The Calculation of Posterior Distributions by Data Augmentation: Comment: A Noniterative Sampling/Importance Resampling Alternative to the Data Augmentation Algorithm for Creating a Few Imputations When Fractions of Missing Information Are Modest: The SIR. Journal of the American Statistical Association, 82(398):543, jun 1987.

Gaussian multiplicative noise is commonly used as a stochastic regularisation technique in training of deterministic neural networks. A recent paper reinterpreted the technique as a specific algorithm for approximate inference in Bayesian neural networks; several extensions ensued. We show that the log-uniform prior used in all the above publications does not generally induce a proper posterior, and thus Bayesian inference in such models is ill-posed. Independent of the log-uniform prior, the correlated weight noise approximation has further issues leading to either infinite objective or high risk of overfitting. The above implies that the reported sparsity of obtained solutions cannot be explained by Bayesian or the related minimum description length arguments. We thus study the objective from a non-Bayesian perspective, provide its previously unknown analytical form which allows exact gradient evaluation, and show that the later proposed additive reparametrisation introduces minima not present in the original multiplicative parametrisation. Implications and future research directions are discussed.

We establish the consistency of an algorithm of Mondrian Forests, a randomized classification algorithm that can be implemented online. First, we amend the original Mondrian Forest algorithm, that considers a fixed lifetime parameter. Indeed, the fact that this parameter is fixed hinders the statistical consistency of the original procedure. Our modified Mondrian Forest algorithm grows trees with increasing lifetime parameters $\lambda_n$, and uses an alternative updating rule, allowing to work also in an online fashion. Second, we provide a theoretical analysis establishing simple conditions for consistency. Our theoretical analysis also exhibits a surprising fact: our algorithm achieves the minimax rate (optimal rate) for the estimation of a Lipschitz regression function, which is a strong extension of previous results to an arbitrary dimension.

This work brings together two powerful concepts in Gaussian processes: the variational approach to sparse approximation and the spectral representation of Gaussian processes. This gives rise to an approximation that inherits the benefits of the variational approach but with the representational power and computational scalability of spectral representations. The work hinges on a key result that there exist spectral features related to a finite domain of the Gaussian process which exhibit almost-independent covariances. We derive these expressions for Matern kernels in one dimension, and generalize to more dimensions using kernels with specific structures. Under the assumption of additive Gaussian noise, our method requires only a single pass through the dataset, making for very fast and accurate computation. We fit a model to 4 million training points in just a few minutes on a standard laptop. With non-conjugate likelihoods, our MCMC scheme reduces the cost of computation from O(NM2) (for a sparse Gaussian process) to O(NM) per iteration, where N is the number of data and M is the number of features.

This study presents an efficient incremental/decremental approach for big streams based on Kernel Ridge Regression (KRR), a frequently used data analysis in cloud centers. To avoid reanalyzing the whole dataset whenever sensors receive new training data, typical incremental KRR used a single-instance mechanism for updating an existing system. However, this inevitably increased redundant computational time, not to mention applicability to big streams. To this end, the proposed mechanism supports incremental/decremental processing for both single and multiple samples (i.e., batch processing). A large scale of data can be divided into batches, processed by a machine, without sacrificing the accuracy. Moreover, incremental/decremental analyses in empirical and intrinsic space are also proposed in this study to handle different types of data either with a large number of samples or high feature dimensions, whereas typical methods focused only on one type. At the end of this study, we further the proposed mechanism to statistical Kernelized Bayesian Regression, so that uncertainty modeling with incremental/decremental computation becomes applicable. Experimental results showed that computational time was significantly reduced, better than the original nonincremental design and the typical single incremental method. Furthermore, the accuracy of the proposed method remained the same as the baselines. This implied that the system enhanced efficiency without sacrificing the accuracy. These findings proved that the proposed method was appropriate for variable streaming data analysis, thereby demonstrating the effectiveness of the proposed method.

Bayes' theorem finds many uses in the probability theory and statistics. There's a micro chance that you have never heard about this theorem in your life. Turns out that this theorem has found its way into the world of machine learning, to form one of the highly decorated algorithms. In this article, we will learn all about the Naive Bayes Algorithm, along with its variations for different purposes in machine learning. As you might have guessed, this requires us to view things from a probabilistic point of view.

Canonical correlation analysis is a family of multivariate statistical methods for the analysis of paired sets of variables. Since its proposition, canonical correlation analysis has for instance been extended to extract relations between two sets of variables when the sample size is insufficient in relation to the data dimensionality, when the relations have been considered to be non-linear, and when the dimensionality is too large for human interpretation. This tutorial explains the theory of canonical correlation analysis including its regularised, kernel, and sparse variants. Additionally, the deep and Bayesian CCA extensions are briefly reviewed. Together with the numerical examples, this overview provides a coherent compendium on the applicability of the variants of canonical correlation analysis. By bringing together techniques for solving the optimisation problems, evaluating the statistical significance and generalisability of the canonical correlation model, and interpreting the relations, we hope that this article can serve as a hands-on tool for applying canonical correlation methods in data analysis.

Accurate diagnosis of tool wear in metal turning process remains an open challenge for both scientists and industrial practitioners because of inhomogeneities in workpiece material, nonstationary machining settings to suit production requirements, and nonlinear relations between measured variables and tool wear. Common methodologies for tool condition monitoring still rely on batch approaches which cannot cope with a fast sampling rate of metal cutting process. Furthermore they require a retraining process to be completed from scratch when dealing with a new set of machining parameters. This paper presents an online tool condition monitoring approach based on Parsimonious Ensemble+, pENsemble+. The unique feature of pENsemble+ lies in its highly flexible principle where both ensemble structure and base-classifier structure can automatically grow and shrink on the fly based on the characteristics of data streams. Moreover, the online feature selection scenario is integrated to actively sample relevant input attributes. The paper presents advancement of a newly developed ensemble learning algorithm, pENsemble+, where online active learning scenario is incorporated to reduce operator labelling effort. The ensemble merging scenario is proposed which allows reduction of ensemble complexity while retaining its diversity. Experimental studies utilising real-world manufacturing data streams and comparisons with well known algorithms were carried out. Furthermore, the efficacy of pENsemble was examined using benchmark concept drift data streams. It has been found that pENsemble+ incurs low structural complexity and results in a significant reduction of operator labelling effort.

Duan 1,2 1 Center for Quantum Information, IIIS, Tsinghua University, Beijing 100084, PR China 2 Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA A central task in the field of quantum computing is to find applications where quantum computer could provide exponential speedup over any classical computer [1-3]. Machine learning represents an important field with broad applications where quantum computer may offer significant speedup [4-8]. Several quantum algorithms for discriminative machine learning [9] have been found based on efficient solving of linear algebraic problems [10-15], with potential exponential speedup in runtime under the assumption of effective input from a quantum random access memory [16]. In machine learning, generative models represent another large class [9] which is widely used for both supervised and unsupervised learning [17, 18]. Here, we propose an efficient quantum algorithm for machine learning based on a quantum generative model. We prove that our proposed model is exponentially more powerful to represent probability distributions compared with classical generative models and has exponential speedup in training and inference at least for some instances under a reasonable assumption in computational complexity theory. Our result opens a new direction for quantum machine learning and offers a remarkable example in which a quantum algorithm shows exponential improvement over any classical algorithm in an important application field. Machine learning and artificial intelligence represent a very important application area which could be revolutionized by quantum computers with clever algorithms that offer exponential speedup [4, 5]. The candidate algorithms with potential exponential speedup so far rely on efficient quantum solution of linear system of equations or linear algebraic problems [12-15]. Those algorithms require quantum random access memory (QRAM) as a critical component in addition to a quantum computer. In a QRAM, the number of required quantum routers scales up exponentially with the number of qubits in those algorithms [16, 19]. This exponential overhead in resource requirement poses a significant challenge for its experimental implementation and is a caveat for fair comparison with corresponding classical algorithms [5, 20]. In this paper, we propose a quantum algorithm with potential exponential speedup for machine learning basedFigure 1: Classical and quantum generative models. A factor graph is a bipartite graph where one group of the vertices represent variables (denoted by circles) and the other group of vertices represent positive functions (denoted by squares) acting on connected variables. The corresponding probability distribution is given by the product of all these functions. Each variable connects to at most a constant number of functions which introduce correlations in the probability distribution.b,

Mechanistic models of single-neuron dynamics have been extensively studied in computational neuroscience. However, identifying which models can quantitatively reproduce empirically measured data has been challenging. We propose to overcome this limitation by using likelihood-free inference approaches (also known as Approximate Bayesian Computation, ABC) to perform full Bayesian inference on single-neuron models. Our approach builds on recent advances in ABC by learning a neural network which maps features of the observed data to the posterior distribution over parameters. We learn a Bayesian mixture-density network approximating the posterior over multiple rounds of adaptively chosen simulations. Furthermore, we propose an efficient approach for handling missing features and parameter settings for which the simulator fails, as well as a strategy for automatically learning relevant features using recurrent neural networks. On synthetic data, our approach efficiently estimates posterior distributions and recovers ground-truth parameters. On in-vitro recordings of membrane voltages, we recover multivariate posteriors over biophysical parameters, which yield model-predicted voltage traces that accurately match empirical data. Our approach will enable neuroscientists to perform Bayesian inference on complex neuron models without having to design model-specific algorithms, closing the gap between mechanistic and statistical approaches to single-neuron modelling.