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.
Bayesian approaches for learning deep neural networks (BNN) have been received much attention and successfully applied to various applications. Particularly, BNNs have the merit of having better generalization ability as well as better uncertainty quantification. For the success of BNN, search an appropriate architecture of the neural networks is an important task, and various algorithms to find good sparse neural networks have been proposed. In this paper, we propose a new node-sparse BNN model which has good theoretical properties and is computationally feasible. We prove that the posterior concentration rate to the true model is near minimax optimal and adaptive to the smoothness of the true model. In particular the adaptiveness is the first of its kind for node-sparse BNNs. In addition, we develop a novel MCMC algorithm which makes the Bayesian inference of the node-sparse BNN model feasible in practice.
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.
Gaussian process regression is widely applied in computational science and engineering for surrogate modeling owning to its kernel-based and probabilistic nature. In this work, we propose a Bayesian approach that integrates the variability of input data into the Gaussian process regression for function and partial differential equation approximation. Leveraging two types of observables -- noise-corrupted outputs with certain inputs and those with prior-distribution-defined uncertain inputs, a posterior distribution of uncertain inputs is estimated via Bayesian inference. Thereafter, such quantified uncertainties of inputs are incorporated into Gaussian process predictions by means of marginalization. The setting of two types of data aligned with common scenarios of constructing surrogate models for the solutions of partial differential equations, where the data of boundary conditions and initial conditions are typically known while the data of solution may involve uncertainties due to the measurement or stochasticity. The effectiveness of the proposed method is demonstrated through several numerical examples including multiple one-dimensional functions, the heat equation and Allen-Cahn equation. A consistently good performance of generalization is observed, and a substantial reduction in the predictive uncertainties is achieved by the Bayesian inference of uncertain inputs.
Estimating the distribution over failures is a key step in validating autonomous systems. Existing approaches focus on finding failures for a small range of initial conditions or make restrictive assumptions about the properties of the system under test. We frame estimating the distribution over failure trajectories for sequential systems as Bayesian inference. Our model-based approach represents the distribution over failure trajectories using rollouts of system dynamics and computes trajectory gradients using automatic differentiation. Our approach is demonstrated in an inverted pendulum control system, an autonomous vehicle driving scenario, and a partially observable lunar lander. Sampling is performed using an off-the-shelf implementation of Hamiltonian Monte Carlo with multiple chains to capture multimodality and gradient smoothing for safe trajectories. In all experiments, we observed improvements in sample efficiency and parameter space coverage compared to black-box baseline approaches. This work is open sourced.
We present CosmoPower-JAX, a JAX-based implementation of the CosmoPower framework, which accelerates cosmological inference by building neural emulators of cosmological power spectra. We show how, using the automatic differentiation, batch evaluation and just-in-time compilation features of JAX, and running the inference pipeline on graphics processing units (GPUs), parameter estimation can be accelerated by orders of magnitude with advanced gradient-based sampling techniques. These can be used to efficiently explore high-dimensional parameter spaces, such as those needed for the analysis of next-generation cosmological surveys. We showcase the accuracy and computational efficiency of CosmoPower-JAX on two simulated Stage IV configurations. We first consider a single survey performing a cosmic shear analysis totalling 37 model parameters. We validate the contours derived with CosmoPower-JAX and a Hamiltonian Monte Carlo sampler against those derived with a nested sampler and without emulators, obtaining a speed-up factor of $\mathcal{O}(10^3)$. We then consider a combination of three Stage IV surveys, each performing a joint cosmic shear and galaxy clustering (3x2pt) analysis, for a total of 157 model parameters. Even with such a high-dimensional parameter space, CosmoPower-JAX provides converged posterior contours in 3 days, as opposed to the estimated 6 years required by standard methods. CosmoPower-JAX is fully written in Python, and we make it publicly available to help the cosmological community meet the accuracy requirements set by next-generation surveys.
Visual Word Sense Disambiguation (VWSD) is a task to find the image that most accurately depicts the correct sense of the target word for the given context. Previously, image-text matching models often suffered from recognizing polysemous words. This paper introduces an unsupervised VWSD approach that uses gloss information of an external lexical knowledge-base, especially the sense definitions. Specifically, we suggest employing Bayesian inference to incorporate the sense definitions when sense information of the answer is not provided. In addition, to ameliorate the out-of-dictionary (OOD) issue, we propose a context-aware definition generation with GPT-3. Experimental results show that the VWSD performance significantly increased with our Bayesian inference-based approach. In addition, our context-aware definition generation achieved prominent performance improvement in OOD examples exhibiting better performance than the existing definition generation method.
Bayesian neural networks (BNNs) have recently regained a significant amount of attention in the deep learning community due to the development of scalable approximate Bayesian inference techniques. There are several advantages of using a Bayesian approach: Parameter and prediction uncertainties become easily available, facilitating rigorous statistical analysis. Furthermore, prior knowledge can be incorporated. However, so far, there have been no scalable techniques capable of combining both structural and parameter uncertainty. In this paper, we apply the concept of model uncertainty as a framework for structural learning in BNNs and hence make inference in the joint space of structures/models and parameters. Moreover, we suggest an adaptation of a scalable variational inference approach with reparametrization of marginal inclusion probabilities to incorporate the model space constraints. Experimental results on a range of benchmark datasets show that we obtain comparable accuracy results with the competing models, but based on methods that are much more sparse than ordinary BNNs.
We study Individual Fairness (IF) for Bayesian neural networks (BNNs). Specifically, we consider the $\epsilon$-$\delta$-individual fairness notion, which requires that, for any pair of input points that are $\epsilon$-similar according to a given similarity metrics, the output of the BNN is within a given tolerance $\delta>0.$ We leverage bounds on statistical sampling over the input space and the relationship between adversarial robustness and individual fairness to derive a framework for the systematic estimation of $\epsilon$-$\delta$-IF, designing Fair-FGSM and Fair-PGD as global,fairness-aware extensions to gradient-based attacks for BNNs. We empirically study IF of a variety of approximately inferred BNNs with different architectures on fairness benchmarks, and compare against deterministic models learnt using frequentist techniques. Interestingly, we find that BNNs trained by means of approximate Bayesian inference consistently tend to be markedly more individually fair than their deterministic counterparts.
This paper introduces a stochastic plug-and-play (PnP) sampling algorithm that leverages variable splitting to efficiently sample from a posterior distribution. The algorithm based on split Gibbs sampling (SGS) draws inspiration from the alternating direction method of multipliers (ADMM). It divides the challenging task of posterior sampling into two simpler sampling problems. The first problem depends on the likelihood function, while the second is interpreted as a Bayesian denoising problem that can be readily carried out by a deep generative model. Specifically, for an illustrative purpose, the proposed method is implemented in this paper using state-of-the-art diffusion-based generative models. Akin to its deterministic PnP-based counterparts, the proposed method exhibits the great advantage of not requiring an explicit choice of the prior distribution, which is rather encoded into a pre-trained generative model. However, unlike optimization methods (e.g., PnP-ADMM) which generally provide only point estimates, the proposed approach allows conventional Bayesian estimators to be accompanied by confidence intervals at a reasonable additional computational cost. Experiments on commonly studied image processing problems illustrate the efficiency of the proposed sampling strategy. Its performance is compared to recent state-of-the-art optimization and sampling methods.
Brain-computer interfaces (BCIs), particularly the P300 BCI, facilitate direct communication between the brain and computers. The fundamental statistical problem in P300 BCIs lies in classifying target and non-target stimuli based on electroencephalogram (EEG) signals. However, the low signal-to-noise ratio (SNR) and complex spatial/temporal correlations of EEG signals present challenges in modeling and computation, especially for individuals with severe physical disabilities-BCI's primary users. To address these challenges, we introduce a novel Gaussian Latent channel model with Sparse time-varying effects (GLASS) under a fully Bayesian framework. GLASS is built upon a constrained multinomial logistic regression particularly designed for the imbalanced target and non-target stimuli. The novel latent channel decomposition efficiently alleviates strong spatial correlations between EEG channels, while the soft-thresholded Gaussian process (STGP) prior ensures sparse and smooth time-varying effects. We demonstrate GLASS substantially improves BCI's performance in participants with amyotrophic lateral sclerosis (ALS) and identifies important EEG channels (PO8, Oz, PO7, and Pz) in parietal and occipital regions that align with existing literature. For broader accessibility, we develop an efficient gradient-based variational inference (GBVI) algorithm for posterior computation and provide a user-friendly Python module available at https://github.com/BangyaoZhao/GLASS.
Bayesian methods for solving inverse problems are a powerful alternative to classical methods since the Bayesian approach offers the ability to quantify the uncertainty in the solution. In recent years, data-driven techniques for solving inverse problems have also been remarkably successful, due to their superior representation ability. In this work, we incorporate data-based models into a class of Langevin-based sampling algorithms for Bayesian inference in imaging inverse problems. In particular, we introduce NF-ULA (Normalizing Flow-based Unadjusted Langevin algorithm), which involves learning a normalizing flow (NF) as the image prior. We use NF to learn the prior because a tractable closed-form expression for the log prior enables the differentiation of it using autograd libraries. Our algorithm only requires a normalizing flow-based generative network, which can be pre-trained independently of the considered inverse problem and the forward operator. We perform theoretical analysis by investigating the well-posedness and non-asymptotic convergence of the resulting NF-ULA algorithm. The efficacy of the proposed NF-ULA algorithm is demonstrated in various image restoration problems such as image deblurring, image inpainting, and limited-angle X-ray computed tomography (CT) reconstruction. NF-ULA is found to perform better than competing methods for severely ill-posed inverse problems.
Meta-learning is a framework for learning learning algorithms through repeated interactions with an environment as opposed to designing them by hand. In recent years, this framework has established itself as a promising tool for building models of human cognition. Yet, a coherent research program around meta-learned models of cognition is still missing. The purpose of this article is to synthesize previous work in this field and establish such a research program. We rely on three key pillars to accomplish this goal. We first point out that meta-learning can be used to construct Bayes-optimal learning algorithms. This result not only implies that any behavioral phenomenon that can be explained by a Bayesian model can also be explained by a meta-learned model but also allows us to draw strong connections to the rational analysis of cognition. We then discuss several advantages of the meta-learning framework over traditional Bayesian methods. In particular, we argue that meta-learning can be applied to situations where Bayesian inference is impossible and that it enables us to make rational models of cognition more realistic, either by incorporating limited computational resources or neuroscientific knowledge. Finally, we reexamine prior studies from psychology and neuroscience that have applied meta-learning and put them into the context of these new insights. In summary, our work highlights that meta-learning considerably extends the scope of rational analysis and thereby of cognitive theories more generally.
Recent advances have shown that GP priors, or their finite realisations, can be encoded using deep generative models such as variational autoencoders (VAEs). These learned generators can serve as drop-in replacements for the original priors during MCMC inference. While this approach enables efficient inference, it loses information about the hyperparameters of the original models, and consequently makes inference over hyperparameters impossible and the learned priors indistinct. To overcome this limitation, we condition the VAE on stochastic process hyperparameters. This allows the joint encoding of hyperparameters with GP realizations and their subsequent estimation during inference. Further, we demonstrate that our proposed method, PriorCVAE, is agnostic to the nature of the models which it approximates, and can be used, for instance, to encode solutions of ODEs. It provides a practical tool for approximate inference and shows potential in real-life spatial and spatiotemporal applications.
Bayesian inference provides a methodology for parameter estimation and uncertainty quantification in machine learning and deep learning methods. Variational inference and Markov Chain Monte-Carlo (MCMC) sampling methods are used to implement Bayesian inference. In the past three decades, MCMC sampling methods have faced some challenges in being adapted to larger models (such as in deep learning) and big data problems. Advanced proposal distributions that incorporate gradients, such as a Langevin proposal distribution, provide a means to address some of the limitations of MCMC sampling for Bayesian neural networks. Furthermore, MCMC methods have typically been constrained to statisticians and currently not well-known among deep learning researchers. We present a tutorial for MCMC methods that covers simple Bayesian linear and logistic models, and Bayesian neural networks. The aim of this tutorial is to bridge the gap between theory and implementation via coding, given a general sparsity of libraries and tutorials to this end. This tutorial provides code in Python with data and instructions that enable their use and extension. We provide results for some benchmark problems showing the strengths and weaknesses of implementing the respective Bayesian models via MCMC. We highlight the challenges in sampling multi-modal posterior distributions for the case of Bayesian neural networks and the need for further improvement of convergence diagnosis methods.
Learning-based outlier (mismatched correspondence) rejection for robust 3D registration generally formulates the outlier removal as an inlier/outlier classification problem. The core for this to be successful is to learn the discriminative inlier/outlier feature representations. In this paper, we develop a novel variational non-local network-based outlier rejection framework for robust alignment. By reformulating the non-local feature learning with variational Bayesian inference, the Bayesian-driven long-range dependencies can be modeled to aggregate discriminative geometric context information for inlier/outlier distinction. Specifically, to achieve such Bayesian-driven contextual dependencies, each query/key/value component in our non-local network predicts a prior feature distribution and a posterior one. Embedded with the inlier/outlier label, the posterior feature distribution is label-dependent and discriminative. Thus, pushing the prior to be close to the discriminative posterior in the training step enables the features sampled from this prior at test time to model high-quality long-range dependencies. Notably, to achieve effective posterior feature guidance, a specific probabilistic graphical model is designed over our non-local model, which lets us derive a variational low bound as our optimization objective for model training. Finally, we propose a voting-based inlier searching strategy to cluster the high-quality hypothetical inliers for transformation estimation. Extensive experiments on 3DMatch, 3DLoMatch, and KITTI datasets verify the effectiveness of our method.
We present a hierarchical Bayesian learning approach to infer jointly sparse parameter vectors from multiple measurement vectors. Our model uses separate conditionally Gaussian priors for each parameter vector and common gamma-distributed hyper-parameters to enforce joint sparsity. The resulting joint-sparsity-promoting priors are combined with existing Bayesian inference methods to generate a new family of algorithms. Our numerical experiments, which include a multi-coil magnetic resonance imaging application, demonstrate that our new approach consistently outperforms commonly used hierarchical Bayesian methods.
Pretrained language models often generate outputs that are not in line with human preferences, such as harmful text or factually incorrect summaries. Recent work approaches the above issues by learning from a simple form of human feedback: comparisons between pairs of model-generated outputs. However, comparison feedback only conveys limited information about human preferences. In this paper, we introduce Imitation learning from Language Feedback (ILF), a new approach that utilizes more informative language feedback. ILF consists of three steps that are applied iteratively: first, conditioning the language model on the input, an initial LM output, and feedback to generate refinements. Second, selecting the refinement incorporating the most feedback. Third, finetuning the language model to maximize the likelihood of the chosen refinement given the input. We show theoretically that ILF can be viewed as Bayesian Inference, similar to Reinforcement Learning from human feedback. We evaluate ILF's effectiveness on a carefully-controlled toy task and a realistic summarization task. Our experiments demonstrate that large language models accurately incorporate feedback and that finetuning with ILF scales well with the dataset size, even outperforming finetuning on human summaries. Learning from both language and comparison feedback outperforms learning from each alone, achieving human-level summarization performance.
Phylogenetic and discrete-trait evolutionary inference depend heavily on an appropriate characterization of the underlying character substitution process. In this paper, we present random-effects substitution models that extend common continuous-time Markov chain models into a richer class of processes capable of capturing a wider variety of substitution dynamics. As these random-effects substitution models often require many more parameters than their usual counterparts, inference can be both statistically and computationally challenging. Thus, we also propose an efficient approach to compute an approximation to the gradient of the data likelihood with respect to all unknown substitution model parameters. We demonstrate that this approximate gradient enables scaling of sampling-based inference, namely Bayesian inference via Hamiltonian Monte Carlo, under random-effects substitution models across large trees and state-spaces. Applied to a dataset of 583 SARS-CoV-2 sequences, an HKY model with random-effects shows strong signals of nonreversibility in the substitution process, and posterior predictive model checks clearly show that it is a more adequate model than a reversible model. When analyzing the pattern of phylogeographic spread of 1441 influenza A virus (H3N2) sequences between 14 regions, a random-effects phylogeographic substitution model infers that air travel volume adequately predicts almost all dispersal rates. A random-effects state-dependent substitution model reveals no evidence for an effect of arboreality on the swimming mode in the tree frog subfamily Hylinae. Simulations reveal that random-effects substitution models can accommodate both negligible and radical departures from the underlying base substitution model. We show that our gradient-based inference approach is over an order of magnitude more time efficient than conventional approaches.
The Mean Field Variational Bayes (MFVB) method is one of the most computationally efficient techniques for Bayesian inference. However, its use has been restricted to models with conjugate priors or those that require analytical calculations. This paper proposes a novel particle-based MFVB approach that greatly expands the applicability of the MFVB method. We establish the theoretical basis of the new method by leveraging the connection between Wasserstein gradient flows and Langevin diffusion dynamics, and demonstrate the effectiveness of this approach using Bayesian logistic regression, stochastic volatility, and deep neural networks.