Sparseness and robustness are two important properties for many machine learning scenarios. In the present study, regarding the maximum correntropy criterion (MCC) based robust regression algorithm, we investigate to integrate the MCC method with the automatic relevance determination (ARD) technique in a Bayesian framework, so that MCC-based robust regression could be implemented with adaptive sparseness. To be specific, we use an inherent noise assumption from the MCC to derive an explicit likelihood function, and realize the maximum a posteriori (MAP) estimation with the ARD prior by variational Bayesian inference. Compared to the existing robust and sparse L1-regularized MCC regression, the proposed MCC-ARD regression can eradicate the troublesome tuning for the regularization hyper-parameter which controls the regularization strength. Further, MCC-ARD achieves superior prediction performance and feature selection capability than L1-regularized MCC, as demonstrated by a noisy and high-dimensional simulation study.
We establish estimations for the parameters of the output distribution for the softmax activation function using the probit function. As an application, we develop a new efficient Bayesian learning algorithm for fully connected neural networks, where training and predictions are performed within the Bayesian inference framework in closed-form. This approach allows sequential learning and requires no computationally expensive gradient calculation and Monte Carlo sampling. Our work generalizes the Bayesian algorithm for a single perceptron for binary classification in \cite{H} to multi-layer perceptrons for multi-class classification.
We propose a novel Bayesian inference framework for distributed differentially private linear regression. We consider a distributed setting where multiple parties hold parts of the data and share certain summary statistics of their portions in privacy-preserving noise. We develop a novel generative statistical model for privately shared statistics, which exploits a useful distributional relation between the summary statistics of linear regression. Bayesian estimation of the regression coefficients is conducted mainly using Markov chain Monte Carlo algorithms, while we also provide a fast version to perform Bayesian estimation in one iteration. The proposed methods have computational advantages over their competitors. We provide numerical results on both real and simulated data, which demonstrate that the proposed algorithms provide well-rounded estimation and prediction.
Likelihood-free inference methods typically make use of a distance between simulated and real data. A common example is the maximum mean discrepancy (MMD), which has previously been used for approximate Bayesian computation, minimum distance estimation, generalised Bayesian inference, and within the nonparametric learning framework. The MMD is commonly estimated at a root-$m$ rate, where $m$ is the number of simulated samples. This can lead to significant computational challenges since a large $m$ is required to obtain an accurate estimate, which is crucial for parameter estimation. In this paper, we propose a novel estimator for the MMD with significantly improved sample complexity. The estimator is particularly well suited for computationally expensive smooth simulators with low- to mid-dimensional inputs. This claim is supported through both theoretical results and an extensive simulation study on benchmark simulators.
In recent years, particle-based variational inference (ParVI) methods such as Stein variational gradient descent (SVGD) have grown in popularity as scalable methods for Bayesian inference. Unfortunately, the properties of such methods invariably depend on hyperparameters such as the learning rate, which must be carefully tuned by the practitioner in order to ensure convergence to the target measure at a suitable rate. In this paper, we introduce a suite of new particle-based methods for scalable Bayesian inference based on coin betting, which are entirely learning-rate free. We illustrate the performance of our approach on a range of numerical examples, including several high-dimensional models and datasets, demonstrating comparable performance to other ParVI algorithms with no need to tune a learning rate.
Learning accurate predictive models of real-world dynamic phenomena (e.g., climate, biological) remains a challenging task. One key issue is that the data generated by both natural and artificial processes often comprise time series that are irregularly sampled and/or contain missing observations. In this work, we propose the Neural Continuous-Discrete State Space Model (NCDSSM) for continuous-time modeling of time series through discrete-time observations. NCDSSM employs auxiliary variables to disentangle recognition from dynamics, thus requiring amortized inference only for the auxiliary variables. Leveraging techniques from continuous-discrete filtering theory, we demonstrate how to perform accurate Bayesian inference for the dynamic states. We propose three flexible parameterizations of the latent dynamics and an efficient training objective that marginalizes the dynamic states during inference. Empirical results on multiple benchmark datasets across various domains show improved imputation and forecasting performance of NCDSSM over existing models.
Variational inference is an approximation framework for Bayesian inference that seeks to improve quantified uncertainty in predictions by optimizing a simplified distribution over parameters to stand in for the full posterior. Capturing model variations that remain consistent with training data enables more robust predictions by reducing parameter sensitivity. This work introduces a fixed-point optimization for variational inference that is applicable when every feasible log density can be expressed as a linear combination of functions from a given basis. In such cases, the optimizer becomes a fixed-point of projective integral updates. When the basis spans univariate quadratics in each parameter, feasible densities are Gaussian and the projective integral updates yield quasi-Newton variational Bayes (QNVB). Other bases and updates are also possible. As these updates require high-dimensional integration, this work first proposes an efficient quasirandom quadrature sequence for mean-field distributions. Each iterate of the sequence contains two evaluation points that combine to correctly integrate all univariate quadratics and, if the mean-field factors are symmetric, all univariate cubics. More importantly, averaging results over short subsequences achieves periodic exactness on a much larger space of multivariate quadratics. The corresponding variational updates require 4 loss evaluations with standard (not second-order) backpropagation to eliminate error terms from over half of all multivariate quadratic basis functions. This integration technique is motivated by first proposing stochastic blocked mean-field quadratures, which may be useful in other contexts. A PyTorch implementation of QNVB allows for better control over model uncertainty during training than competing methods. Experiments demonstrate superior generalizability for multiple learning problems and architectures.
Gaussian process regression in its most simplified form assumes normal homoscedastic noise and utilizes analytically tractable mean and covariance functions of predictive posterior distribution using Gaussian conditioning. Its hyperparameters are estimated by maximizing the evidence, commonly known as type II maximum likelihood estimation. Unfortunately, Bayesian inference based on Gaussian likelihood is not robust to outliers, which are often present in the observational training data sets. To overcome this problem, we propose a robust process model in the Gaussian process framework with the likelihood of observed data expressed as the Huber probability distribution. The proposed model employs weights based on projection statistics to scale residuals and bound the influence of vertical outliers and bad leverage points on the latent functions estimates while exhibiting a high statistical efficiency at the Gaussian and thick tailed noise distributions. The proposed method is demonstrated by two real world problems and two numerical examples using datasets with additive errors following thick tailed distributions such as Students t, Laplace, and Cauchy distribution.
Many functional magnetic resonance imaging (fMRI) studies rely on estimates of hierarchically organised brain networks whose segregation and integration reflect the dynamic transitions of latent cognitive states. However, most existing methods for estimating the community structure of networks from both individual and group-level analysis neglect the variability between subjects and lack validation. In this paper, we develop a new multilayer community detection method based on Bayesian latent block modelling. The method can robustly detect the group-level community structure of weighted functional networks that give rise to hidden brain states with an unknown number of communities and retain the variability of individual networks. For validation, we propose a new community structure-based multivariate Gaussian generative model to simulate synthetic signal. Our result shows that the inferred community memberships using hierarchical Bayesian analysis are consistent with the predefined node labels in the generative model. The method is also tested using real working memory task-fMRI data of 100 unrelated healthy subjects from the Human Connectome Project. The results show distinctive community structure patterns between 2-back, 0-back, and fixation conditions, which may reflect cognitive and behavioural states under working memory task conditions.
Machine learning algorithms have achieved superhuman performance in specific complex domains. However, learning online from few examples and compositional learning for efficient generalization across domains remain elusive. In humans, such learning includes specific declarative memory formation and is closely associated with consciousness. Predictive processing has been advanced as a principled Bayesian framework for understanding the cortex as implementing deep generative models for both sensory perception and action control. However, predictive processing offers little direct insight into fast compositional learning or of the separation between conscious and unconscious contents. Here, propose that access consciousness arises as a consequence of a particular learning mechanism operating within a predictive processing system. We extend predictive processing by adding online, single-example new structure learning via hierarchical binding of unpredicted inferences. This system learns new causes by quickly connecting together novel combinations of perceptions, which manifests as working memories that can become short- and long-term declarative memories retrievable by associative recall. The contents of such bound representations are unified yet differentiated, can be maintained by selective attention and are globally available. The proposed learning process explains contrast and masking manipulations, postdictive perceptual integration, and other paradigm cases of consciousness research. 'Phenomenal conscious experience' is how the learning system transparently models its own functioning, giving rise to perceptual illusions underlying the meta-problem of consciousness. Our proposal naturally unifies the feature binding, recurrent processing, predictive processing, and global workspace theories of consciousness.
Sequential Bayesian inference can be used for continual learning to prevent catastrophic forgetting of past tasks and provide an informative prior when learning new tasks. We revisit sequential Bayesian inference and test whether having access to the true posterior is guaranteed to prevent catastrophic forgetting in Bayesian neural networks. To do this we perform sequential Bayesian inference using Hamiltonian Monte Carlo. We propagate the posterior as a prior for new tasks by fitting a density estimator on Hamiltonian Monte Carlo samples. We find that this approach fails to prevent catastrophic forgetting demonstrating the difficulty in performing sequential Bayesian inference in neural networks. From there we study simple analytical examples of sequential Bayesian inference and CL and highlight the issue of model misspecification which can lead to sub-optimal continual learning performance despite exact inference. Furthermore, we discuss how task data imbalances can cause forgetting. From these limitations, we argue that we need probabilistic models of the continual learning generative process rather than relying on sequential Bayesian inference over Bayesian neural network weights. In this vein, we also propose a simple baseline called Prototypical Bayesian Continual Learning, which is competitive with state-of-the-art Bayesian continual learning methods on class incremental continual learning vision benchmarks.
Riemannian manifold Hamiltonian (RMHMC) and Lagrangian Monte Carlo (LMC) have emerged as powerful methods of Bayesian inference. Unlike Euclidean Hamiltonian Monte Carlo (EHMC) and the Metropolis-adjusted Langevin algorithm (MALA), the geometric ergodicity of these Riemannian algorithms has not been extensively studied. On the other hand, the manifold Metropolis-adjusted Langevin algorithm (MMALA) has recently been shown to exhibit geometric ergodicity under certain conditions. This work investigates the mixture of the LMC and RMHMC transition kernels with MMALA in order to equip the resulting method with an "inherited" geometric ergodicity theory. We motivate this mixture kernel based on an analogy between single-step HMC and MALA. We then proceed to evaluate the original and modified transition kernels on several benchmark Bayesian inference tasks.
This work proposes a view of probability as a relative measure rather than an absolute one. To demonstrate this concept, we focus on finite outcome spaces and develop three fundamental axioms that establish requirements for relative probability functions. We then provide a library of examples of these functions and a system for composing them. Additionally, we discuss a relative version of Bayesian inference and its digital implementation. Finally, we prove the topological closure of the relative probability space, highlighting its ability to preserve information under limits.
Characterizing how neural network depth, width, and dataset size jointly impact model quality is a central problem in deep learning theory. We give here a complete solution in the special case of linear networks with output dimension one trained using zero noise Bayesian inference with Gaussian weight priors and mean squared error as a negative log-likelihood. For any training dataset, network depth, and hidden layer widths, we find non-asymptotic expressions for the predictive posterior and Bayesian model evidence in terms of Meijer-G functions, a class of meromorphic special functions of a single complex variable. Through novel asymptotic expansions of these Meijer-G functions, a rich new picture of the joint role of depth, width, and dataset size emerges. We show that linear networks make provably optimal predictions at infinite depth: the posterior of infinitely deep linear networks with data-agnostic priors is the same as that of shallow networks with evidence-maximizing data-dependent priors. This yields a principled reason to prefer deeper networks when priors are forced to be data-agnostic. Moreover, we show that with data-agnostic priors, Bayesian model evidence in wide linear networks is maximized at infinite depth, elucidating the salutary role of increased depth for model selection. Underpinning our results is a novel emergent notion of effective depth, given by the number of hidden layers times the number of data points divided by the network width; this determines the structure of the posterior in the large-data limit.
The problem of detecting the Out-of-Distribution (OoD) inputs is of paramount importance for Deep Neural Networks. It has been previously shown that even Deep Generative Models that allow estimating the density of the inputs may not be reliable and often tend to make over-confident predictions for OoDs, assigning to them a higher density than to the in-distribution data. This over-confidence in a single model can be potentially mitigated with Bayesian inference over the model parameters that take into account epistemic uncertainty. This paper investigates three approaches to Bayesian inference: stochastic gradient Markov chain Monte Carlo, Bayes by Backpropagation, and Stochastic Weight Averaging-Gaussian. The inference is implemented over the weights of the deep neural networks that parameterize the likelihood of the Variational Autoencoder. We empirically evaluate the approaches against several benchmarks that are often used for OoD detection: estimation of the marginal likelihood utilizing sampled model ensemble, typicality test, disagreement score, and Watanabe-Akaike Information Criterion. Finally, we introduce two simple scores that demonstrate the state-of-the-art performance.
Linear partial differential equations (PDEs) are an important, widely applied class of mechanistic models, describing physical processes such as heat transfer, electromagnetism, and wave propagation. In practice, specialized numerical methods based on discretization are used to solve PDEs. They generally use an estimate of the unknown model parameters and, if available, physical measurements for initialization. Such solvers are often embedded into larger scientific models with a downstream application and thus error quantification plays a key role. However, by ignoring parameter and measurement uncertainty, classical PDE solvers may fail to produce consistent estimates of their inherent approximation error. In this work, we approach this problem in a principled fashion by interpreting solving linear PDEs as physics-informed Gaussian process (GP) regression. Our framework is based on a key generalization of the Gaussian process inference theorem to observations made via an arbitrary bounded linear operator. Crucially, this probabilistic viewpoint allows to (1) quantify the inherent discretization error; (2) propagate uncertainty about the model parameters to the solution; and (3) condition on noisy measurements. Demonstrating the strength of this formulation, we prove that it strictly generalizes methods of weighted residuals, a central class of PDE solvers including collocation, finite volume, pseudospectral, and (generalized) Galerkin methods such as finite element and spectral methods. This class can thus be directly equipped with a structured error estimate. In summary, our results enable the seamless integration of mechanistic models as modular building blocks into probabilistic models by blurring the boundaries between numerical analysis and Bayesian inference.
We build on the view of the Exact Renormalization Group (ERG) as an instantiation of Optimal Transport described by a functional convection-diffusion equation. We provide a new information theoretic perspective for understanding the ERG through the intermediary of Bayesian Statistical Inference. This connection is facilitated by the Dynamical Bayesian Inference scheme, which encodes Bayesian inference in the form of a one parameter family of probability distributions solving an integro-differential equation derived from Bayes' law. In this note, we demonstrate how the Dynamical Bayesian Inference equation is, itself, equivalent to a diffusion equation which we dub Bayesian Diffusion. Identifying the features that define Bayesian Diffusion, and mapping them onto the features that define the ERG, we obtain a dictionary outlining how renormalization can be understood as the inverse of statistical inference.
Bistable perception follows from observing a static, ambiguous, (visual) stimulus with two possible interpretations. Here, we present an active (Bayesian) inference account of bistable perception and posit that perceptual transitions between different interpretations (i.e., inferences) of the same stimulus ensue from specific eye movements that shift the focus to a different visual feature. Formally, these inferences are a consequence of precision control that determines how confident beliefs are and change the frequency with which one can perceive - and alternate between - two distinct percepts. We hypothesised that there are multiple, but distinct, ways in which precision modulation can interact to give rise to a similar frequency of bistable perception. We validated this using numerical simulations of the Necker's cube paradigm and demonstrate the multiple routes that underwrite the frequency of perceptual alternation. Our results provide an (enactive) computational account of the intricate precision balance underwriting bistable perception. Importantly, these precision parameters can be considered the computational homologues of particular neurotransmitters - i.e., acetylcholine, noradrenaline, dopamine - that have been previously implicated in controlling bistable perception, providing a computational link between the neurochemistry and perception.
We introduce ensembles of stochastic neural networks to approximate the Bayesian posterior, combining stochastic methods such as dropout with deep ensembles. The stochastic ensembles are formulated as families of distributions and trained to approximate the Bayesian posterior with variational inference. We implement stochastic ensembles based on Monte Carlo dropout, DropConnect and a novel non-parametric version of dropout and evaluate them on a toy problem and CIFAR image classification. For both tasks, we test the quality of the posteriors directly against Hamiltonian Monte Carlo simulations. Our results show that stochastic ensembles provide more accurate posterior estimates than other popular baselines for Bayesian inference.
The integration of data and knowledge from several sources is known as data fusion. When data is only available in a distributed fashion or when different sensors are used to infer a quantity of interest, data fusion becomes essential. In Bayesian settings, a priori information of the unknown quantities is available and, possibly, present among the different distributed estimators. When the local estimates are fused, the prior knowledge used to construct several local posteriors might be overused unless the fusion node accounts for this and corrects it. In this paper, we analyze the effects of shared priors in Bayesian data fusion contexts. Depending on different common fusion rules, our analysis helps to understand the performance behavior as a function of the number of collaborative agents and as a consequence of different types of priors. The analysis is performed by using two divergences which are common in Bayesian inference, and the generality of the results allows to analyze very generic distributions. These theoretical results are corroborated through experiments in a variety of estimation and classification problems, including linear and nonlinear models, and federated learning schemes.