I was reading the Naive Bayes article on Wikipedia and I read that, In Naive Bayes, the naive assumption that Naive Bayes make is "each feature is conditionally independent of every other feature, given the category Ck". So, I was wondering if'Conditional Independence' means there should be no multicollinearity among features?

We present a deep generative model, named Monge-Amp\`ere flow, which builds on continuous-time gradient flow arising from the Monge-Amp\`ere equation in optimal transport theory. The generative map from the latent space to the data space follows a dynamical system, where a learnable potential function guides a compressible fluid to flow towards the target density distribution. Training of the model amounts to solving an optimal control problem. The Monge-Amp\`ere flow has tractable likelihoods and supports efficient sampling and inference. One can easily impose symmetry constraints in the generative model by designing suitable scalar potential functions. We apply the approach to unsupervised density estimation of the MNIST dataset and variational calculation of the two-dimensional Ising model at the critical point. This approach brings insights and techniques from Monge-Amp\`ere equation, optimal transport, and fluid dynamics into reversible flow-based generative models.

Abstract--Principal component analysis (PCA) is a popular method for projecting data onto uncorrelated components in lower dimension, although the optimal number of components is not specified. Likewise, multiple signal classification (MUSIC) algorithm is a popular PCA-based method for estimating directions of arrival (DOAs) of sinusoidal sources, yet it requires the number of sources to be known a priori. The accurate estimation of the number of sources is hence a crucial issue for performance of these algorithms. In this paper, we will show that both PCA and MUSIC actually return the exact joint maximum-a-posteriori (MAP) estimate for uncorrelated steering vectors, although they can only compute this MAP estimate approximately in correlated case. We then use Bayesian method to, for the first time, compute the MAP estimate for the number of sources in PCA and MUSIC algorithms. Intuitively, this MAP estimate corresponds to the highest probability that signal-plus- noise's variance still dominates projected noise's variance on signal subspace. In simulations of overlapping multi-tone sources for linear sensor array, our exact MAP estimate is far superior to the asymptotic Akaike information criterion (AIC), which is a popular method for estimating the number of components in PCA and MUSIC algorithms. In many systems of array signal processing, e.g. in radar, sonar and antenna systems, linear sensor array is the most basic and universal mathematical model. Because far distant sources with different directions of arrival (DOAs) will oscillate the steering sensor array with different angular frequencies, the array's output data is then a superposition of sinusoidal signals [1]. Hence, a common problem of array systems is to detect the number of sources, as well as their tone frequencies and DOAs, from noisy sinusoidal signals. In literature, most papers only consider the case of single-tone or narrowband sources (i.e. When the number of sources is small, the DOA's line spectra are sparse and can be estimated effectively via sparse techniques like atomic norm (also known as total variation norm) [1], [2], LASSO [4], [5] and Bayesian compressed sensing [6], [7].

Question: Why do you square the error in a regression machine learning task? Ans: "Why, of course, it turns out all the errors (residuals) into positive quantities!" Question: "OK, why not use a simpler absolute value function x to make all the errors positive?" Ans: "Aha, you are trying to trick me. Absolute value function is not differentiable everywhere!" Question: "That should not matter much for numerical algorithms. LASSO regression uses a term with absolute value and it can be handled.

This article follows my previous one on Bayesian probability & probabilistic programming that I published few months ago on LinkedIn. And for the purpose of this article, I am going to assume that most this article readers have some idea what a Neural Network or Artificial Neural Network is. Neural Network is a non-linear function approximator. We can think of it as a parameterized function where the parameters are the weights & biases of Neural Network through which we will be typically passing our data (inputs), that will be converted to a probability between 0 and 1, to some kind of non-linearity such as a sigmoid function and help make our predictions or estimations. These non-linear functions can be composed together hence Deep Learning Neural Network with multiple layers of this function compositions.

Multiple generalized additive models (GAMs) are a type of distributional regression wherein parameters of probability distributions depend on predictors through smooth functions, with selection of the degree of smoothness via $L_2$ regularization. Multiple GAMs allow finer statistical inference by incorporating explanatory information in any or all of the parameters of the distribution. Owing to their nonlinearity, flexibility and interpretability, GAMs are widely used, but reliable and fast methods for automatic smoothing in large datasets are still lacking, despite recent advances. We develop a general methodology for automatically learning the optimal degree of $L_2$ regularization for multiple GAMs using an empirical Bayes approach. The smooth functions are penalized by different amounts, which are learned simultaneously by maximization of a marginal likelihood through an approximate expectation-maximization algorithm that involves a double Laplace approximation at the E-step, and leads to an efficient M-step. Empirical analysis shows that the resulting algorithm is numerically stable, faster than all existing methods and achieves state-of-the-art accuracy. For illustration, we apply it to an important and challenging problem in the analysis of extremal data.

We present a Bayesian method for feature selection in the presence of grouping information with sparsity on the between- and within group level. Instead of using a stochastic algorithm for parameter inference, we employ expectation propagation, which is a deterministic and fast algorithm. Available methods for feature selection in the presence of grouping information have a number of short-comings: on one hand, lasso methods, while being fast, underestimate the regression coefficients and do not make good use of the grouping information, and on the other hand, Bayesian approaches, while accurate in parameter estimation, often rely on the stochastic and slow Gibbs sampling procedure to recover the parameters, rendering them infeasible e.g. for gene network reconstruction. Our approach of a Bayesian sparse-group framework with expectation propagation enables us to not only recover accurate parameter estimates in signal recovery problems, but also makes it possible to apply this Bayesian framework to large-scale network reconstruction problems. The presented method is generic but in terms of application we focus on gene regulatory networks. We show on simulated and experimental data that the method constitutes a good choice for network reconstruction regarding the number of correctly selected features, prediction on new data and reasonable computing time.

The era of big data provides researchers with convenient access to copious data. However, people often have little knowledge about it. The increasing prevalence of big data is challenging the traditional methods of learning causality because they are developed for the cases with limited amount of data and solid prior causal knowledge. This survey aims to close the gap between big data and learning causality with a comprehensive and structured review of traditional and frontier methods and a discussion about some open problems of learning causality. We begin with preliminaries of learning causality. Then we categorize and revisit methods of learning causality for the typical problems and data types. After that, we discuss the connections between learning causality and machine learning. At the end, some open problems are presented to show the great potential of learning causality with data.

This paper addresses challenges in flexibly modeling multimodal data that lie on constrained spaces. Applications include climate or crime measurements in a geographical area, or flow-cytometry experiments, where unsuitable recordings are discarded. A simple approach to modeling such data is through the use of mixture models, with each component following an appropriate truncated distribution. Problems arise when the truncation involves complicated constraints, leading to difficulties in specifying the component distributions, and in evaluating their normalization constants. Bayesian inference over the parameters of these models results in posterior distributions that are doubly-intractable. We address this problem via an algorithm based on rejection sampling and data augmentation. We view samples from a truncated distribution as outcomes of a rejection sampling scheme, where proposals are made from a simple mixture model, and are rejected if they violate the constraints. Our scheme proceeds by imputing the rejected samples given mixture parameters, and then resampling parameters given all samples. We study two modeling approaches: mixtures of truncated components and truncated mixtures of components. In both situations, we describe exact Markov chain Monte Carlo sampling algorithms, as well as approximations that bound the number of rejected samples, achieving computational efficiency and lower variance at the cost of asymptotic bias. Overall, our methodology only requires practitioners to provide an indicator function for the set of interest. We present results on simulated data and apply our algorithm to two problems, one involving flow-cytometry data, and the other, crime recorded in the city of Chicago.

This investigation aims to study different adaptive fuzzy inference algorithms capable of real-time sequential learning and prediction of time-series data. A brief qualitative description of these algorithms namely meta-cognitive fuzzy inference system (McFIS), sequential adaptive fuzzy inference system (SAFIS) and evolving Takagi-Sugeno (ETS) model provide a comprehensive comparison of their working principle, especially their unique characteristics are discussed. These algorithms are then simulated with dataset collected at one of the academic buildings at Nanyang Technological University, Singapore. The performance are compared by means of the root mean squared error (RMSE) and non-destructive error index (NDEI) of the predicted output. Analysis shows that McFIS shows promising results either with lower RMSE and NDEI or with lower architectural complexity over ETS and SAFIS. Statistical Analysis also reveals the significance of the outcome of these algorithms.