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Bayesian Networks (BNs) have become increasingly popular over the last few decades as a tool for reasoning under uncertainty in fields as diverse as medicine, biology, epidemiology, economics and the social sciences. This is especially true in real-world areas where we seek to answer complex questions based on hypothetical evidence to determine actions for intervention. However, determining the graphical structure of a BN remains a major challenge, especially when modelling a problem under causal assumptions. Solutions to this problem include the automated discovery of BN graphs from data, constructing them based on expert knowledge, or a combination of the two. This paper provides a comprehensive review of combinatoric algorithms proposed for learning BN structure from data, describing 61 algorithms including prototypical, well-established and state-of-the-art approaches. The basic approach of each algorithm is described in consistent terms, and the similarities and differences between them highlighted. Methods of evaluating algorithms and their comparative performance are discussed including the consistency of claims made in the literature. Approaches for dealing with data noise in real-world datasets and incorporating expert knowledge into the learning process are also covered.

Order effects occur when judgments about a hypothesis's probability given a sequence of information do not equal the probability of the same hypothesis when the information is reversed. Different experiments have been performed in the literature that supports evidence of order effects. We proposed a Bayesian update model for order effects where each question can be thought of as a mini-experiment where the respondents reflect on their beliefs. We showed that order effects appear, and they have a simple cognitive explanation: the respondent's prior belief that two questions are correlated. The proposed Bayesian model allows us to make several predictions: (1) we found certain conditions on the priors that limit the existence of order effects; (2) we show that, for our model, the QQ equality is not necessarily satisfied (due to symmetry assumptions); and (3) the proposed Bayesian model has the advantage of possessing fewer parameters than its quantum counterpart.

We present two methods to reduce the complexity of Bayesian network (BN) classifiers. First, we introduce quantization-aware training using the straight-through gradient estimator to quantize the parameters of BNs to few bits. Second, we extend a recently proposed differentiable tree-augmented naive Bayes (TAN) structure learning approach by also considering the model size. Both methods are motivated by recent developments in the deep learning community, and they provide effective means to trade off between model size and prediction accuracy, which is demonstrated in extensive experiments. Furthermore, we contrast quantized BN classifiers with quantized deep neural networks (DNNs) for small-scale scenarios which have hardly been investigated in the literature. We show Pareto optimal models with respect to model size, number of operations, and test error and find that both model classes are viable options.

Distances between data points are widely used in point cloud representation learning. Yet, it is no secret that under the effect of noise, these distances-and thus the models based upon them-may lose their usefulness in high dimensions. Indeed, the small marginal effects of the noise may then accumulate quickly, shifting empirical closest and furthest neighbors away from the ground truth. In this paper, we characterize such effects in high-dimensional data using an asymptotic probabilistic expression. Furthermore, while it has been previously argued that neighborhood queries become meaningless and unstable when there is a poor relative discrimination between the furthest and closest point, we conclude that this is not necessarily the case when explicitly separating the ground truth data from the noise. More specifically, we derive that under particular conditions, empirical neighborhood relations affected by noise are still likely to be true even when we observe this discrimination to be poor. We include thorough empirical verification of our results, as well as experiments that interestingly show our derived phase shift where neighbors become random or not is identical to the phase shift where common dimensionality reduction methods perform poorly or well for finding low-dimensional representations of high-dimensional data with dense noise.

Recent works have investigated deep learning models trained by optimising PAC-Bayes bounds, with priors that are learnt on subsets of the data. This combination has been shown to lead not only to accurate classifiers, but also to remarkably tight risk certificates, bearing promise towards self-certified learning (i.e. use all the data to learn a predictor and certify its quality). In this work, we empirically investigate the role of the prior. We experiment on 6 datasets with different strategies and amounts of data to learn data-dependent PAC-Bayes priors, and we compare them in terms of their effect on test performance of the learnt predictors and tightness of their risk certificate. We ask what is the optimal amount of data which should be allocated for building the prior and show that the optimum may be dataset dependent. We demonstrate that using a small percentage of the prior-building data for validation of the prior leads to promising results. We include a comparison of underparameterised and overparameterised models, along with an empirical study of different training objectives and regularisation strategies to learn the prior distribution.

We propose two generic methods for improving semi-supervised learning (SSL). The first integrates weight perturbation (WP) into existing "consistency regularization" (CR) based methods. We implement WP by leveraging variational Bayesian inference (VBI). The second method proposes a novel consistency loss called "maximum uncertainty regularization" (MUR). While most consistency losses act on perturbations in the vicinity of each data point, MUR actively searches for "virtual" points situated beyond this region that cause the most uncertain class predictions. This allows MUR to impose smoothness on a wider area in the input-output manifold. Our experiments show clear improvements in classification errors of various CR based methods when they are combined with VBI or MUR or both.

Graph neural networks (GNNs) are a popular class of machine learning models whose major advantage is their ability to incorporate a sparse and discrete dependency structure between data points. Unfortunately, GNNs can only be used when such a graph-structure is available. In practice, however, real-world graphs are often noisy and incomplete or might not be available at all. With this work, we propose to jointly learn the graph structure and the parameters of graph convolutional networks (GCNs) by approximately solving a bilevel program that learns a discrete probability distribution on the edges of the graph. This allows one to apply GCNs not only in scenarios where the given graph is incomplete or corrupted but also in those where a graph is not available. We conduct a series of experiments that analyze the behavior of the proposed method and demonstrate that it outperforms related methods by a significant margin.

Data augmentation has been widely used for training deep learning systems for medical image segmentation and plays an important role in obtaining robust and transformation-invariant predictions. However, it has seldom been used at test time for segmentation and not been formulated in a consistent mathematical framework. In this paper, we first propose a theoretical formulation of test-time augmentation for deep learning in image recognition, where the prediction is obtained through estimating its expectation by Monte Carlo simulation with prior distributions of parameters in an image acquisition model that involves image transformations and noise. We then propose a novel uncertainty estimation method based on the formulated test-time augmentation. Experiments with segmentation of fetal brains and brain tumors from 2D and 3D Magnetic Resonance Images (MRI) showed that 1) our test-time augmentation outperforms a single-prediction baseline and dropout-based multiple predictions, and 2) it provides a better uncertainty estimation than calculating the model-based uncertainty alone and helps to reduce overconfident incorrect predictions.

Stochastic gradient Markov chain Monte Carlo (SGMCMC) has become a popular method for scalable Bayesian inference. These methods are based on sampling a discrete-time approximation to a continuous time process, such as the Langevin diffusion. When applied to distributions defined on a constrained space, such as the simplex, the time-discretisation error can dominate when we are near the boundary of the space. We demonstrate that while current SGMCMC methods for the simplex perform well in certain cases, they struggle with sparse simplex spaces; when many of the components are close to zero. However, most popular large-scale applications of Bayesian inference on simplex spaces, such as network or topic models, are sparse. We argue that this poor performance is due to the biases of SGMCMC caused by the discretization error. To get around this, we propose the stochastic CIR process, which removes all discretization error and we prove that samples from the stochastic CIR process are asymptotically unbiased. Use of the stochastic CIR process within a SGMCMC algorithm is shown to give substantially better performance for a topic model and a Dirichlet process mixture model than existing SGMCMC approaches.

During recent years, active learning has evolved into a popular paradigm for utilizing user's feedback to improve accuracy of learning algorithms. Active learning works by selecting the most informative sample among unlabeled data and querying the label of that point from user. Many different methods such as uncertainty sampling and minimum risk sampling have been utilized to select the most informative sample in active learning. Although many active learning algorithms have been proposed so far, most of them work with binary or multi-class classification problems and therefore can not be applied to problems in which only samples from one class as well as a set of unlabeled data are available. Such problems arise in many real-world situations and are known as the problem of learning from positive and unlabeled data. In this paper we propose an active learning algorithm that can work when only samples of one class as well as a set of unlabelled data are available. Our method works by separately estimating probability desnity of positive and unlabeled points and then computing expected value of informativeness to get rid of a hyper-parameter and have a better measure of informativeness./ Experiments and empirical analysis show promising results compared to other similar methods.