Through sequential construction of posteriors on observing data online, Bayes' theorem provides a natural framework for continual learning. We develop Variational Auto-Regressive Gaussian Processes (VAR-GPs), a principled posterior updating mechanism to solve sequential tasks in continual learning. By relying on sparse inducing point approximations for scalable posteriors, we propose a novel auto-regressive variational distribution which reveals two fruitful connections to existing results in Bayesian inference, expectation propagation and orthogonal inducing points. Mean predictive entropy estimates show VAR-GPs prevent catastrophic forgetting, which is empirically supported by strong performance on modern continual learning benchmarks against competitive baselines. A thorough ablation study demonstrates the efficacy of our modeling choices.
Much of human learning and inference can be framed within the computational problem of relational generalization. In this project, we propose a Bayesian model that generalizes relational knowledge to novel environments by analogically weighting predictions from previously encountered relational structures. First, we show that this learner outperforms a naive, theory-based learner on relational data derived from random- and Wikipedia-based systems when experience with the environment is small. Next, we show how our formalization of analogical similarity translates to the selection and weighting of analogies. Finally, we combine the analogy- and theory-based learners in a single nonparametric Bayesian model, and show that optimal relational generalization transitions from relying on analogies to building a theory of the novel system with increasing experience in it. Beyond predicting unobserved interactions better than either baseline, this formalization gives a computational-level perspective on the formation and abstraction of analogies themselves.
In this paper, we employ variational arguments to establish a connection between ensemble methods for Neural Networks and Bayesian inference. We consider an ensemble-based scheme where each model/particle corresponds to a perturbation of the data by means of parametric bootstrap and a perturbation of the prior. We derive conditions under which any optimization steps of the particles makes the associated distribution reduce its divergence to the posterior over model parameters. Such conditions do not require any particular form for the approximation and they are purely geometrical, giving insights on the behavior of the ensemble on a number of interesting models such as Neural Networks with ReLU activations. Experiments confirm that ensemble methods can be a valid alternative to approximate Bayesian inference; the theoretical developments in the paper seek to explain this behavior.
Functional Distributional Semantics provides a computationally tractable framework for learning truth-conditional semantics from a corpus. Previous work in this framework has provided a probabilistic version of first-order logic, recasting quantification as Bayesian inference. In this paper, I show how the previous formulation gives trivial truth values when a precise quantifier is used with vague predicates. I propose an improved account, avoiding this problem by treating a vague predicate as a distribution over precise predicates. I connect this account to recent work in the Rational Speech Acts framework on modelling generic quantification, and I extend this to modelling donkey sentences. Finally, I explain how the generic quantifier can be both pragmatically complex and yet computationally simpler than precise quantifiers.
This paper provides the foundations of a unified cognitive decision-making framework (QulBIT) which is derived from quantum theory. The main advantage of this framework is that it can cater for paradoxical and irrational human decision making. Although quantum approaches for cognition have demonstrated advantages over classical probabilistic approaches and bounded rationality models, they still lack explanatory power. To address this, we introduce a novel explanatory analysis of the decision-maker's belief space. This is achieved by exploiting quantum interference effects as a way of both quantifying and explaining the decision-maker's uncertainty. We detail the main modules of the unified framework, the explanatory analysis method, and illustrate their application in situations violating the Sure Thing Principle.
Numerical integration and emulation are fundamental topics across scientific fields. We propose novel adaptive quadrature schemes based on an active learning procedure. We consider an interpolative approach for building a surrogate posterior density, combining it with Monte Carlo sampling methods and other quadrature rules. The nodes of the quadrature are sequentially chosen by maximizing a suitable acquisition function, which takes into account the current approximation of the posterior and the positions of the nodes. This maximization does not require additional evaluations of the true posterior. We introduce two specific schemes based on Gaussian and Nearest Neighbors (NN) bases. For the Gaussian case, we also provide a novel procedure for fitting the bandwidth parameter, in order to build a suitable emulator of a density function. With both techniques, we always obtain a positive estimation of the marginal likelihood (a.k.a., Bayesian evidence). An equivalent importance sampling interpretation is also described, which allows the design of extended schemes. Several theoretical results are provided and discussed. Numerical results show the advantage of the proposed approach, including a challenging inference problem in an astronomic dynamical model, with the goal of revealing the number of planets orbiting a star.
Crowdsourcing has emerged as an effective means for performing a number of machine learning tasks such as annotation and labelling of images and other data sets. In most early settings of crowdsourcing, the task involved classification, that is assigning one of a discrete set of labels to each task. Recently, however, more complex tasks have been attempted including asking crowdsource workers to assign continuous labels, or predictions. In essence, this involves the use of crowdsourcing for function estimation. We are motivated by this problem to drive applications such as collaborative prediction, that is, harnessing the wisdom of the crowd to predict quantities more accurately. To do so, we propose a Bayesian approach aimed specifically at alleviating overfitting, a typical impediment to accurate prediction models in practice. In particular, we develop a variational Bayesian technique for two different worker noise models - one that assumes workers' noises are independent and the other that assumes workers' noises have a latent low-rank structure. Our evaluations on synthetic and real-world datasets demonstrate that these Bayesian approaches perform significantly better than existing non-Bayesian approaches and are thus potentially useful for this class of crowdsourcing problems.
Deep networks are currently the state-of-the-art for sensory perception in autonomous driving and robotics. However, deep models often generate overconfident predictions precluding proper probabilistic interpretation which we argue is due to the nature of the SoftMax layer. To reduce the overconfidence without compromising the classification performance, we introduce a CNN probabilistic approach based on distributions calculated in the network's Logit layer. The approach enables Bayesian inference by means of ML and MAP layers. Experiments with calibrated and the proposed prediction layers are carried out on object classification using data from the KITTI database. Results are reported for camera ($RGB$) and LiDAR (range-view) modalities, where the new approach shows promising performance compared to SoftMax.
We propose VarFA, a variational inference factor analysis framework that extends existing factor analysis models for educational data mining to efficiently output uncertainty estimation in the model's estimated factors. Such uncertainty information is useful, for example, for an adaptive testing scenario, where additional tests can be administered if the model is not quite certain about a students' skill level estimation. Traditional Bayesian inference methods that produce such uncertainty information are computationally expensive and do not scale to large data sets. VarFA utilizes variational inference which makes it possible to efficiently perform Bayesian inference even on very large data sets. We use the sparse factor analysis model as a case study and demonstrate the efficacy of VarFA on both synthetic and real data sets. VarFA is also very general and can be applied to a wide array of factor analysis models.
Deep neural networks have revolutionized medical image analysis and disease diagnosis. Despite their impressive performance, it is difficult to generate well-calibrated probabilistic outputs for such networks, which makes them uninterpretable black boxes. Bayesian neural networks provide a principled approach for modelling uncertainty and increasing patient safety, but they have a large computational overhead and provide limited improvement in calibration. In this work, by taking skin lesion classification as an example task, we show that by shifting Bayesian inference to the functional space we can craft meaningful priors that give better calibrated uncertainty estimates at a much lower computational cost.
High dimensional integration is essential to many areas of science, ranging from particle physics to Bayesian inference. Approximating these integrals is hard, due in part to the difficulty of locating and sampling from regions of the integration domain that make significant contributions to the overall integral. Here, we present a new algorithm called Tree Quadrature (TQ) that separates this sampling problem from the problem of using those samples to produce an approximation of the integral. TQ places no qualifications on how the samples provided to it are obtained, allowing it to use state-of-the-art sampling algorithms that are largely ignored by existing integration algorithms. Given a set of samples, TQ constructs a surrogate model of the integrand in the form of a regression tree, with a structure optimised to maximise integral precision. The tree divides the integration domain into smaller containers, which are individually integrated and aggregated to estimate the overall integral. Any method can be used to integrate each individual container, so existing integration methods, like Bayesian Monte Carlo, can be combined with TQ to boost their performance. On a set of benchmark problems, we show that TQ provides accurate approximations to integrals in up to 15 dimensions; and in dimensions 4 and above, it outperforms simple Monte Carlo and the popular Vegas method.
Variational Bayesian inference is an important machine-learning tool that finds application from statistics to robotics. The goal is to find an approximate probability density function (PDF) from a chosen family that is in some sense 'closest' to the full Bayesian posterior. Closeness is typically defined through the selection of an appropriate loss functional such as the Kullback-Leibler (KL) divergence. In this paper, we explore a new formulation of variational inference by exploiting the fact that (most) PDFs are members of a Bayesian Hilbert space under careful definitions of vector addition, scalar multiplication and an inner product. We show that, under the right conditions, variational inference based on KL divergence can amount to iterative projection, in the Euclidean sense, of the Bayesian posterior onto a subspace corresponding to the selected approximation family. We work through the details of this general framework for the specific case of the Gaussian approximation family and show the equivalence to another Gaussian variational inference approach. We furthermore discuss the implications for systems that exhibit sparsity, which is handled naturally in Bayesian space, and give an example of a high-dimensional robotic state estimation problem that can be handled as a result. We provide some preliminary examples of how the approach could be applied to non-Gaussian inference and discuss the limitations of the approach in detail to encourage follow-on work along these lines.
The ability to recognize objects despite there being differences in appearance, known as Core Object Recognition, forms a critical part of human perception. While it is understood that the brain accomplishes Core Object Recognition through feedforward, hierarchical computations through the visual stream, the underlying algorithms that allow for invariant representations to form downstream is still not well understood. (DiCarlo et al., 2012) Various computational perceptual models have been built to attempt and tackle the object identification task in an artificial perceptual setting. Artificial Neural Networks, computational graphs consisting of weighted edges and mathematical operations at vertices, are loosely inspired by neural networks in the brain and have proven effective at various visual perceptual tasks, including object characterization and identification. (Pinto et al., 2008) (DiCarlo et al., 2012) For many data analysis tasks, learning representations where each dimension is statistically independent and thus disentangled from the others is useful. If the underlying generative factors of the data are also statistically independent, Bayesian inference of latent variables can form disentangled representations. This thesis constitutes a research project exploring a generalization of the Variational Autoencoder (VAE), $\beta$-VAE, that aims to learn disentangled representations using variational inference. $\beta$-VAE incorporates the hyperparameter $\beta$, and enforces conditional independence of its bottleneck neurons, which is in general not compatible with the statistical independence of latent variables. This text examines this architecture, and provides analytical and numerical arguments, with the goal of demonstrating that this incompatibility leads to a non-monotonic inference performance in $\beta$-VAE with a finite optimal $\beta$.
The COVID-19 pandemic has highlighted the importance of in-silico epidemiological modelling in predicting the dynamics of infectious diseases to inform health policy and decision makers about suitable prevention and containment strategies. Work in this setting involves solving challenging inference and control problems in individual-based models of ever increasing complexity. Here we discuss recent breakthroughs in machine learning, specifically in simulation-based inference, and explore its potential as a novel venue for model calibration to support the design and evaluation of public health interventions. To further stimulate research, we are developing software interfaces that turn two cornerstone COVID-19 and malaria epidemiology models COVID-sim, (https://github.com/mrc-ide/covid-sim/) and OpenMalaria (https://github.com/SwissTPH/openmalaria) into probabilistic programs, enabling efficient interpretable Bayesian inference within those simulators.
We consider systems under uncertainty whose dynamics are partially unknown. Our aim is to study satisfaction of temporal logic properties by trajectories of such systems. We express these properties as signal temporal logic formulas and check if the probability of satisfying the property is at least a given threshold. Since the dynamics are parameterized and partially unknown, we collect data from the system and employ Bayesian inference techniques to associate a confidence value to the satisfaction of the property. The main novelty of our approach is to combine both data-driven and model-based techniques in order to have a two-layer probabilistic reasoning over the behavior of the system: one layer is related to the stochastic noise inside the system and the next layer is related to the noisy data collected from the system. We provide approximate algorithms for computing the confidence for linear dynamical systems.
As models of cognition grow in complexity and number of parameters, Bayesian inference with standard methods can become intractable, especially when the data-generating model is of unknown analytic form. Recent advances in simulation-based inference using specialized neural network architectures circumvent many previous problems of approximate Bayesian computation. Moreover, due to the properties of these special neural network estimators, the effort of training the networks via simulations amortizes over subsequent evaluations which can re-use the same network for multiple datasets and across multiple researchers. However, these methods have been largely underutilized in cognitive science and psychology so far, even though they are well suited for tackling a wide variety of modeling problems. With this work, we provide a general introduction to amortized Bayesian parameter estimation and model comparison and demonstrate the applicability of the proposed methods on a well-known class of intractable response-time models.
Incorporating environmental, social, and governance (ESG) considerations into systematic investments has drawn numerous attention recently. In this paper, we focus on the ESG events in financial news flow and exploring the predictive power of ESG related financial news on stock volatility. In particular, we develop a pipeline of ESG news extraction, news representations, and Bayesian inference of deep learning models. Experimental evaluation on real data and different markets demonstrates the superior predicting performance as well as the relation of high volatility prediction to stocks with potential high risk and low return. It also shows the prospect of the proposed pipeline as a flexible predicting framework for various textual data and target variables.
Group testing is a method of identifying infected patients by performing tests on a pool of specimens collected from patients. For the case in which the test returns a false result with finite probability, we propose Bayesian inference and a corresponding belief propagation (BP) algorithm to identify the infected patients from the results of tests performed on the pool. We show that the true-positive rate is improved by taking into account the credible interval of a point estimate of each patient. Further, the prevalence and the error probability in the test are estimated by combining an expectation-maximization method with the BP algorithm. As another approach, we introduce a hierarchical Bayes model to identify the infected patients and estimate the prevalence. By comparing these methods, we formulate a guide for practical usage.
The concept of free energy has its origins in 19th century thermodynamics, but has recently found its way into the behavioral and neural sciences, where it has been promoted for its wide applicability and has even been suggested as a fundamental principle of understanding intelligent behavior and brain function. We argue that there are essentially two different notions of free energy in current models of intelligent agency, that can both be considered as applications of Bayesian inference to the problem of action selection: one that appears when trading off accuracy and uncertainty based on a general maximum entropy principle, and one that formulates action selection in terms of minimizing an error measure that quantifies deviations of beliefs and policies from given reference models. The first approach provides a normative rule for action selection in the face of model uncertainty or when information processing capabilities are limited. The second approach directly aims to formulate the action selection problem as an inference problem in the context of Bayesian brain theories, also known as Active Inference in the literature. We elucidate the main ideas and discuss critical technical and conceptual issues revolving around these two notions of free energy that both claim to apply at all levels of decision-making, from the high-level deliberation of reasoning down to the low-level information processing of perception.
We revisit Zadeh's notion of "evidence of the second kind" and show that it provides the foundation for a general theory of epistemic random fuzzy sets, which generalizes both the Dempster-Shafer theory of belief functions and possibility theory. In this perspective, Dempster-Shafer theory deals with belief functions generated by random sets, while possibility theory deals with belief functions induced by fuzzy sets. The more general theory allows us to represent and combine evidence that is both uncertain and fuzzy. We demonstrate the application of this formalism to statistical inference, and show that it makes it possible to reconcile the possibilistic interpretation of likelihood with Bayesian inference.