arXiv subjects

Augustin Marignier

Publications and source records attributed to Augustin Marignier.

5 recordsLinked to original sources

arXiv2026

A note on conditional densities, Bayes' rule, and recent criticisms of Bayesian inference

When performing Bayesian inference, we frequently need to work with conditional probability densities. For example, the posterior function is the conditional density of the parameters given the data. Some might worry that conditional densities are ill-defined, considering that for a continuous random variable $Y$, the event $\{Y=y\}$ has probability zero, meaning the formula $\mathbb{P}(A|B)=\mathbb{P}(A\cap B)/\mathbb{P}(B)$ is inapplicable. In reality, when we work with conditional densities, we never condition directly on the zero-probability event $\{Y=y\}$; rather, we first condition on the random variable $Y$, and then we may plug in an observed value $y$. The first purpose of our article is to provide an exposition on conditional densities that elaborates on this point. While we have aimed to make this explanation accessible, we follow it with a roadmap of the measure theory needed to make it rigorous. A recent preprint (arXiv:2411.13570) has expressed the concern that probability densities are ill-defined and that as a result Bayes' theorem cannot be used, and they provide examples that allegedly demonstrate inconsistencies in the Bayesian framework. The second purpose of our article is to investigate their claims. We contend that the examples given in their work do not demonstrate any inconsistencies; we find that there are mathematical errors and that they deviate significantly from the Bayesian framework.

stat.ME↗

arXiv2024

Fast emulation of anisotropies induced in the cosmic microwave background by cosmic strings

Cosmic strings are linear topological defects that may have been produced during symmetry-breaking phase transitions in the very early Universe. In an expanding Universe the existence of causally separate regions prevents such symmetries from being broken uniformly, with a network of cosmic string inevitably forming as a result. To faithfully generate observables of such processes requires computationally expensive numerical simulations, which prohibits many types of analyses. We propose a technique to instead rapidly emulate observables, thus circumventing simulation. Emulation is a form of generative modelling, often built upon a machine learning backbone. End-to-end emulation often fails due to high dimensionality and insufficient training data. Consequently, it is common to instead emulate a latent representation from which observables may readily be synthesised. Wavelet phase harmonics are an excellent latent representations for cosmological fields, both as a summary statistic and for emulation, since they do not require training and are highly sensitive to non-Gaussian information. Leveraging wavelet phase harmonics as a latent representation, we develop techniques to emulate string induced CMB anisotropies over a 7.2 degree field of view, with sub-arcminute resolution, in under a minute on a single GPU. Beyond generating high fidelity emulations, we provide a technique to ensure these observables are distributed correctly, providing a more representative ensemble of samples. The statistics of our emulations are commensurate with those calculated on comprehensive Nambu-Goto simulations. Our findings indicate these fast emulation approaches may be suitable for wide use in, e.g., simulation based inference pipelines. We make our code available to the community so that researchers may rapidly emulate cosmic string induced CMB anisotropies for their own analysis.

astro-ph.CO↗

arXiv2023

Sparse Bayesian mass-mapping using trans-dimensional MCMC

Uncertainty quantification is a crucial step of cosmological mass-mapping that is often ignored. Suggested methods are typically only approximate or make strong assumptions of Gaussianity of the shear field. Probabilistic sampling methods, such as Markov chain Monte Carlo (MCMC), draw samples form a probability distribution, allowing for full and flexible uncertainty quantification, however these methods are notoriously slow and struggle in the high-dimensional parameter spaces of imaging problems. In this work we use, for the first time, a trans-dimensional MCMC sampler for mass-mapping, promoting sparsity in a wavelet basis. This sampler gradually grows the parameter space as required by the data, exploiting the extremely sparse nature of mass maps in wavelet space. The wavelet coefficients are arranged in a tree-like structure, which adds finer scale detail as the parameter space grows. We demonstrate the trans-dimensional sampler on galaxy cluster-scale images where the planar modelling approximation is valid. In high-resolution experiments, this method produces naturally parsimonious solutions, requiring less than 1% of the potential maximum number of wavelet coefficients and still producing a good fit to the observed data. In the presence of noisy data, trans-dimensional MCMC produces a better reconstruction of mass-maps than the standard smoothed Kaiser-Squires method, with the addition that uncertainties are fully quantified. This opens up the possibility for new mass maps and inferences about the nature of dark matter using the new high-resolution data from upcoming weak lensing surveys such as Euclid.

astro-ph.CO↗

arXiv2022

Posterior sampling for inverse imaging problems on the sphere in seismology and cosmology

Inverse problems defined on the sphere arise in many fields, including seismology and cosmology where problems are defined on the globe and the cosmic sphere. These are generally high-dimensional and computationally very complex and, as a result, sampling the posterior of spherical inverse problems is a challenging task. In this work, we describe a framework that leverages a proximal Markov chain Monte Carlo (MCMC) algorithm to efficiently sample the high-dimensional space of spherical inverse problems with a sparsity-promoting wavelet prior. We detail the modifications needed for the algorithm to be applied to spherical problems, and give special consideration to the crucial forward modelling step which contains spherical harmonic transforms that are computationally expensive. By sampling the posterior, our framework allows for full and flexible uncertainty quantification, something which is not possible with other methods based on, for example, convex optimisation. We demonstrate our framework in practice on full-sky cosmological mass-mapping and on a common problem in global seismic tomography. We find that our approach is potentially useful at moderate resolutions, such as those of interest in seismology. Our framework is generally limited by resolution requirements, such as those required for astrophysical applications, due to the poor scaling of the complexity of spherical harmonic transforms with resolution. A new Python package, pxmcmc, containing the proximal MCMC sampler, measurement operators, wavelet transforms and sparse priors is made publicly available.

physics.data-an↗

arXiv2021

Efficient Generalized Spherical CNNs

Many problems across computer vision and the natural sciences require the analysis of spherical data, for which representations may be learned efficiently by encoding equivariance to rotational symmetries. We present a generalized spherical CNN framework that encompasses various existing approaches and allows them to be leveraged alongside each other. The only existing non-linear spherical CNN layer that is strictly equivariant has complexity $\mathcal{O}(C^2L^5)$, where $C$ is a measure of representational capacity and $L$ the spherical harmonic bandlimit. Such a high computational cost often prohibits the use of strictly equivariant spherical CNNs. We develop two new strictly equivariant layers with reduced complexity $\mathcal{O}(CL^4)$ and $\mathcal{O}(CL^3 \log L)$, making larger, more expressive models computationally feasible. Moreover, we adopt efficient sampling theory to achieve further computational savings. We show that these developments allow the construction of more expressive hybrid models that achieve state-of-the-art accuracy and parameter efficiency on spherical benchmark problems.

cs.CV↗