In this work we explore a new framework for approximate Bayesian inference in large datasets based on stochastic control (i.e. Schr\"odinger bridges). We advocate stochastic control as a finite time and low variance alternative to popular steady-state methods such as stochastic gradient Langevin dynamics (SGLD). Furthermore, we discuss and adapt the existing theoretical guarantees of this framework and establish connections to already existing VI routines in SDE-based models.
The Bayesian brain hypothesis postulates that the brain accurately operates on statistical distributions according to Bayes' theorem. The random failure of presynaptic vesicles to release neurotransmitters may allow the brain to sample from posterior distributions of network parameters, interpreted as epistemic uncertainty. It has not been shown previously how random failures might allow networks to sample from observed distributions, also known as aleatoric or residual uncertainty. Sampling from both distributions enables probabilistic inference, efficient search, and creative or generative problem solving. We demonstrate that under a population-code based interpretation of neural activity, both types of distribution can be represented and sampled with synaptic failure alone. We first define a biologically constrained neural network and sampling scheme based on synaptic failure and lateral inhibition. Within this framework, we derive drop-out based epistemic uncertainty, then prove an analytic mapping from synaptic efficacy to release probability that allows networks to sample from arbitrary, learned distributions represented by a receiving layer. Second, our result leads to a local learning rule by which synapses adapt their release probabilities. Our result demonstrates complete Bayesian inference, related to the variational learning method of dropout, in a biologically constrained network using only locally-learned synaptic failure rates.
The idea that the brain is a probabilistic (Bayesian) inference machine, continuously trying to figure out the hidden causes of its inputs, has become very influential in cognitive (neuro)science over recent decades. Here I present a relatively straightforward generalization of this idea: the primary computational task that the brain is faced with is to track the probabilistic structure of observations themselves, without recourse to hidden states. Taking this starting point seriously turns out to have considerable explanatory power, and several key ideas are developed from it: (1) past experience, encoded in prior expectations, has an influence over the future that is analogous to regularization as known from machine learning; (2) action generation (interpreted as constraint satisfaction) is a special case of such regularization; (3) the concept of attractors in dynamical systems provides a useful lens through which prior expectations, regularization, and action induction can be viewed; these thus appear as different perspectives on the same phenomenon; (4) the phylogenetically ancient imperative of acting to ensure and thereby observe conditions beneficial for survival is likely the same as that which underlies perceptual inference. The Bayesian brain hypothesis has been touted as promising to deliver a "unified science of mind and action". In this paper, I sketch an informal step towards fulfilling that promise, while avoiding some pitfalls that other such attempts have fallen prey to.
Climbing grades are used to classify a climbing route based on its perceived difficulty, and have come to play a central role in the sport of rock climbing. Recently, the first statistically rigorous method for estimating climbing grades from whole-history ascent data was described, based on the dynamic Bradley-Terry model for games between players of time-varying ability. In this paper, we implement inference under the whole-history rating model using Markov chain Monte Carlo and apply the method to a curated data set made up of climbers who climb regularly. We use these data to get an estimate of the model's fundamental scale parameter m, which defines the proportional increase in difficulty associated with an increment of grade. We show that the data conform to assumptions that the climbing grade scale is a logarithmic scale of difficulty, like decibels or stellar magnitude. We estimate that an increment in Ewbank, French and UIAA climbing grade systems corresponds to 2.1, 2.09 and 2.13 times increase in difficulty respectively, assuming a logistic model of probability of success as a function of grade. Whereas we find that the Vermin scale for bouldering (V-grade scale) corresponds to a 3.17 increase in difficulty per grade increment. In addition, we highlight potential connections between the logarithmic properties of climbing grade scales and the psychophysical laws of Weber and Fechner.
Bayesian Neural Networks (BNNs), unlike Traditional Neural Networks (TNNs) are robust and adept at handling adversarial attacks by incorporating randomness. This randomness improves the estimation of uncertainty, a feature lacking in TNNs. Thus, we investigate the robustness of BNNs to white-box attacks using multiple Bayesian neural architectures. Furthermore, we create our BNN model, called BNN-DenseNet, by fusing Bayesian inference (i.e., variational Bayes) to the DenseNet architecture, and BDAV, by combining this intervention with adversarial training. Experiments are conducted on the CIFAR-10 and FGVC-Aircraft datasets. We attack our models with strong white-box attacks ($l_\infty$-FGSM, $l_\infty$-PGD, $l_2$-PGD, EOT $l_\infty$-FGSM, and EOT $l_\infty$-PGD). In all experiments, at least one BNN outperforms traditional neural networks during adversarial attack scenarios. An adversarially-trained BNN outperforms its non-Bayesian, adversarially-trained counterpart in most experiments, and often by significant margins. Lastly, we investigate network calibration and find that BNNs do not make overconfident predictions, providing evidence that BNNs are also better at measuring uncertainty.
We propose MC-CIM, a compute-in-memory (CIM) framework for robust, yet low power, Bayesian edge intelligence. Deep neural networks (DNN) with deterministic weights cannot express their prediction uncertainties, thereby pose critical risks for applications where the consequences of mispredictions are fatal such as surgical robotics. To address this limitation, Bayesian inference of a DNN has gained attention. Using Bayesian inference, not only the prediction itself, but the prediction confidence can also be extracted for planning risk-aware actions. However, Bayesian inference of a DNN is computationally expensive, ill-suited for real-time and/or edge deployment. An approximation to Bayesian DNN using Monte Carlo Dropout (MC-Dropout) has shown high robustness along with low computational complexity. Enhancing the computational efficiency of the method, we discuss a novel CIM module that can perform in-memory probabilistic dropout in addition to in-memory weight-input scalar product to support the method. We also propose a compute-reuse reformulation of MC-Dropout where each successive instance can utilize the product-sum computations from the previous iteration. Even more, we discuss how the random instances can be optimally ordered to minimize the overall MC-Dropout workload by exploiting combinatorial optimization methods. Application of the proposed CIM-based MC-Dropout execution is discussed for MNIST character recognition and visual odometry (VO) of autonomous drones. The framework reliably gives prediction confidence amidst non-idealities imposed by MC-CIM to a good extent. Proposed MC-CIM with 16x31 SRAM array, 0.85 V supply, 16nm low-standby power (LSTP) technology consumes 27.8 pJ for 30 MC-Dropout instances of probabilistic inference in its most optimal computing and peripheral configuration, saving 43% energy compared to typical execution.
Neural processes have recently emerged as a class of powerful neural latent variable models that combine the strengths of neural networks and stochastic processes. As they can encode contextual data in the network's function space, they offer a new way to model task relatedness in multi-task learning. To study its potential, we develop multi-task neural processes, a new variant of neural processes for multi-task learning. In particular, we propose to explore transferable knowledge from related tasks in the function space to provide inductive bias for improving each individual task. To do so, we derive the function priors in a hierarchical Bayesian inference framework, which enables each task to incorporate the shared knowledge provided by related tasks into its context of the prediction function. Our multi-task neural processes methodologically expand the scope of vanilla neural processes and provide a new way of exploring task relatedness in function spaces for multi-task learning. The proposed multi-task neural processes are capable of learning multiple tasks with limited labeled data and in the presence of domain shift. We perform extensive experimental evaluations on several benchmarks for the multi-task regression and classification tasks. The results demonstrate the effectiveness of multi-task neural processes in transferring useful knowledge among tasks for multi-task learning and superior performance in multi-task classification and brain image segmentation.
Multi-task learning aims to explore task relatedness to improve individual tasks, which is of particular significance in the challenging scenario that only limited data is available for each task. To tackle this challenge, we propose variational multi-task learning (VMTL), a general probabilistic inference framework for learning multiple related tasks. We cast multi-task learning as a variational Bayesian inference problem, in which task relatedness is explored in a unified manner by specifying priors. To incorporate shared knowledge into each task, we design the prior of a task to be a learnable mixture of the variational posteriors of other related tasks, which is learned by the Gumbel-Softmax technique. In contrast to previous methods, our VMTL can exploit task relatedness for both representations and classifiers in a principled way by jointly inferring their posteriors. This enables individual tasks to fully leverage inductive biases provided by related tasks, therefore improving the overall performance of all tasks. Experimental results demonstrate that the proposed VMTL is able to effectively tackle a variety of challenging multi-task learning settings with limited training data for both classification and regression. Our method consistently surpasses previous methods, including strong Bayesian approaches, and achieves state-of-the-art performance on five benchmark datasets.
Neural bandits have been shown to provide an efficient solution to practical sequential decision tasks that have nonlinear reward functions. The main contributor to that success is approximate Bayesian inference, which enables neural network (NN) training with uncertainty estimates. However, Bayesian NNs often suffer from a prohibitive computational overhead or operate on a subset of parameters. Alternatively, certain classes of infinite neural networks were shown to directly correspond to Gaussian processes (GP) with neural kernels (NK). NK-GPs provide accurate uncertainty estimates and can be trained faster than most Bayesian NNs. We propose to guide common bandit policies with NK distributions and show that NK bandits achieve state-of-the-art performance on nonlinear structured data. Moreover, we propose a framework for measuring independently the ability of a bandit algorithm to learn representations and explore, and use it to analyze the impact of NK distributions w.r.t.~those two aspects. We consider policies based on a GP and a Student's t-process (TP). Furthermore, we study practical considerations, such as training frequency and model partitioning. We believe our work will help better understand the impact of utilizing NKs in applied settings.
Probabilistic context-free grammars (PCFGs) and dynamic Bayesian networks (DBNs) are widely used sequence models with complementary strengths and limitations. While PCFGs allow for nested hierarchical dependencies (tree structures), their latent variables (non-terminal symbols) have to be discrete. In contrast, DBNs allow for continuous latent variables, but the dependencies are strictly sequential (chain structure). Therefore, neither can be applied if the latent variables are assumed to be continuous and also to have a nested hierarchical dependency structure. In this paper, we present Recursive Bayesian Networks (RBNs), which generalise and unify PCFGs and DBNs, combining their strengths and containing both as special cases. RBNs define a joint distribution over tree-structured Bayesian networks with discrete or continuous latent variables. The main challenge lies in performing joint inference over the exponential number of possible structures and the continuous variables. We provide two solutions: 1) For arbitrary RBNs, we generalise inside and outside probabilities from PCFGs to the mixed discrete-continuous case, which allows for maximum posterior estimates of the continuous latent variables via gradient descent, while marginalising over network structures. 2) For Gaussian RBNs, we additionally derive an analytic approximation, allowing for robust parameter optimisation and Bayesian inference. The capacity and diverse applications of RBNs are illustrated on two examples: In a quantitative evaluation on synthetic data, we demonstrate and discuss the advantage of RBNs for segmentation and tree induction from noisy sequences, compared to change point detection and hierarchical clustering. In an application to musical data, we approach the unsolved problem of hierarchical music analysis from the raw note level and compare our results to expert annotations.
We formulate natural gradient variational inference (VI), expectation propagation (EP), and posterior linearisation (PL) as extensions of Newton's method for optimising the parameters of a Bayesian posterior distribution. This viewpoint explicitly casts inference algorithms under the framework of numerical optimisation. We show that common approximations to Newton's method from the optimisation literature, namely Gauss-Newton and quasi-Newton methods (e.g., the BFGS algorithm), are still valid under this 'Bayes-Newton' framework. This leads to a suite of novel algorithms which are guaranteed to result in positive semi-definite (PSD) covariance matrices, unlike standard VI and EP. Our unifying viewpoint provides new insights into the connections between various inference schemes. All the presented methods apply to any model with a Gaussian prior and non-conjugate likelihood, which we demonstrate with (sparse) Gaussian processes and state space models.
Several problems in neuroimaging and beyond require inference on the parameters of multi-task sparse hierarchical regression models. Examples include M/EEG inverse problems, neural encoding models for task-based fMRI analyses, and climate science. In these domains, both the model parameters to be inferred and the measurement noise may exhibit a complex spatio-temporal structure. Existing work either neglects the temporal structure or leads to computationally demanding inference schemes. Overcoming these limitations, we devise a novel flexible hierarchical Bayesian framework within which the spatio-temporal dynamics of model parameters and noise are modeled to have Kronecker product covariance structure. Inference in our framework is based on majorization-minimization optimization and has guaranteed convergence properties. Our highly efficient algorithms exploit the intrinsic Riemannian geometry of temporal autocovariance matrices. For stationary dynamics described by Toeplitz matrices, the theory of circulant embeddings is employed. We prove convex bounding properties and derive update rules of the resulting algorithms. On both synthetic and real neural data from M/EEG, we demonstrate that our methods lead to improved performance.
Fast, highly accurate, and reliable inference of the sky origin of gravitational waves would enable real-time multi-messenger astronomy. Current Bayesian inference methodologies, although highly accurate and reliable, are slow. Deep learning models have shown themselves to be accurate and extremely fast for inference tasks on gravitational waves, but their output is inherently questionable due to the blackbox nature of neural networks. In this work, we join Bayesian inference and deep learning by applying importance sampling on an approximate posterior generated by a multi-headed convolutional neural network. The neural network parametrizes Von Mises-Fisher and Gaussian distributions for the sky coordinates and two masses for given simulated gravitational wave injections in the LIGO and Virgo detectors. We generate skymaps for unseen gravitational-wave events that highly resemble predictions generated using Bayesian inference in a few minutes. Furthermore, we can detect poor predictions from the neural network, and quickly flag them.
Multi-task learning requires accurate identification of the correlations between tasks. In real-world time-series, tasks are rarely perfectly temporally aligned; traditional multi-task models do not account for this and subsequent errors in correlation estimation will result in poor predictive performance and uncertainty quantification. We introduce a method that automatically accounts for temporal misalignment in a unified generative model that improves predictive performance. Our method uses Gaussian processes (GPs) to model the correlations both within and between the tasks. Building on the previous work by Kazlauskaiteet al. [2019], we include a separate monotonic warp of the input data to model temporal misalignment. In contrast to previous work, we formulate a lower bound that accounts for uncertainty in both the estimates of the warping process and the underlying functions. Also, our new take on a monotonic stochastic process, with efficient path-wise sampling for the warp functions, allows us to perform full Bayesian inference in the model rather than MAP estimates. Missing data experiments, on synthetic and real time-series, demonstrate the advantages of accounting for misalignments (vs standard unaligned method) as well as modelling the uncertainty in the warping process(vs baseline MAP alignment approach).
In recent years, local differential privacy (LDP) has emerged as a technique of choice for privacy-preserving data collection in several scenarios when the aggregator is not trustworthy. LDP provides client-side privacy by adding noise at the user's end. Thus, clients need not rely on the trustworthiness of the aggregator. In this work, we provide a noise-aware probabilistic modeling framework, which allows Bayesian inference to take into account the noise added for privacy under LDP, conditioned on locally perturbed observations. Stronger privacy protection (compared to the central model) provided by LDP protocols comes at a much harsher privacy-utility trade-off. Our framework tackles several computational and statistical challenges posed by LDP for accurate uncertainty quantification under Bayesian settings. We demonstrate the efficacy of our framework in parameter estimation for univariate and multi-variate distributions as well as logistic and linear regression.
The nature of available economic data has changed fundamentally in the last decade due to the economy's digitisation. With the prevalence of often black box data-driven machine learning methods, there is a necessity to develop interpretable machine learning methods that can conduct econometric inference, helping policymakers leverage the new nature of economic data. We therefore present a novel Variational Bayesian Inference approach to incorporate a time-varying parameter auto-regressive model which is scalable for big data. Our model is applied to a large blockchain dataset containing prices, transactions of individual actors, analyzing transactional flows and price movements on a very granular level. The model is extendable to any dataset which can be modelled as a dynamical system. We further improve the simple state-space modelling by introducing non-linearities in the forward model with the help of machine learning architectures.
Current diagnostic practice in psychiatry is not relying on objective biophysical evidence. Recent pandemic emphasized the need to address the rising number of mood disorders (in particular, depression) cases in a more efficient way. We are proposing several already developed practices that can help improve that diagnostic process: detection based on electrophysiological signals (both electroencephalogram and electrocardiogram based) that were shown to be accurate for clinical practice and several modalities of electromagnetic stimulation that were proven to ameliorate symptoms of depression. In this work, we are connecting the two with explanations coming from physiological complexity studies (and our own work) as well as advanced statistical methods like machine learning and the Bayesian inference approach. It is shown that fractal and nonlinear measures can adequately quantify previously undetected changes in intrinsic dynamics of physiological systems, providing the basis for early detection of depression. We are also advocating for early screening of cardiovascular risks in depression which is in connection to previously described decomplexification of the autonomous nervous system resulting in symptoms recognized clinically. All that said, additional information about the level of complexity can help clinicians make a better decisions in the therapeutic process, increase the overall effectiveness of the treatment, and finally increase the quality of life of the patient.
This work develops a class of probabilistic algorithms for the numerical solution of nonlinear, time-dependent partial differential equations (PDEs). Current state-of-the-art PDE solvers treat the space- and time-dimensions separately, serially, and with black-box algorithms, which obscures the interactions between spatial and temporal approximation errors and misguides the quantification of the overall error. To fix this issue, we introduce a probabilistic version of a technique called method of lines. The proposed algorithm begins with a Gaussian process interpretation of finite difference methods, which then interacts naturally with filtering-based probabilistic ordinary differential equation (ODE) solvers because they share a common language: Bayesian inference. Joint quantification of space- and time-uncertainty becomes possible without losing the performance benefits of well-tuned ODE solvers. Thereby, we extend the toolbox of probabilistic programs for differential equation simulation to PDEs.
Minimizing functionals in the space of probability distributions can be done with Wasserstein gradient flows. To solve them numerically, a possible approach is to rely on the Jordan-Kinderlehrer-Otto (JKO) scheme which is analogous to the proximal scheme in Euclidean spaces. However, it requires solving a nested optimization problem at each iteration, and is known for its computational challenges, especially in high dimension. To alleviate it, very recent works propose to approximate the JKO scheme leveraging Brenier's theorem, and using gradients of Input Convex Neural Networks to parameterize the density (JKO-ICNN). However, this method comes with a high computational cost and stability issues. Instead, this work proposes to use gradient flows in the space of probability measures endowed with the sliced-Wasserstein (SW) distance. We argue that this method is more flexible than JKO-ICNN, since SW enjoys a closed-form differentiable approximation. Thus, the density at each step can be parameterized by any generative model which alleviates the computational burden and makes it tractable in higher dimensions.
Probabilistic numerical solvers for ordinary differential equations compute posterior distributions over the solution of an initial value problem via Bayesian inference. In this paper, we leverage their probabilistic formulation to seamlessly include additional information as general likelihood terms. We show that second-order differential equations should be directly provided to the solver, instead of transforming the problem to first order. Additionally, by including higher-order information or physical conservation laws in the model, solutions become more accurate and more physically meaningful. Lastly, we demonstrate the utility of flexible information operators by solving differential-algebraic equations. In conclusion, the probabilistic formulation of numerical solvers offers a flexible way to incorporate various types of information, thus improving the resulting solutions.