Solving Bayesian inverse problems with diffusion priors and off-policy RL

This paper presents a practical application of Relative Trajectory Balance (RTB), a recently introduced off-policy reinforcement learning (RL) objective that can asymptotically solve Bayesian inverse problems optimally. We extend the original work by using RTB to train conditional diffusion model posteriors from pretrained unconditional priors for challenging linear and non-linear inverse problems in vision, and science. We use the objective alongside techniques such as off-policy backtracking exploration to improve training. Importantly, our results show that existing training-free diffusion posterior methods struggle to perform effective posterior inference in latent space due to inherent biases.

IRIS: A Bayesian Approach for Image Reconstruction in Radio Interferometry with expressive Score-Based priors

Inferring sky surface brightness distributions from noisy interferometric data in a principled statistical framework has been a key challenge in radio astronomy. In this work, we introduce Imaging for Radio Interferometry with Score-based models (IRIS). We use score-based models trained on optical images of galaxies as an expressive prior in combination with a Gaussian likelihood in the uv-space to infer images of protoplanetary disks from visibility data of the DSHARP survey conducted by ALMA. We demonstrate the advantages of this framework compared with traditional radio interferometry imaging algorithms, showing that it produces plausible posterior samples despite the use of a misspecified galaxy prior. Through coverage testing on simulations, we empirically evaluate the accuracy of this approach to generate calibrated posterior samples.

Tackling the Problem of Distributional Shifts: Correcting Misspecified, High-Dimensional Data-Driven Priors for Inverse Problems

Bayesian inference for inverse problems hinges critically on the choice of priors. In the absence of specific prior information, population-level distributions can serve as effective priors for parameters of interest. With the advent of machine learning, the use of data-driven population-level distributions (encoded, e.g., in a trained deep neural network) as priors is emerging as an appealing alternative to simple parametric priors in a variety of inverse problems. However, in many astrophysical applications, it is often difficult or even impossible to acquire independent and identically distributed samples from the underlying data-generating process of interest to train these models. In these cases, corrupted data or a surrogate, e.g. a simulator, is often used to produce training samples, meaning that there is a risk of obtaining misspecified priors. This, in turn, can bias the inferred posteriors in ways that are difficult to quantify, which limits the potential applicability of these models in real-world scenarios. In this work, we propose addressing this issue by iteratively updating the population-level distributions by retraining the model with posterior samples from different sets of observations and showcase the potential of this method on the problem of background image reconstruction in strong gravitational lensing when score-based models are used as data-driven priors. We show that starting from a misspecified prior distribution, the updated distribution becomes progressively closer to the underlying population-level distribution, and the resulting posterior samples exhibit reduced bias after several updates.

PQMass: Probabilistic Assessment of the Quality of Generative Models using Probability Mass Estimation

We propose a comprehensive sample-based method for assessing the quality of generative models. The proposed approach enables the estimation of the probability that two sets of samples are drawn from the same distribution, providing a statistically rigorous method for assessing the performance of a single generative model or the comparison of multiple competing models trained on the same dataset. This comparison can be conducted by dividing the space into non-overlapping regions and comparing the number of data samples in each region. The method only requires samples from the generative model and the test data. It is capable of functioning directly on high-dimensional data, obviating the need for dimensionality reduction. Significantly, the proposed method does not depend on assumptions regarding the density of the true distribution, and it does not rely on training or fitting any auxiliary models. Instead, it focuses on approximating the integral of the density (probability mass) across various sub-regions within the data space.

Improving Gradient-guided Nested Sampling for Posterior Inference

We present a performant, general-purpose gradient-guided nested sampling algorithm, ${\tt GGNS}$, combining the state of the art in differentiable programming, Hamiltonian slice sampling, clustering, mode separation, dynamic nested sampling, and parallelization. This unique combination allows ${\tt GGNS}$ to scale well with dimensionality and perform competitively on a variety of synthetic and real-world problems. We also show the potential of combining nested sampling with generative flow networks to obtain large amounts of high-quality samples from the posterior distribution. This combination leads to faster mode discovery and more accurate estimates of the partition function.

Echoes in the Noise: Posterior Samples of Faint Galaxy Surface Brightness Profiles with Score-Based Likelihoods and Priors

Examining the detailed structure of galaxy populations provides valuable insights into their formation and evolution mechanisms. Significant barriers to such analysis are the non-trivial noise properties of real astronomical images and the point spread function (PSF) which blurs structure. Here we present a framework which combines recent advances in score-based likelihood characterization and diffusion model priors to perform a Bayesian analysis of image deconvolution. The method, when applied to minimally processed \emph{Hubble Space Telescope} (\emph{HST}) data, recovers structures which have otherwise only become visible in next-generation \emph{James Webb Space Telescope} (\emph{JWST}) imaging.

Bayesian Imaging for Radio Interferometry with Score-Based Priors

The inverse imaging task in radio interferometry is a key limiting factor to retrieving Bayesian uncertainties in radio astronomy in a computationally effective manner. We use a score-based prior derived from optical images of galaxies to recover images of protoplanetary disks from the DSHARP survey. We demonstrate that our method produces plausible posterior samples despite the misspecified galaxy prior. We show that our approach produces results which are competitive with existing radio interferometry imaging algorithms.

Lie Point Symmetry and Physics Informed Networks

Symmetries have been leveraged to improve the generalization of neural networks through different mechanisms from data augmentation to equivariant architectures. However, despite their potential, their integration into neural solvers for partial differential equations (PDEs) remains largely unexplored. We explore the integration of PDE symmetries, known as Lie point symmetries, in a major family of neural solvers known as physics-informed neural networks (PINNs). We propose a loss function that informs the network about Lie point symmetries in the same way that PINN models try to enforce the underlying PDE through a loss function. Intuitively, our symmetry loss ensures that the infinitesimal generators of the Lie group conserve the PDE solutions. Effectively, this means that once the network learns a solution, it also learns the neighbouring solutions generated by Lie point symmetries. Empirical evaluations indicate that the inductive bias introduced by the Lie point symmetries of the PDEs greatly boosts the sample efficiency of PINNs.

Sampling-Based Accuracy Testing of Posterior Estimators for General Inference

Parameter inference, i.e. inferring the posterior distribution of the parameters of a statistical model given some data, is a central problem to many scientific disciplines. Posterior inference with generative models is an alternative to methods such as Markov Chain Monte Carlo, both for likelihood-based and simulation-based inference. However, assessing the accuracy of posteriors encoded in generative models is not straightforward. In this paper, we introduce `distance to random point' (DRP) coverage testing as a method to estimate coverage probabilities of generative posterior estimators. Our method differs from previously-existing coverage-based methods, which require posterior evaluations. We prove that our approach is necessary and sufficient to show that a posterior estimator is optimal. We demonstrate the method on a variety of synthetic examples, and show that DRP can be used to test the results of posterior inference analyses in high-dimensional spaces. We also show that our method can detect non-optimal inferences in cases where existing methods fail.

Pixelated Reconstruction of Foreground Density and Background Surface Brightness in Gravitational Lensing Systems using Recurrent Inference Machines

Modeling strong gravitational lenses in order to quantify the distortions in the images of background sources and to reconstruct the mass density in the foreground lenses has been a difficult computational challenge. As the quality of gravitational lens images increases, the task of fully exploiting the information they contain becomes computationally and algorithmically more difficult. In this work, we use a neural network based on the Recurrent Inference Machine (RIM) to simultaneously reconstruct an undistorted image of the background source and the lens mass density distribution as pixelated maps. The method iteratively reconstructs the model parameters (the image of the source and a pixelated density map) by learning the process of optimizing the likelihood given the data using the physical model (a ray-tracing simulation), regularized by a prior implicitly learned by the neural network through its training data. When compared to more traditional parametric models, the proposed method is significantly more expressive and can reconstruct complex mass distributions, which we demonstrate by using realistic lensing galaxies taken from the IllustrisTNG cosmological hydrodynamic simulation.