy

ABSTRACT Observations of gravitational waves emitted by merging compact binaries have provided tantalizing hints about stellar astrophysics, cosmology, and fundamental physics. However, the physical parameters describing the systems (mass, spin, distance) used to extract these inferences about the Universe are subject to large uncertainties. The most widely used method of performing these analyses requires performing many Monte Carlo integrals to marginalize over the uncertainty in the properties of the individual binaries and the survey selection bias. These Monte Carlo integrals are subject to fundamental statistical uncertainties. Previous treatments of this statistical uncertainty have focused on ensuring that the precision of the inferred inference is unaffected; however, these works have neglected the question of whether sufficient accuracy can also be achieved. In this work, we provide a practical exploration of the impact of uncertainty in our analyses and provide a suggested framework for verifying that astrophysical inferences made with the gravitational-wave transient catalogue are accurate. Applying our framework to models used by the LIGO–Virgo–KAGRA collaboration and in the wider literature, we find that Monte Carlo uncertainty in estimating the survey selection bias is the limiting factor in our ability to probe narrow population models and this will rapidly grow more problematic as the size of the observed population increases.

Distributions of the Density and Kinetic Temperature of the Molecular Gas in the Central Region of NGC 613 Using Hierarchical Bayesian Inference

We present position–position–velocity (PPV) cubes of the physical and chemical properties of the molecular medium in the central 1.2 kpc region of the active galaxy NGC 613 at a PPV resolution of 0.″8 × 0.″8 × 10 km s−1 (0.″8 = ∼68 pc). We used eight molecular lines (13CO(1–0), C18O(1–0), HCN(1–0), HCO+(1–0), CS(2–1), HCN(4–3), HCO+(4–3), and CS(7–6)) obtained with the Atacama Large Millimeter/submillimeter Array. Non-LTE calculation with hierarchical Bayesian inference was used to construct PPV cubes of the gas kinetic temperature (T kin), molecular hydrogen volume density (), column densities (), and fractional abundances of four molecules (12C18O, HCN, HCO+, and CS). The derived , , and T kin ranged from 103.21−3.85 cm−3, 1020.8−22.1 cm−2, and 102.33−2.64 K, respectively. Our first application of the non-LTE method with the hierarchical Bayesian inference to external galaxies yielded compatible results compared with the previous studies of this galaxy, demonstrating the efficacy of this method for application to other galaxies. We examined the correlation between gas surface density (converted from ) and the star formation rate ΣSFR obtained from the 110 GHz continuum flux map and found two distinct sequences in the –ΣSFR diagram; the southwestern subregion of the star-forming ring exhibited a ∼0.5 dex higher star formation efficiency (SFE; ΣSFR/) than the eastern subregion. However, they exhibited no systematic difference in , which is often argued as a driver of SFE variation. We suggest that the deficiency of molecular gas in the southwestern subregion, where no significant gas supply is evident along the offset ridges in the bar, is responsible for the elevated SFE.

Weak lensing tomographic redshift distribution inference for the Hyper Suprime-Cam Subaru Strategic Program three-year shape catalogue

ABSTRACTWe present posterior sample redshift distributions for the Hyper Suprime-Cam Subaru Strategic Program Weak Lensing three-year (HSC Y3) analysis. Using the galaxies’ photometry and spatial cross-correlations, we conduct a combined Bayesian Hierarchical Inference of the sample redshift distributions. The spatial cross-correlations are derived using a subsample of Luminous Red Galaxies (LRGs) with accurate redshift information available up to a photometric redshift of z < 1.2. We derive the photometry-based constraints using a combination of two empirical techniques calibrated on spectroscopic and multiband photometric data that cover a spatial subset of the shear catalogue. The limited spatial coverage induces a cosmic variance error budget that we include in the inference. Our cross-correlation analysis models the photometric redshift error of the LRGs to correct for systematic biases and statistical uncertainties. We demonstrate consistency between the sample redshift distributions derived using the spatial cross-correlations, the photometry, and the posterior of the combined analysis. Based on this assessment, we recommend conservative priors for sample redshift distributions of tomographic bins used in the three-year cosmological Weak Lensing analyses.

Probing the delay time of supermassive black hole binary mergers with gravitational waves

ABSTRACT Merging supermassive black hole binaries are expected as a consequence of galaxy mergers, yet the detailed evolution path and underlying merging mechanisms of these binaries are still subject to large theoretical uncertainties. In this work, we propose to combine the (future) gravitational wave measurements of supermassive black hole binary merger events with the galaxy merger rate distributions from galaxy surveys/cosmological simulations, to infer the delay time of binary mergers, as a function of binary mass. The delay time encodes key information about binary evolution, which can be used to test the predictions of various evolution models. With a Mock data set of supermassive black hole binary merger events, we discuss how to infer the distribution of delay time with hierarchical Bayesian inference and test evolution models with the Bayesian model selection method. The astrophysical model uncertainties are also considered in the hierarchical Bayesian inference and Bayesian model selection.

Ponderings on the Possible Preponderance of Perpendicular Planets

Misalignments between planetary orbits and the equatorial planes of their host stars are clues about the formation and evolution of planetary systems. Earlier work found evidence for a peak near 90° in the distribution of stellar obliquities, based on frequentist tests. We performed hierarchical Bayesian inference on a sample of 174 planets for which either the full three-dimensional stellar obliquity has been measured (72 planets) or for which only the sky-projected stellar obliquity has been measured (102 planets). We investigated whether the obliquities are best described by a Rayleigh distribution or by a mixture of a Rayleigh distribution representing well-aligned systems and a different distribution representing misaligned systems. The mixture models are strongly favored over the single-component distribution. For the misaligned component, we tried an isotropic distribution and a distribution peaked at 90° and found the evidence to be essentially the same for both models. Thus, our Bayesian inference engine did not find strong evidence favoring a “perpendicular peak,” unlike the frequentist tests. We also investigated selection biases that affect the inferred obliquity distribution, such as the bias of the gravity-darkening method against obliquities near 0° or 180°. Further progress in characterizing the obliquity distribution will probably require the construction of a more homogeneous and complete sample of measurements.

Model exploration in gravitational-wave astronomy with the maximum population likelihood

Hierarchical Bayesian inference is an essential tool for studying the population properties of compact binaries with gravitational waves. The basic premise is to infer the unknown prior distribution of binary black hole and/or neutron star parameters such component masses, spin vectors, and redshift. These distributions shed light on the fate of massive stars, how and where binaries are assembled, and the evolution of the Universe over cosmic time. Hierarchical analyses model the binary black hole population using a prior distribution conditioned on hyperparameters, which are inferred from the data. However, a misspecified model can lead to faulty astrophysical inferences. In this paper we answer the question: given some data, which prior distribution---from the set of all possible prior distributions---produces the largest possible population likelihood? This distribution (which is not a true prior) is $\overline{)\ensuremath{\pi}}$ (pronounced ``pi stroke''), and the associated maximum population likelihood is $\overline{)\mathcal{L}}$ (pronounced ``L stroke''). The structure of $\overline{)\ensuremath{\pi}}$ is a linear superposition of delta functions, a result which follows from Carath\'eodory's theorem. We show how $\overline{)\ensuremath{\pi}}$ and $\overline{)\mathcal{L}}$ can be used for model exploration/criticism. We apply this $\overline{)\mathcal{L}}$ formalism to study the population of binary black hole mergers observed in the LIGO-Virgo-KAGRA Collaboration's third gravitational-wave transient catalog. Based on our results, we discuss possible improvements for gravitational-wave population models.

Learning to Learn Functions.

Humans can learn complex functional relationships between variables from small amounts of data. In doing so, they draw on prior expectations about the form of these relationships. In three experiments, we show that people learn to adjust these expectations through experience, learning about the likely forms of the functions they will encounter. Previous work has used Gaussian processes-a statistical framework that extends Bayesian nonparametric approaches to regression-to model human function learning. We build on this work, modeling the process of learning to learn functions as a form of hierarchical Bayesian inference about the Gaussian processhyperparameters.

Rapid hierarchical inference of neutron star equation of state from multiple gravitational wave observations of binary neutron star coalescences

Bayesian hierarchical inference of phenomenological parametrized neutron star equations of state (EoS) from multiple gravitational wave observations of binary neutron star mergers is of fundamental importance in improving our understanding of neutron star structure, the general properties of matter at supra nuclear densities, and the strong nuclear force. However, such an analysis is computationally costly, as it is unable to reuse single-event EoS agnostic parameter estimation runs that are carried out regardless for generating gravitational wave transient catalogs. With the number of events expected to be observable during the fourth observing run (O4) of LIGO/Virgo/KAGRA, this problem can only be expected to worsen. We develop a novel and robust algorithm for rapid and computationally cheap hierarchical inference of parametrized EoS from gravitational wave data which reuses single-event EoS agnostic parameter estimation samples to significantly reduce computational cost. We efficiently include a priori knowledge of neutron star physics as Bayesian priors on the EoS parameters. The high speed and low computational cost of our method allow for efficient recomputation of EoS inference every time a new binary neutron star event is discovered or whenever new observations and theoretical discoveries change the prior on EoS parameters. We test our method on both real and simulated gravitational wave data to demonstrate its accuracy. We show that our computationally cheap method produces EoS constraints that are completely consistent with existing analysis for real data, the chosen fiducial EoS for simulated data. Armed with our fast analysis scheme, we also study the variability of EoS constraints with binary neutron star properties for sets of simulated events drawn in different signal-to-noise ratio and mass ranges.

Theoretical challenges in single molecule.

Biological processes are directed by molecular actors conspiring, often in small numbers, to achieve tasks relevant to life. Yet these actors are not directly observable. Within the fluorescence paradigm, multiple layers of stochasticity lie between what we care about (molecular actor behavior) and what we observe (detector counts). These hierarchically organized layers include, but are not limited to, incomplete labeling, label photo-physics, optical aberrations, and detector noise. As a direct consequence, it is not possible to speak of discovering unique models of molecular behavior from data. Indeed many models are consistent with the data and our goal as theorists is to assign probabilities to candidate models based on observation. Here we give an overview of hierarchical Bayesian inference models and MCMC strategies for single molecule experiments with an emphasis on nonparametric Bayesian analysis of single photon smFRET highlighting a sequence of three papers to appear in Biophysical Reports.

Towards a reliable reconstruction of the power spectrum of primordial curvature perturbation on small scales from GWTC-3

Primordial black holes (PBHs) can be both candidates of dark matter and progenitors of binary black holes (BBHs) detected by the LIGO-Virgo-KAGRA collaboration. Since PBHs could form in the very early Universe through the gravitational collapse of primordial density perturbations, the population of BBHs detected by gravitational waves encodes much information on primordial curvature perturbation. In this work, we take a reliable and systematic approach to reconstruct the power spectrum of the primordial curvature perturbation from GWTC-3, under the hierarchical Bayesian inference framework, by accounting for the measurement uncertainties and selection effects. In addition to just considering the single PBH population model, we also report the results considering the multi-population model, i.e., the mixed PBH and astrophysical black hole binaries model. We find that the maximum amplitude of the reconstructed power spectrum of primordial curvature perturbation can be ∼2.5×10−2 at O(105)Mpc−1 scales, which is consistent with the PBH formation scenario from inflation at small scales.