In this work, we study non-parametric estimation of joint probabilities of a given set of discrete and continuous random variables from their (empirically estimated) 2D marginals, under the assumption that the joint probability could be decomposed and approximated by a mixture of product densities/mass functions. The problem of estimating the joint probability density function (PDF) using semi-parametric techniques such as Gaussian Mixture Models (GMMs) is widely studied. However such techniques yield poor results when the underlying densities are mixtures of various other families of distributions such as Laplacian or generalized Gaussian, uniform, Cauchy, etc. Further, GMMs are not the best choice to estimate joint distributions which are hybrid in nature, i.e., some random variables are discrete while others are continuous. We present a novel approach for estimating the PDF using ideas from dictionary representations in signal processing coupled with low rank tensor decompositions. To the best our knowledge, this is the first work on estimating joint PDFs employing dictionaries alongside tensor decompositions. We create a dictionary of various families of distributions by inspecting the data, and use it to approximate each decomposed factor of the product in the mixture. Our approach can naturally handle hybrid $N$-dimensional distributions. We test our approach on a variety of synthetic and real datasets to demonstrate its effectiveness in terms of better classification rates and lower error rates, when compared to state of the art estimators.

We consider stochastic approximation for the least squares regression problem in the non-strongly convex setting. We present the first practical algorithm that achieves the optimal prediction error rates in terms of dependence on the noise of the problem, as $O(d/t)$ while accelerating the forgetting of the initial conditions to $O(d/t^2)$. Our new algorithm is based on a simple modification of the accelerated gradient descent. We provide convergence results for both the averaged and the last iterate of the algorithm. In order to describe the tightness of these new bounds, we present a matching lower bound in the noiseless setting and thus show the optimality of our algorithm.

We consider the problem of discovering causal relations from independence constraints selection bias in addition to confounding is present. While the seminal FCI algorithm is sound and complete in this setup, no criterion for the causal interpretation of its output under selection bias is presently known. We focus instead on local patterns of independence relations, where we find no sound method for only three variable that can include background knowledge. Y-Structure patterns (Mani et al., 2006; Mooij and Cremers, 2015) are shown to be sound in predicting causal relations from data under selection bias, where cycles may be present. We introduce a finite-sample scoring rule for Y-Structures that is shown to successfully predict causal relations in simulation experiments that include selection mechanisms. On real-world microarray data, we show that a Y-Structure variant performs well across different datasets, potentially circumventing spurious correlations due to selection bias.

You have correctly summarized the 0-1 loss function as effectively looking at accuracy. Your 1's become indicators for misclassified items, regardless of how they were misclassified. Since you have three 1's out of 10 items, your classification accuracy is 70%. If you change the weighting on the loss function, this interpretation doesn't apply anymore. For example, in disease classification, it might be more costly to miss a positive case of disease (false negative) than to falsely diagnose disease (false positive).

While attempting to solve a ML problem, we do a train_test split. If this split is done randomly than it might be possible that some dataset might be completely present in test set and absent from training set or vice versa. This reduces the accuracy of model. So Cross Validation comes into picture. Cross-validation is a step in the process of building a machine learning model which helps us ensure that our models fit the data accurately and also ensures that we do not overfit.Cross-validation is dividing training data into a few parts.

We present simple methods for out-of-distribution detection using a trained generative model. These techniques, based on classical statistical tests, are model-agnostic in the sense that they can be applied to any differentiable generative model. The idea is to combine a classical parametric test (Rao's score test) with the recently introduced typicality test. These two test statistics are both theoretically well-founded and exploit different sources of information based on the likelihood for the typicality test and its gradient for the score test. We show that combining them using Fisher's method overall leads to a more accurate out-of-distribution test. We also discuss the benefits of casting out-of-distribution detection as a statistical testing problem, noting in particular that false positive rate control can be valuable for practical out-of-distribution detection. Despite their simplicity and generality, these methods can be competitive with model-specific out-of-distribution detection algorithms without any assumptions on the out-distribution.

This article was written, edited and designed on laptop computers. Such foldable, transportable devices would have astounded computer scientists just a few decades ago, and seemed like sheer magic before that. The machines contain billions of tiny computing elements, running millions of lines of software instructions, collectively written by countless people across the globe. You click or tap or type or speak, and the result seamlessly appears on the screen. Computers were once so large they filled rooms. Now they're everywhere and invisible, embedded in watches, car engines, cameras, televisions and toys. They manage electrical grids, analyze scientific data and predict the weather. The modern world would be impossible without them. Scientists aim to make computers faster and programs more intelligent, while deploying technology in an ethical manner. Their efforts build on more than a century of innovation. In 1833, English mathematician Charles Babbage conceived a programmable machine that presaged today's computing architecture, featuring a "store" for holding numbers, a "mill" for operating on them, an instruction reader and a printer. This Analytical Engine also had logical functions like branching (if X, then Y).

In most binary classification problems we use the ROC Curve and ROC AUC score as measurements of how well the model separates the predictions of the two different classes. I explain this mechanism in another article, but the intuition is easy: if the model gives lower probability scores for the negative class, and higher scores for the positive class, we can say that this is a good model. Now here's the catch: we can also use the KS-2samp test to do that! The KS statistic for two samples is simply the highest distance between their two CDFs, so if we measure the distance between the positive and negative class distributions, we can have another metric to evaluate classifiers. There is a benefit for this approach: the ROC AUC score goes from 0.5 to 1.0, while KS statistics range from 0.0 to 1.0.

The simple image captured in camera consists colors of different wavelengths (visible spectrum) which can be represented with combination of three colors — Red,Green and Blue (RGB). Thus digital…

This paper discusses techniques for merging data files where no key field exists between the files. The paper will illustrate an approach to resolve two issues that are common to most fuzzy matching techniques: 1) how to weight proxy identifier fields, and 2) how to measure the Type One and Type Two errors of the merge estimation algorithm. A common requirement in analytics is to merge records in two or more large sets of information (i.e., thousands if not millions of records) where no exact key exists to match records between the information sets. When no exact key between the two data sets exists, a common merging solution is to use "fuzzy" matching. "Fuzzy" matching uses proxy keys as substitute keys to match records between the two data files.