Semi-supervised learning is a model training method that uses both labeled and unlabeled data. This paper proposes a fully Bayes semi-supervised learning algorithm that can be applied to any multi-category classification problem. We assume the labels are missing at random when using unlabeled data in a semi-supervised setting. Suppose we have $K$ classes in the data. We assume that the observations follow $K$ multivariate normal distributions depending on their true class labels after some common unknown transformation is applied to each component of the observation vector. The function is expanded in a B-splines series, and a prior is added to the coefficients. We consider a normal prior on the coefficients and constrain the values to meet the normality and identifiability constraints requirement. The precision matrices of the Gaussian distributions are given a conjugate Wishart prior, while the means are given the improper uniform prior. The resulting posterior is still conditionally conjugate, and the Gibbs sampler aided by a data-augmentation technique can thus be adopted. An extensive simulation study compares the proposed method with several other available methods. The proposed method is also applied to real datasets on diagnosing breast cancer and classification of signals. We conclude that the proposed method has a better prediction accuracy in various cases.
Universidad Carlos III de Madrid（马德里卡洛斯三世大学）；Gregorio Marañón Health Research Institute（格雷戈里奥·马拉尼翁健康研究所）
The Bayesian approach to feature extraction, known as factor analysis (FA), has been widely studied in machine learning to obtain a latent representation of the data. An adequate selection of the probabilities and priors of these bayesian models allows the model to better adapt to the data nature (i.e. heterogeneity, sparsity), obtaining a more representative latent space. The objective of this article is to propose a general FA framework capable of modelling any problem. To do so, we start from the Bayesian Inter-Battery Factor Analysis (BIBFA) model, enhancing it with new functionalities to …
Beijing Institute of Technology（北京理工大学）；University of Technology Sydney（悉尼科技大学）
Bayesian optimization (BO) has been broadly applied to computational expensive problems, but it is still challenging to extend BO to high dimensions. Existing works are usually under strict assumption of an additive or a linear embedding structure for objective functions. This paper directly introduces a supervised dimension reduction method, Sliced Inverse Regression (SIR), to high dimensional Bayesian optimization, which could effectively learn the intrinsic sub-structure of objective function during the optimization. Furthermore, a kernel trick is developed to reduce computational complexit…
Given a set of possible models (e.g., Bayesian network structures) and a data sample, in the unsupervised model selection problem the task is to choose the most accurate model with respect to the domain joint probability distribution. In contrast to this, in supervised model selection it is a priori known that the chosen model will be used in the future for prediction tasks involving more ``focused' predictive distributions. Although focused predictive distributions can be produced from the joint probability distribution by marginalization, in practice the best model in the unsupervised sense …
Given a supervised machine learning problem where the training set has been subject to a known sampling bias, how can a model be trained to fit the original dataset? We achieve this through the Bayesian inference framework by altering the posterior distribution to account for the sampling function. We then apply this solution to binary logistic regression, and discuss scenarios where a dataset might be subject to intentional sample bias such as label imbalance. This technique is widely applicable for statistical inference on big data, from the medical sciences to image recognition to marketing…
Virtual screening aims to find desirable compounds from chemical library by using computational methods. For this purpose with machine learning, model outputs that can be interpreted as predictive probability will be beneficial, in that a high prediction score corresponds to high probability of correctness. In this work, we present a study on the prediction performance and reliability of graph neural networks trained with the recently proposed Bayesian learning algorithms. Our work shows that Bayesian learning algorithms allow well-calibrated predictions for various GNN architectures and class…
Many real-world functions are defined over both categorical and category-specific continuous variables and thus cannot be optimized by traditional Bayesian optimization (BO) methods. To optimize such functions, we propose a new method that formulates the problem as a multi-armed bandit problem, wherein each category corresponds to an arm with its reward distribution centered around the optimum of the objective function in continuous variables. Our goal is to identify the best arm and the maximizer of the corresponding continuous function simultaneously. Our algorithm uses a Thompson sampling s…
State Key Laboratory for Novel Software Technology（新型软件技术国家重点实验室）；Nanjing University（南京大学）
Bayesian optimization is a broadly applied methodology to optimize the expensive black-box function. Despite its success, it still faces the challenge from the high-dimensional search space. To alleviate this problem, we propose a novel Bayesian optimization framework (termed SILBO), which finds a low-dimensional space to perform Bayesian optimization iteratively through semi-supervised dimension reduction. SILBO incorporates both labeled points and unlabeled points acquired from the acquisition function to guide the embedding space learning. To accelerate the learning procedure, we present a …
Predicting bioactivity and physical properties of small molecules is a central challenge in drug discovery. Deep learning is becoming the method of choice but studies to date focus on mean accuracy as the main metric. However, to replace costly and mission-critical experiments by models, a high mean accuracy is not enough: Outliers can derail a discovery campaign, thus models need reliably predict when it will fail, even when the training data is biased; experiments are expensive, thus models need to be data-efficient and suggest informative training sets using active learning. We show that un…
School of Engineering, University of Warwick, UK（华威大学工程学院，英国）；Warwick Center for Predictive Modelling, University of Warwick, UK（华威大学预测建模中心，英国）；Department of Statistics, University of Warwick, UK（华威大学统计系，英国）；Division of Translational Medicine, Warwick Medical School, UK（华威医学院转化医学部，英国）；Dept of Clinical Biochemistry, Division of Pathology, University Hospital Coventry and Warwickshire, Coventry, UK（考文垂和华威郡大学医院病理学部临床生物化学系，考文垂，英国）
We introduce a Bayesian framework for inference with a supervised version of the Gaussian process latent variable model. The framework overcomes the high correlations between latent variables and hyperparameters by using an unbiased pseudo estimate for the marginal likelihood that approximately integrates over the latent variables. This is used to construct a Markov Chain to explore the posterior of the hyperparameters. We demonstrate the procedure on simulated and real examples, showing its ability to capture uncertainty and multimodality of the hyperparameters and improved uncertainty quanti…
University College London（伦敦大学学院）；The Alan Turing Institute（艾伦·图灵研究所）
We propose a data-efficient Gaussian process-based Bayesian approach to the semi-supervised learning problem on graphs. The proposed model shows extremely competitive performance when compared to the state-of-the-art graph neural networks on semi-supervised learning benchmark experiments, and outperforms the neural networks in active learning experiments where labels are scarce. Furthermore, the model does not require a validation data set for early stopping to control over-fitting. Our model can be viewed as an instance of empirical distribution regression weighted locally by network connecti…
Although deep neural network (DNN) has achieved many state-of-the-art results, estimating the uncertainty presented in the DNN model and the data is a challenging task. Problems related to uncertainty such as classifying unknown classes (class which does not appear in the training data) data as known class with high confidence, is critically concerned in the safety domain area (e.g, autonomous driving, medical diagnosis). In this paper, we show that applying current Bayesian Neural Network (BNN) techniques alone does not effectively capture the uncertainty. To tackle this problem, we introduce…
Nonnegative matrix factorization (NMF) with group sparsity constraints is formulated as a probabilistic graphical model and, assuming some observed data have been generated by the model, a feasible variational Bayesian algorithm is derived for learning model parameters. When used in a supervised learning scenario, NMF is most often utilized as an unsupervised feature extractor followed by classification in the obtained feature subspace. Having mapped the class labels to a more general concept of groups which underlie sparsity of the coefficients, what the proposed group sparse NMF model allows…
Semi-supervised clustering is the task of clustering data points into clusters where only a fraction of the points are labelled. The true number of clusters in the data is often unknown and most models require this parameter as an input. Dirichlet process mixture models are appealing as they can infer the number of clusters from the data. However, these models do not deal with high dimensional data well and can encounter difficulties in inference. We present a novel nonparameteric Bayesian kernel based method to cluster data points without the need to prespecify the number of clusters or to mo…
Recently, techniques for applying convolutional neural networks to graph-structured data have emerged. Graph convolutional neural networks (GCNNs) have been used to address node and graph classification and matrix completion. Although the performance has been impressive, the current implementations have limited capability to incorporate uncertainty in the graph structure. Almost all GCNNs process a graph as though it is a ground-truth depiction of the relationship between nodes, but often the graphs employed in applications are themselves derived from noisy data or modelling assumptions. Spuri…
University of Ljubljana（卢布尔雅那大学）；West University of Timisoara（蒂米什瓦拉西部大学）
Human annotations are an important source of information in the development of natural language understanding approaches. As under the pressure of productivity annotators can assign different labels to a given text, the quality of produced annotations frequently varies. This is especially the case if decisions are difficult, with high cognitive load, requires awareness of broader context, or careful consideration of background knowledge. To alleviate the problem, we propose two semi-supervised methods to guide the annotation process: a Bayesian deep learning model and a Bayesian ensemble metho…
Self-Supervised Laplace Approximation for Bayesian Uncertainty Quantification
Rational Intelligence Lab, CISPA Helmholtz Center for Information Security Department of Statistics, LMU Munich（理性智能实验室，CISPA海德堡信息安全中心统计学系，慕尼黑大学）；Department of Statistics, LMU Munich（统计学系，慕尼黑大学）；Department of Computer Science, The University of Manchester（计算机科学系，曼彻斯特大学）
University of Toulouse, IRIT/INP-ENSEEIHT（图卢兹大学 IRIT/INP-ENSEEIHT）
Within a supervised classification framework, labeled data are used to learn classifier parameters. Prior to that, it is generally required to perform dimensionality reduction via feature extraction. These preprocessing steps have motivated numerous research works aiming at recovering latent variables in an unsupervised context. This paper proposes a unified framework to perform classification and low-level modeling jointly. The main objective is to use the estimated latent variables as features for classification and to incorporate simultaneously supervised information to help latent variable…
Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）；Shanghai Jiao Tong University（上海交通大学）
In emotion recognition, it is difficult to recognize human's emotional states using just a single modality. Besides, the annotation of physiological emotional data is particularly expensive. These two aspects make the building of effective emotion recognition model challenging. In this paper, we first build a multi-view deep generative model to simulate the generative process of multi-modality emotional data. By imposing a mixture of Gaussians assumption on the posterior approximation of the latent variables, our model can learn the shared deep representation from multiple modalities. To solve…
We generalise an existing construction of Bayesian Lenses to admit lenses between pairs of objects where the backwards object is dependent on states on the forwards object (interpreted as probability distributions). This gives a natural setting for studying stochastic maps with Bayesian inverses restricted to the points supported by a given prior. In order to state this formally we develop a proposed definition by Fritz of a support object in a Markov category and show that these give rise to a section into the category of dependent Bayesian lenses encoding a more canonical notion of Bayesian …
Technical University of Munich（慕尼黑工业大学）；University of Pennsylvania（宾夕法尼亚大学）；Munich Data Science Institute（慕尼黑数据科学研究所）
Neural Operators offer a powerful, data-driven tool for solving parametric PDEs as they can represent maps between infinite-dimensional function spaces. In this work, we employ physics-informed Neural Operators in the context of high-dimensional, Bayesian inverse problems. Traditional solution strategies necessitate an enormous, and frequently infeasible, number of forward model solves, as well as the computation of parametric derivatives. In order to enable efficient solutions, we extend Deep Operator Networks (DeepONets) by employing a RealNVP architecture which yields an invertible and diff…
Hierarchical probabilistic models, such as Gaussian mixture models, are widely used for unsupervised learning tasks. These models consist of observable and latent variables, which represent the observable data and the underlying data-generation process, respectively. Unsupervised learning tasks, such as cluster analysis, are regarded as estimations of latent variables based on the observable ones. The estimation of latent variables in semi-supervised learning, where some labels are observed, will be more precise than that in unsupervised, and one of the concerns is to clarify the effect of the…
We study the accuracy of Bayesian supervised method used to cluster individuals into genetically homogeneous groups on the basis of dominant or codominant molecular markers. We provide a formula relating an error criterion the number of loci used and the number of clusters. This formula is exact and holds for arbitrary number of clusters and markers. Our work suggests that dominant markers studies can achieve an accuracy similar to that of codominant markers studies if the number of markers used in the former is about 1.7 times larger than in the latter.
The University of Adelaide（阿德莱德大学）；Queensland University of Technology（昆士兰科技大学）；CMU（卡内基梅隆大学）
Traditional GANs use a deterministic generator function (typically a neural network) to transform a random noise input $z$ to a sample $\mathbf{x}$ that the discriminator seeks to distinguish. We propose a new GAN called Bayesian Conditional Generative Adversarial Networks (BC-GANs) that use a random generator function to transform a deterministic input $y'$ to a sample $\mathbf{x}$. Our BC-GANs extend traditional GANs to a Bayesian framework, and naturally handle unsupervised learning, supervised learning, and semi-supervised learning problems. Experiments show that the proposed BC-GANs outpe…
Crowdsourcing has emerged as a powerful paradigm for efficiently labeling large datasets and performing various learning tasks, by leveraging crowds of human annotators. When additional information is available about the data, semi-supervised crowdsourcing approaches that enhance the aggregation of labels from human annotators are well motivated. This work deals with semi-supervised crowdsourced classification, under two regimes of semi-supervision: a) label constraints, that provide ground-truth labels for a subset of data; and b) potentially easier to obtain instance-level constraints, that …
Target encoding is an effective encoding technique of categorical variables and is often used in machine learning systems for processing tabular data sets with mixed numeric and categorical variables. Recently en enhanced version of this encoding technique was proposed by using conjugate Bayesian modeling. This paper presents a further development of Bayesian encoding method by using sampling techniques, which helps in extracting information from intra-category distribution of the target variable, improves generalization and reduces target leakage.
Age-disaggregated health data is crucial for effective public health planning and monitoring. Monitoring under-five mortality, for example, requires highly detailed age data since the distribution of potential causes of death varies substantially within the first few years of life. Comparative researchers often have to rely on multiple data sources yet, these sources often have ages aggregated at different levels, making it difficult to combine the data into a single, coherent picture. To address this challenge in the context of under-five cause-specific mortality, we propose a Bayesian approa…
Land cover maps are a vital input variable to many types of environmental research and management. While they can be produced automatically by machine learning techniques, these techniques require substantial training data to achieve high levels of accuracy, which are not always available. One technique researchers use when labelled training data are scarce is domain adaptation (DA) -- where data from an alternate region, known as the source domain, are used to train a classifier and this model is adapted to map the study region, or target domain. The scenario we address in this paper is known…
Understanding sub-cellular protein localisation is an essential component to analyse context specific protein function. Recent advances in quantitative mass-spectrometry (MS) have led to high resolution mapping of thousands of proteins to sub-cellular locations within the cell. Novel modelling considerations to capture the complex nature of these data are thus necessary. We approach analysis of spatial proteomics data in a non-parametric Bayesian framework, using mixtures of Gaussian process regression models. The Gaussian process regression model accounts for correlation structure within a su…
Generalization is essential for deep learning. In contrast to previous works claiming that Deep Neural Networks (DNNs) have an implicit regularization implemented by the stochastic gradient descent, we demonstrate explicitly Bayesian regularizations in a specific category of DNNs, i.e., Convolutional Neural Networks (CNNs). First, we introduce a novel probabilistic representation for the hidden layers of CNNs and demonstrate that CNNs correspond to Bayesian networks with the serial connection. Furthermore, we show that the hidden layers close to the input formulate prior distributions, thus CN…
Neural network based generative models with discriminative components are a powerful approach for semi-supervised learning. However, these techniques a) cannot account for model uncertainty in the estimation of the model's discriminative component and b) lack flexibility to capture complex stochastic patterns in the label generation process. To avoid these problems, we first propose to use a discriminative component with stochastic inputs for increased noise flexibility. We show how an efficient Gibbs sampling procedure can marginalize the stochastic inputs when inferring missing labels in thi…
Georgia Institute of Technology（佐治亚理工学院）；GE Research（通用电气研究中心）
Modern day engineering problems are ubiquitously characterized by sophisticated computer codes that map parameters or inputs to an underlying physical process. In other situations, experimental setups are used to model the physical process in a laboratory, ensuring high precision while being costly in materials and logistics. In both scenarios, only limited amount of data can be generated by querying the expensive information source at a finite number of inputs or designs. This problem is compounded further in the presence of a high-dimensional input space. State-of-the-art parameter space dim…
It has been argued that in supervised classification tasks, in practice it may be more sensible to perform model selection with respect to some more focused model selection score, like the supervised (conditional) marginal likelihood, than with respect to the standard marginal likelihood criterion. However, for most Bayesian network models, computing the supervised marginal likelihood score takes exponential time with respect to the amount of observed data. In this paper, we consider diagnostic Bayesian network classifiers where the significant model parameters represent conditional distributi…
Supervised Bayesian joint graphical model for simultaneous network estimation and subgroup identification
School of Statistics and Data Science, Shanghai University of International Business and Economics（上海对外经贸大学统计与数据科学学院）；School of Statistics and Data Science, Shanghai University of Finance and Economics（上海财经大学统计与数据科学学院）；Department of Biostatistics, Yale School of Public Health（耶鲁公共卫生学院生物统计学系）
Uber AI Labs（Uber人工智能实验室）；Cornell University（康奈尔大学）
Generative adversarial networks (GANs) can implicitly learn rich distributions over images, audio, and data which are hard to model with an explicit likelihood. We present a practical Bayesian formulation for unsupervised and semi-supervised learning with GANs. Within this framework, we use stochastic gradient Hamiltonian Monte Carlo to marginalize the weights of the generator and discriminator networks. The resulting approach is straightforward and obtains good performance without any standard interventions such as feature matching, or mini-batch discrimination. By exploring an expressive pos…
We propose a modification of the improved cross entropy (iCE) method to enhance its performance for network reliability assessment. The iCE method performs a transition from the nominal density to the optimal importance sampling (IS) density via a parametric distribution model whose cross entropy with the optimal IS is minimized. The efficiency and accuracy of the iCE method are largely influenced by the choice of the parametric model. In the context of reliability of systems with independent multi-state components, the obvious choice of the parametric family is the categorical distribution. W…
The main contribution of this article is a new prior distribution over directed acyclic graphs, which gives larger weight to sparse graphs. This distribution is intended for structured Bayesian networks, where the structure is given by an ordered block model. That is, the nodes of the graph are objects which fall into categories (or blocks); the blocks have a natural ordering. The presence of a relationship between two objects is denoted by an arrow, from the object of lower category to the object of higher category. The models considered here were introduced in Kemp et al. (2004) for relation…
In this paper we develop a functorial language of probabilistic morphisms and apply it to some basic problems in Bayesian nonparametrics. First we extend and unify the Kleisli category of probabilistic morphisms proposed by Lawvere and Giry with the category of statistical models proposed by Chentsov and Morse-Sacksteder. Then we introduce the notion of a Bayesian statistical model that formalizes the notion of a parameter space with a given prior distribution in Bayesian statistics. {We revisit the existence of a posterior distribution, using probabilistic morphisms}. In particular, we give a…
A method has been developed for the analysis of images of sentinel lymph nodes generated by a spectral scanning device. The aim is to classify the nodes, excised during surgery for breast cancer, as normal or metastatic. The data from one node constitute spectra at 86 wavelengths for each pixel of a 20*20 grid. For the analysis, the spectra are reduced to scores on two factors, one derived externally from a linear discriminant analysis using spectra taken manually from known normal and metastatic tissue, and one derived from the node under investigation to capture variability orthogonal to the…
We undertake Bayesian learning of the high-dimensional functional relationship between a system parameter vector and an observable, that is in general tensor-valued. The ultimate aim is Bayesian inverse prediction of the system parameters, at which test data is recorded. We attempt such learning given hypercuboidally-shaped data that displays strong discontinuities, rendering learning challenging. We model the sought high-dimensional function, with a tensor-variate Gaussian Process (GP), and use three independent ways for learning covariance matrices of the resulting likelihood, which is Tenso…
In this paper, a scale mixture of Normal distributions model is developed for classification and clustering of data having outliers and missing values. The classification method, based on a mixture model, focuses on the introduction of latent variables that gives us the possibility to handle sensitivity of model to outliers and to allow a less restrictive modelling of missing data. Inference is processed through a Variational Bayesian Approximation and a Bayesian treatment is adopted for model learning, supervised classification and clustering.
This paper explores an approach to Bayesian sample size determination in clinical trials. The approach falls into the category of what is often called "proper Bayesian", in that it does not mix frequentist concepts with Bayesian ones. A criterion for a "successful trial" is defined in terms of a posterior probability, its probability is assessed using the marginal distribution of the data, and this probability forms the basis for choosing sample sizes. We illustrate with a standard problem in clinical trials, that of establishing superiority of a new drug over a control.
Forecasting enterprise-wide revenue is critical to many companies and presents several challenges and opportunities for significant business impact. This case study is based on model developments to address these challenges for forecasting in a large-scale retail company. Focused on multivariate revenue forecasting across collections of supermarkets and product Categories, hierarchical dynamic models are natural: these are able to couple revenue streams in an integrated forecasting model, while allowing conditional decoupling to enable relevant and sensitive analysis together with scalable com…
In high-dimensional prediction problems, where the number of features may greatly exceed the number of training instances, fully Bayesian approach with a sparsifying prior is known to produce good results but is computationally challenging. To alleviate this computational burden, we propose to use a preprocessing step where we first apply a dimension reduction to the original data to reduce the number of features to something that is computationally conveniently handled by Bayesian methods. To do this, we propose a new dimension reduction technique, called iterative supervised principal compon…
Bayes' rule tells us how to invert a causal process in order to update our beliefs in light of new evidence. If the process is believed to have a complex compositional structure, we may ask whether composing the inversions of the component processes gives the same belief update as the inversion of the whole. We answer this question affirmatively, showing that the relevant compositional structure is precisely that of the lens pattern, and that we can think of Bayesian inversion as a particular instance of a state-dependent morphism in a corresponding fibred category. We define a general notion …
Anomaly Detection has several important applications. In this paper, our focus is on detecting anomalies in seller-reviewer data using tensor decomposition. While tensor-decomposition is mostly unsupervised, we formulate Bayesian semi-supervised tensor decomposition to take advantage of sparse labeled data. In addition, we use Polya-Gamma data augmentation for the semi-supervised Bayesian tensor decomposition. Finally, we show that the Pólya-Gamma formulation simplifies calculation of the Fisher information matrix for partial natural gradient learning. Our experimental results show that our se…
Supervised topic models with a logistic likelihood have two issues that potentially limit their practical use: 1) response variables are usually over-weighted by document word counts; and 2) existing variational inference methods make strict mean-field assumptions. We address these issues by: 1) introducing a regularization constant to better balance the two parts based on an optimization formulation of Bayesian inference; and 2) developing a simple Gibbs sampling algorithm by introducing auxiliary Polya-Gamma variables and collapsing out Dirichlet variables. Our augment-and-collapse sampling …
