We describe a formal approach based on graphical causal models to identify the "root causes" of the change in the probability distribution of variables. After factorizing the joint distribution into conditional distributions of each variable, given its parents (the "causal mechanisms"), we attribute the change to changes of these causal mechanisms. This attribution analysis accounts for the fact that mechanisms often change independently and sometimes only some of them change. Through simulations, we study the performance of our distribution change attribution method. We then present a real-world case study identifying the drivers of the difference in the income distribution between men and women.

In this section we will learn - What does Machine Learning mean. What are the meanings or different terms associated with machine learning? You will see some examples so that you understand what machine learning actually is. It also contains steps involved in building a machine learning model, not just linear models, any machine learning model.

First, let us start by the usual Stochastic Oscillator before proceeding with the Stochastic Smoothing Oscillator. An overbought level is an area where the market is perceived to be extremely bullish and is bound to consolidate. An oversold level is an area where market is perceived to be extremely bearish and is bound to bounce. Hence, the Stochastic Oscillator is a contrarian indicator that seeks to signal reactions of extreme movements.

Missing time-series data is a prevalent practical problem. Imputation methods in time-series data often are applied to the full panel data with the purpose of training a model for a downstream out-of-sample task. For example, in finance, imputation of missing returns may be applied prior to training a portfolio optimization model. Unfortunately, this practice may result in a look-ahead-bias in the future performance on the downstream task. There is an inherent trade-off between the look-ahead-bias of using the full data set for imputation and the larger variance in the imputation from using only the training data. By connecting layers of information revealed in time, we propose a Bayesian posterior consensus distribution which optimally controls the variance and look-ahead-bias trade-off in the imputation. We demonstrate the benefit of our methodology both in synthetic and real financial data.

Stein variational gradient descent (SVGD) refers to a class of methods for Bayesian inference based on interacting particle systems. In this paper, we consider the originally proposed deterministic dynamics as well as a stochastic variant, each of which represent one of the two main paradigms in Bayesian computational statistics: variational inference and Markov chain Monte Carlo. As it turns out, these are tightly linked through a correspondence between gradient flow structures and large-deviation principles rooted in statistical physics. To expose this relationship, we develop the cotangent space construction for the Stein geometry, prove its basic properties, and determine the large-deviation functional governing the many-particle limit for the empirical measure. Moreover, we identify the Stein-Fisher information (or kernelised Stein discrepancy) as its leading order contribution in the long-time and many-particle regime in the sense of $\Gamma$-convergence, shedding some light on the finite-particle properties of SVGD. Finally, we establish a comparison principle between the Stein-Fisher information and RKHS-norms that might be of independent interest.

To balance exploration and exploitation, multi-armed bandit algorithms need to conduct inference on the true mean reward of each arm in every time step using the data collected so far. However, the history of arms and rewards observed up to that time step is adaptively collected and there are known challenges in conducting inference with non-iid data. In particular, sample averages, which play a prominent role in traditional upper confidence bound algorithms and traditional Thompson sampling algorithms, are neither unbiased nor asymptotically normal. We propose a variant of a Thompson sampling based algorithm that leverages recent advances in the causal inference literature and adaptively re-weighs the terms of a doubly robust estimator on the true mean reward of each arm -- hence its name doubly-adaptive Thompson sampling. The regret of the proposed algorithm matches the optimal (minimax) regret rate and its empirical evaluation in a semi-synthetic experiment based on data from a randomized control trial of a web service is performed: we see that the proposed doubly-adaptive Thompson sampling has superior empirical performance to existing baselines in terms of cumulative regret and statistical power in identifying the best arm. Further, we extend this approach to contextual bandits, where there are more sources of bias present apart from the adaptive data collection -- such as the mismatch between the true data generating process and the reward model assumptions or the unequal representations of certain regions of the context space in initial stages of learning -- and propose the linear contextual doubly-adaptive Thompson sampling and the non-parametric contextual doubly-adaptive Thompson sampling extensions of our approach.

Analyzing multi-source data, which are multiple views of data on the same subjects, has become increasingly common in molecular biomedical research. Recent methods have sought to uncover underlying structure and relationships within and/or between the data sources, and other methods have sought to build a predictive model for an outcome using all sources. However, existing methods that do both are presently limited because they either (1) only consider data structure shared by all datasets while ignoring structures unique to each source, or (2) they extract underlying structures first without consideration to the outcome. We propose a method called supervised joint and individual variation explained (sJIVE) that can simultaneously (1) identify shared (joint) and source-specific (individual) underlying structure and (2) build a linear prediction model for an outcome using these structures. These two components are weighted to compromise between explaining variation in the multi-source data and in the outcome. Simulations show sJIVE to outperform existing methods when large amounts of noise are present in the multi-source data. An application to data from the COPDGene study reveals gene expression and proteomic patterns that are predictive of lung function. Functions to perform sJIVE are included in the R.JIVE package, available online at http://github.com/lockEF/r.jive .

Computationally efficient contextual bandits are often based on estimating a predictive model of rewards given contexts and arms using past data. However, when the reward model is not well-specified, the bandit algorithm may incur unexpected regret, so recent work has focused on algorithms that are robust to misspecification. We propose a simple family of contextual bandit algorithms that adapt to misspecification error by reverting to a good safe policy when there is evidence that misspecification is causing a regret increase. Our algorithm requires only an offline regression oracle to ensure regret guarantees that gracefully degrade in terms of a measure of the average misspecification level. Compared to prior work, we attain similar regret guarantees, but we do no rely on a master algorithm, and do not require more robust oracles like online or constrained regression oracles (e.g., Foster et al. (2020a); Krishnamurthy et al. (2020)). This allows us to design algorithms for more general function approximation classes.

A number of machine learning tasks entail a high degree of invariance: the data distribution does not change if we act on the data with a certain group of transformations. For instance, labels of images are invariant under translations of the images. Certain neural network architectures -- for instance, convolutional networks -- are believed to owe their success to the fact that they exploit such invariance properties. With the objective of quantifying the gain achieved by invariant architectures, we introduce two classes of models: invariant random features and invariant kernel methods. The latter includes, as a special case, the neural tangent kernel for convolutional networks with global average pooling. We consider uniform covariates distributions on the sphere and hypercube and a general invariant target function. We characterize the test error of invariant methods in a high-dimensional regime in which the sample size and number of hidden units scale as polynomials in the dimension, for a class of groups that we call `degeneracy $\alpha$', with $\alpha \leq 1$. We show that exploiting invariance in the architecture saves a $d^\alpha$ factor ($d$ stands for the dimension) in sample size and number of hidden units to achieve the same test error as for unstructured architectures. Finally, we show that output symmetrization of an unstructured kernel estimator does not give a significant statistical improvement; on the other hand, data augmentation with an unstructured kernel estimator is equivalent to an invariant kernel estimator and enjoys the same improvement in statistical efficiency.

We study the problem of machine unlearning and identify a notion of algorithmic stability, Total Variation (TV) stability, which we argue, is suitable for the goal of exact unlearning. For convex risk minimization problems, we design TV-stable algorithms based on noisy Stochastic Gradient Descent (SGD). Our key contribution is the design of corresponding efficient unlearning algorithms, which are based on constructing a (maximal) coupling of Markov chains for the noisy SGD procedure. To understand the trade-offs between accuracy and unlearning efficiency, we give upper and lower bounds on excess empirical and populations risk of TV stable algorithms for convex risk minimization. Our techniques generalize to arbitrary non-convex functions, and our algorithms are differentially private as well.