Classification problems involving high dimensional data are extensive in many fields such as finance, marketing, and bioinformatics. Unique challenges with high dimensional datasets are numerous and well known, with many classifiers built under traditional low dimensional frameworks simply unable to be applied to such high dimensional data. Discriminant Analysis (DA) is one such example (Fisher, 1936). DA classifiers work by assuming the distribution of the features is strictly Gaussian at the class level, and assign a particular point to the class label which minimises the Mahalanobis (for linear discriminant analysis (LDA)) distance between that point and the mean of the multivariate normal corresponding to such class. Although extraordinary simple and easy to use in low dimensional settings, DA is well known to be unusable in high dimensions due to the maximum likelihood estimate of the corresponding covariance matrix being singular when the number of features is greater than that of the observations.

Our interest in this paper is in the construction of symbolic explanations for predictions made by a deep neural network. We will focus attention on deep relational machines (DRMs, first proposed by H. Lodhi). A DRM is a deep network in which the input layer consists of Boolean-valued functions (features) that are defined in terms of relations provided as domain, or background, knowledge. Our DRMs differ from those proposed by Lodhi, which use an Inductive Logic Programming (ILP) engine to first select features (we use random selections from a space of features that satisfies some approximate constraints on logical relevance and non-redundancy). But why do the DRMs predict what they do? One way of answering this is the LIME setting, in which readable proxies for a black-box predictor. The proxies are intended only to model the predictions of the black-box in local regions of the instance-space. But readability alone may not enough: to be understandable, the local models must use relevant concepts in an meaningful manner. We investigate the use of a Bayes-like approach to identify logical proxies for local predictions of a DRM. We show: (a) DRM's with our randomised propositionalization method achieve state-of-the-art predictive performance; (b) Models in first-order logic can approximate the DRM's prediction closely in a small local region; and (c) Expert-provided relevance information can play the role of a prior to distinguish between logical explanations that perform equivalently on prediction alone.

A subtle difference between propositional and relational data is that in many relational models, marginal probabilities depend on the population or domain size. This paper connects the dependence on population size to the classic notion of projectivity from statistical theory: Projectivity implies that relational predictions are robust with respect to changes in domain size. We discuss projectivity for a number of common SRL systems, and identify syntactic fragments that are guaranteed to yield projective models. The syntactic conditions are restrictive, which suggests that projectivity is difficult to achieve in SRL, and care must be taken when working with different domain sizes.

Data scientist Stefano Cosentino observed in a post that the Bayesian approach leans more towards the distributions associated with each parameter. For instance, he writes that the two parameters depicted below, as shown by the Gaussian curves after a trained Bayesian network has converged. Hence the Bayesian approach, where the parameters are unknown quantities can be considered as random variables. University of Buffalo's paper defines the Bayesian approach to uncertainty, which treats all uncertain quantities as random variables and uses the laws of probability to manipulate those uncertain quantities. Hence, the right Bayesian approach integrates over all uncertain quantities rather than optimise them, states the paper.

In the last few months, I have had several people contact me about their enthusiasm for venturing into the world of data science and using Machine Learning (ML) techniques to probe statistical regularities and build impeccable data-driven products. However, I've observed that some actually lack the necessary mathematical intuition and framework to get useful results. This is the main reason I decided to write this blog post. Recently, there has been an upsurge in the availability of many easy-to-use machine and deep learning packages such as scikit-learn, Weka, Tensorflow etc. Machine Learning theory is a field that intersects statistical, probabilistic, computer science and algorithmic aspects arising from learning iteratively from data and finding hidden insights which can be used to build intelligent applications. Despite the immense possibilities of Machine and Deep Learning, a thorough mathematical understanding of many of these techniques is necessary for a good grasp of the inner workings of the algorithms and getting good results.

Bayesian optimization has emerged as a strong candidate tool for global optimization of functions with expensive evaluation costs. However, due to the dynamic nature of research in Bayesian approaches, and the evolution of computing technology, using Bayesian optimization in a parallel computing environment remains a challenge for the non-expert. In this report, I review the state-of-the-art in parallel and scalable Bayesian optimization methods. In addition, I propose practical ways to avoid a few of the pitfalls of Bayesian optimization, such as oversampling of edge parameters and over-exploitation of high performance parameters. Finally, I provide relatively simple, heuristic algorithms, along with their open source software implementations, that can be immediately and easily deployed in any computing environment.

This paper is based on a previous publication [29]. Our work extends exception mining and outlier detection to the case of object-relational data. Object-relational data represent a complex heterogeneous network [12], which comprises objects of different types, links among these objects, also of different types, and attributes of these links. This special structure prohibits a direct vectorial data representation. We follow the well-established Exceptional Model Mining framework, which leverages machine learning models for exception mining: A object is exceptional to the extent that a model learned for the object data differs from a model learned for the general population. Exceptional objects can be viewed as outliers. We apply state of-the-art probabilistic modelling techniques for object-relational data that construct a graphical model (Bayesian network), which compactly represents probabilistic associations in the data. A new metric, derived from the learned object-relational model, quantifies the extent to which the individual association pattern of a potential outlier deviates from that of the whole population. The metric is based on the likelihood ratio of two parameter vectors: One that represents the population associations, and another that represents the individual associations. Our method is validated on synthetic datasets and on real-world data sets about soccer matches and movies. Compared to baseline methods, our novel transformed likelihood ratio achieved the best detection accuracy on all datasets.

We present a learning theory for the training of a linear system operator having an input compositional variable and propose a Bayesian inversion method for inferring the unknown variable from an output of a noisy linear system. We assume that we have partial or even no knowledge of the operator but have training data of input and ouput. A compositional variable satisfies the constraints that the elements of the variable are all non-negative and sum to unity. We quantified the uncertainty in the trained operator and present the convergence rates of training in explicit forms for several interesting cases under stochastic compositional models. The trained linear operator with the covariance matrix, estimated from the training set of pairs of ground-truth input and noisy output data, is further used in evaluation of posterior uncertainty of the solution. This posterior uncertainty clearly demonstrates uncertainty propagation from noisy training data and addresses possible mismatch between the true operator and the estimated one in the final solution.

Exploiting the appropriate inductive bias based on the knowledge of data is essential for achieving good performance in statistical machine learning. In practice, however, the domain knowledge of interest often provides information on the relationship of data attributes only distantly, which hinders direct utilization of such domain knowledge in popular regularization methods. In this paper, we propose the knowledge-based distant regularization framework, in which we utilize the distant information encoded in a knowledge graph for regularization of probabilistic model estimation. In particular, we propose to impose prior distributions on model parameters specified by knowledge graph embeddings. As an instance of the proposed framework, we present the factor analysis model with the knowledge-based distant regularization. We show the results of preliminary experiments on the improvement of the generalization capability of such model.

Bayesian learning is built on an assumption that the model space contains a true reflection of the data generating mechanism. This assumption is problematic, particularly in complex data environments. Here we present a Bayesian nonparametric approach to learning that makes use of statistical models, but does not assume that the model is true. Our approach has provably better properties than using a parametric model and admits a trivially parallelizable Monte Carlo sampling scheme that affords massive scalability on modern computer architectures. The model-based aspect of learning is particularly attractive for regularizing nonparametric inference when the sample size is small, and also for correcting approximate approaches such as variational Bayes (VB). We demonstrate the approach on a number of examples including VB classifiers and Bayesian random forests.