Motivated by A/B/n testing applications, we consider a finite set of distributions (called \emph{arms}), one of which is treated as a \emph{control}. We assume that the population is stratified into homogeneous subpopulations. At every time step, a subpopulation is sampled and an arm is chosen: the resulting observation is an independent draw from the arm conditioned on the subpopulation. The quality of each arm is assessed through a weighted combination of its subpopulation means. We propose a strategy for sequentially choosing one arm per time step so as to discover as fast as possible which arms, if any, have higher weighted expectation than the control. This strategy is shown to be asymptotically optimal in the following sense: if $\tau_\delta$ is the first time when the strategy ensures that it is able to output the correct answer with probability at least $1-\delta$, then $\mathbb{E}[\tau_\delta]$ grows linearly with $\log(1/\delta)$ at the exact optimal rate. This rate is identified in the paper in three different settings: (1) when the experimenter does not observe the subpopulation information, (2) when the subpopulation of each sample is observed but not chosen, and (3) when the experimenter can select the subpopulation from which each response is sampled. We illustrate the efficiency of the proposed strategy with numerical simulations on synthetic and real data collected from an A/B/n experiment.
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Efficient contact tracing and isolation is an effective strategy to control epidemics. It was used effectively during the Ebola epidemic and successfully implemented in several parts of the world during the ongoing COVID-19 pandemic. An important consideration while implementing manual contact tracing is the number of contact tracers available -- the number of such individuals is limited for socioeconomic reasons. In this paper, we present a Markov Decision Process (MDP) framework to formulate the problem of efficient contact tracing that reduces the size of the outbreak while using a limited number of contact tracers. We formulate each step of the MDP as a combinatorial problem, MinExposed. We demonstrate that MinExposed is NP-Hard, so we develop an LP-based approximation algorithm. Though this algorithm directly solves MinExposed, it is often impractical in the real world due to information constraints. To this end, we develop a greedy approach based on insights from the analysis of the previous algorithm, which we show is more interpretable. A key feature of the greedy algorithm is that it does not need complete information of the underlying social contact network. This makes the heuristic implementable in practice and is an important consideration. Finally, we carry out experiments on simulations of the MDP run on real-world networks, and show how the algorithms can help in bending the epidemic curve while limiting the number of isolated individuals. Our experimental results demonstrate that the greedy algorithm and its variants are especially effective, robust, and practical in a variety of realistic scenarios, such as when the contact graph and specific transmission probabilities are not known. All code can be found in our GitHub repository: //github.com/gzli929/ContactTracing.
We investigate multiple testing and variable selection using the Least Angle Regression (LARS) algorithm in high dimensions under the assumption of Gaussian noise. LARS is known to produce a piecewise affine solution path with change points referred to as the knots of the LARS path. The key to our results is an expression in closed form of the exact joint law of a $K$-tuple of knots conditional on the variables selected by LARS, namely the so-called post-selection joint law of the LARS knots. Numerical experiments demonstrate the perfect fit of our findings. This paper makes three main contributions. First, we build testing procedures on variables entering the model along the LARS path in the general design case when the noise level can be unknown. These testing procedures are referred to as the Generalized $t$-Spacing tests (GtSt) and we prove that they have an exact non-asymptotic level (i.e., the Type I error is exactly controlled). This extends work of (Taylor et al., 2014) where the spacing test works for consecutive knots and known variance. Second, we introduce a new exact multiple false negatives test after model selection in the general design case when the noise level may be unknown. We prove that this testing procedure has exact non-asymptotic level for general design and unknown noise level. Third, we give an exact control of the false discovery rate under orthogonal design assumption. Monte Carlo simulations and a real data experiment are provided to illustrate our results in this case. Of independent interest, we introduce an equivalent formulation of the LARS algorithm based on a recursive function.
Given many popular functional forms for the Lorenz curve do not have a closed-form expression for the Gini index and no study has utilized the observed Gini index to estimate parameter(s) associated with the corresponding parametric functional form, a simple method for estimating the Lorenz curve is introduced. It utilizes 3 indicators, namely, the Gini index and the income shares of the bottom and the top in order to calculate the values of parameters associated with the specified functional form which has a closed-form expression for the Gini index. No error minimization technique is required in order to estimate the Lorenz curve. The data on the Gini index and the income shares of 4 countries that have different level of income inequality, economic, sociological, and regional backgrounds from the United Nations University-World Income Inequality Database are used to illustrate how the simple method works. The overall results indicate that the estimated Lorenz curves fit the actual observations practically well. This simple method could be useful in the situation where the availability of data on income distribution is low. However, if more data on income distribution are available, this study shows that the specified functional form could be used to directly estimate the Lorenz curve. Moreover, the estimated values of the Gini index calculated based on the specified functional form are virtually identical to their actual observations.
We propose two-stage and sequential procedures to construct prescribed proportional closeness confidence intervals for the unknown parameter N of a binomial distribution with unknown parameter p, when we reinforce data with an independent sample of a negative-binomial experiment having the same p
The goal of this paper is to investigate a control theoretic analysis of linear stochastic iterative algorithm and temporal difference (TD) learning. TD-learning is a linear stochastic iterative algorithm to estimate the value function of a given policy for a Markov decision process, which is one of the most popular and fundamental reinforcement learning algorithms. While there has been a series of successful works in theoretical analysis of TD-learning, it was not until recently that researchers found some guarantees on its statistical efficiency. In this paper, we propose a control theoretic finite-time analysis TD-learning, which exploits standard notions in linear system control communities. Therefore, the proposed work provides additional insights on TD-learning and reinforcement learning with simple concepts and analysis tools in control theory.
Variance estimation is important for statistical inference. It becomes non-trivial when observations are masked by serial dependence structures and time-varying mean structures. Existing methods either ignore or sub-optimally handle these nuisance structures. This paper develops a general framework for the estimation of the long-run variance for time series with non-constant means. The building blocks are difference statistics. The proposed class of estimators is general enough to cover many existing estimators. Necessary and sufficient conditions for consistency are investigated. The first asymptotically optimal estimator is derived. Our proposed estimator is theoretically proven to be invariant to arbitrary mean structures, which may include trends and a possibly divergent number of discontinuities.
A/B testing, or online experiment is a standard business strategy to compare a new product with an old one in pharmaceutical, technological, and traditional industries. Major challenges arise in online experiments of two-sided marketplace platforms (e.g., Uber) where there is only one unit that receives a sequence of treatments over time. In those experiments, the treatment at a given time impacts current outcome as well as future outcomes. The aim of this paper is to introduce a reinforcement learning framework for carrying A/B testing in these experiments, while characterizing the long-term treatment effects. Our proposed testing procedure allows for sequential monitoring and online updating. It is generally applicable to a variety of treatment designs in different industries. In addition, we systematically investigate the theoretical properties (e.g., size and power) of our testing procedure. Finally, we apply our framework to both simulated data and a real-world data example obtained from a technological company to illustrate its advantage over the current practice. A Python implementation of our test is available at //github.com/callmespring/CausalRL.
We determine the exact value of the optimal symmetric rate point $(r, r)$ in the Dueck zero-error capacity region of the binary adder channel with complete feedback. We proved that the average zero-error capacity $r = h(1/2-\delta) \approx 0.78974$, where $h(\cdot)$ is the binary entropy function and $\delta = 1/(2\log_2(2+\sqrt3))$. Our motivation is a problem in quantitative group testing. Given a set of $n$ elements two of which are defective, the quantitative group testing problem asks for the identification of these two defectives through a series of tests. Each test gives the number of defectives contained in the tested subset, and the outcomes of previous tests are assumed known at the time of designing the current test. We establish that the minimum number of tests is asymptotic to $(\log_2 n) / r$ as $n \to \infty$.
We propose a new method of estimation in topic models, that is not a variation on the existing simplex finding algorithms, and that estimates the number of topics K from the observed data. We derive new finite sample minimax lower bounds for the estimation of A, as well as new upper bounds for our proposed estimator. We describe the scenarios where our estimator is minimax adaptive. Our finite sample analysis is valid for any number of documents (n), individual document length (N_i), dictionary size (p) and number of topics (K), and both p and K are allowed to increase with n, a situation not handled well by previous analyses. We complement our theoretical results with a detailed simulation study. We illustrate that the new algorithm is faster and more accurate than the current ones, although we start out with a computational and theoretical disadvantage of not knowing the correct number of topics K, while we provide the competing methods with the correct value in our simulations.
This is full length article (draft version) where problem number of topics in Topic Modeling is discussed. We proposed idea that Renyi and Tsallis entropy can be used for identification of optimal number in large textual collections. We also report results of numerical experiments of Semantic stability for 4 topic models, which shows that semantic stability play very important role in problem topic number. The calculation of Renyi and Tsallis entropy based on thermodynamics approach.
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