This is a complete Free course for statistics. In this course, you will learn how to estimate parameters of a population using sample statistics, hypothesis testing and confidence intervals, t-tests and ANOVA, correlation and regression, and chi-squared test. This course is taught by industry professionals and you will learn by doing various exercises.

This is a completely free course and a good first step towards understanding the data analysis process. In this course, you will learn the entire data analysis process including posing a question, data wrangling, exploring the data, drawing conclusions, and communicating your findings. This course will also teach Python libraries NumPy, Pandas, and Matplotlib.

Distributed optimization has been widely used as one of the most efficient approaches for model training with massive samples. However, large-scale learning problems with both massive samples and high-dimensional features widely exist in the era of big data. Safe screening is a popular technique to speed up high-dimensional models by discarding the inactive features with zero coefficients. Nevertheless, existing safe screening methods are limited to the sequential setting. In this paper, we propose a new distributed dynamic safe screening (DDSS) method for sparsity regularized models and apply it on shared-memory and distributed-memory architecture respectively, which can achieve significant speedup without any loss of accuracy by simultaneously enjoying the sparsity of the model and dataset. To the best of our knowledge, this is the first work of distributed safe dynamic screening method. Theoretically, we prove that the proposed method achieves the linear convergence rate with lower overall complexity and can eliminate almost all the inactive features in a finite number of iterations almost surely. Finally, extensive experimental results on benchmark datasets confirm the superiority of our proposed method.

Evidence from observational studies has become increasingly important for supporting healthcare policy making via cost-effectiveness (CE) analyses. Similar as in comparative effectiveness studies, health economic evaluations that consider subject-level heterogeneity produce individualized treatment rules (ITRs) that are often more cost-effective than one-size-fits-all treatment. Thus, it is of great interest to develop statistical tools for learning such a cost-effective ITR (CE-ITR) under the causal inference framework that allows proper handling of potential confounding and can be applied to both trials and observational studies. In this paper, we use the concept of net-monetary-benefit (NMB) to assess the trade-off between health benefits and related costs. We estimate CE-ITR as a function of patients' characteristics that, when implemented, optimizes the allocation of limited healthcare resources by maximizing health gains while minimizing treatment-related costs. We employ the conditional random forest approach and identify the optimal CE-ITR using NMB-based classification algorithms, where two partitioned estimators are proposed for the subject-specific weights to effectively incorporate information from censored individuals. We conduct simulation studies to evaluate the performance of our proposals. We apply our top-performing algorithm to the NIH-funded Systolic Blood Pressure Intervention Trial (SPRINT) to illustrate the CE gains of assigning customized intensive blood pressure therapy.

In this paper, by introducing Generalized Bernstein condition, we propose the first $\mathcal{O}\big(\frac{\sqrt{p}}{n\epsilon}\big)$ high probability excess population risk bound for differentially private algorithms under the assumptions $G$-Lipschitz, $L$-smooth, and Polyak-{\L}ojasiewicz condition, based on gradient perturbation method. If we replace the properties $G$-Lipschitz and $L$-smooth by $\alpha$-H{\"o}lder smoothness (which can be used in non-smooth setting), the high probability bound comes to $\mathcal{O}\big(n^{-\frac{\alpha}{1+2\alpha}}\big)$ w.r.t $n$, which cannot achieve $\mathcal{O}\left(1/n\right)$ when $\alpha\in(0,1]$. To solve this problem, we propose a variant of gradient perturbation method, \textbf{max$\{1,g\}$-Normalized Gradient Perturbation} (m-NGP). We further show that by normalization, the high probability excess population risk bound under assumptions $\alpha$-H{\"o}lder smooth and Polyak-{\L}ojasiewicz condition can achieve $\mathcal{O}\big(\frac{\sqrt{p}}{n\epsilon}\big)$, which is the first $\mathcal{O}\left(1/n\right)$ high probability excess population risk bound w.r.t $n$ for differentially private algorithms under non-smooth conditions. Moreover, we evaluate the performance of the new proposed algorithm m-NGP, the experimental results show that m-NGP improves the performance of the differentially private model over real datasets. It demonstrates that m-NGP improves the utility bound and the accuracy of the DP model on real datasets simultaneously.

Conventional methods for query autocompletion aim to predict which completed query a user will select from a list. A shortcoming of this approach is that users often do not know which query will provide the best retrieval performance on the current information retrieval system, meaning that any query autocompletion methods trained to mimic user behavior can lead to suboptimal query suggestions. To overcome this limitation, we propose a new approach that explicitly optimizes the query suggestions for downstream retrieval performance. We formulate this as a problem of ranking a set of rankings, where each query suggestion is represented by the downstream item ranking it produces. We then present a learning method that ranks query suggestions by the quality of their item rankings. The algorithm is based on a counterfactual learning approach that is able to leverage feedback on the items (e.g., clicks, purchases) to evaluate query suggestions through an unbiased estimator, thus avoiding the assumption that users write or select optimal queries. We establish theoretical support for the proposed approach and provide learning-theoretic guarantees. We also present empirical results on publicly available datasets, and demonstrate real-world applicability using data from an online shopping store.

Gaussian Processes (GPs) are a versatile and popular method in Bayesian Machine Learning. A common modification are Sparse Variational Gaussian Processes (SVGPs) which are well suited to deal with large datasets. While GPs allow to elegantly deal with Gaussian-distributed target variables in closed form, their applicability can be extended to non-Gaussian data as well. These extensions are usually impossible to treat in closed form and hence require approximate solutions. This paper proposes to approximate the inverse-link function, which is necessary when working with non-Gaussian likelihoods, by a piece-wise constant function. It will be shown that this yields a closed form solution for the corresponding SVGP lower bound. In addition, it is demonstrated how the piece-wise constant function itself can be optimized, resulting in an inverse-link function that can be learnt from the data at hand.

In this note, we provide an elementary analysis of the prediction error of ridge regression with random design. The proof is short and self-contained. In particular, it bypasses the use of Rudelson's deviation inequality for covariance matrices, through a combination of exchangeability arguments, matrix perturbation and operator convexity.

More and more businesses are adopting machine learning, from predictive analysis to improving the overall workflow of the organization. Machine Learning has now become a critical element of business functionality in the past few years. Businesses are curious about how implementing technology can benefit them, whilst Machine Learning professionals are eager to learn how far Machine Learning can take us. In order for this to be successful, the operations behind it need to become proficient in understanding the concept of Machine Learning, being able to analyse the data, tweaking algorithms, solving problems, and more. That seems like a lot of work, that's umbrellaed under one topic.

Two types of states are widely used in quantum mechanics, namely (deterministic-coefficient) pure states and statistical mixtures. A density operator can be associated with each of them. We here address a third type of states, that we previously introduced in a more restricted framework. These states generalize pure ones by replacing each of their deterministic ket coefficients by a random variable. We therefore call them Random-Coefficient Pure States, or RCPS. We analyze their properties and their relationships with both types of usual states. We show that RCPS contain much richer information than the density operator and mean of observables that we associate with them. This occurs because the latter operator only exploits the second-order statistics of the random state coefficients, whereas their higher-order statistics contain additional information. That information can be accessed in practice with the multiple-preparation procedure that we propose for RCPS, by using second-order and higher-order statistics of associated random probabilities of measurement outcomes. Exploiting these higher-order statistics opens the way to a very general approach for performing advanced quantum information processing tasks. We illustrate the relevance of this approach with a generic example, dealing with the estimation of parameters of a quantum process and thus related to quantum process tomography. This parameter estimation is performed in the non-blind (i.e. supervised) or blind (i.e. unsupervised) mode. We show that this problem cannot be solved by using only the density operator \rho of an RCPS and the associated mean value Tr(\rho A) of the operator A that corresponds to the considered physical quantity. We succeed in solving this problem by exploiting a fourth-order statistical parameter of state coefficients, in addition to second-order statistics. Numerical tests validate this result.