A common statistical problem in econometrics is to estimate the impact of a treatment on a treated unit given a control sample with untreated outcomes. Here we develop a generative learning approach to this problem, learning the probability distribution of the data, which can be used for downstream tasks such as post-treatment counterfactual prediction and hypothesis testing. We use control samples to transform the data to a Gaussian and homoschedastic form and then perform Gaussian process analysis in Fourier space, evaluating the optimal Gaussian kernel via non-parametric power spectrum estimation. We combine this Gaussian prior with the data likelihood given by the pre-treatment data of the single unit, to obtain the synthetic prediction of the unit post-treatment, which minimizes the error variance of synthetic prediction. Given the generative model the minimum variance counterfactual is unique, and comes with an associated error covariance matrix. We extend this basic formalism to include correlations of primary variable with other covariates of interest. Given the probabilistic description of generative model we can compare synthetic data prediction with real data to address the question of whether the treatment had a statistically significant impact. For this purpose we develop a hypothesis testing approach and evaluate the Bayes factor. We apply the method to the well studied example of California (CA) tobacco sales tax of 1988. We also perform a placebo analysis using control states to validate our methodology. Our hypothesis testing method suggests 5.8:1 odds in favor of CA tobacco sales tax having an impact on the tobacco sales, a value that is at least three times higher than any of the 38 control states.
We propose a two-sample testing procedure based on learned deep neural network representations. To this end, we define two test statistics that perform an asymptotic location test on data samples mapped onto a hidden layer. The tests are consistent and asymptotically control the type-1 error rate. Their test statistics can be evaluated in linear time (in the sample size). Suitable data representations are obtained in a data-driven way, by solving a supervised or unsupervised transfer-learning task on an auxiliary (potentially distinct) data set. If no auxiliary data is available, we split the data into two chunks: one for learning representations and one for computing the test statistic. In experiments on audio samples, natural images and three-dimensional neuroimaging data our tests yield significant decreases in type-2 error rate (up to 35 percentage points) compared to state-of-the-art two-sample tests such as kernel-methods and classifier two-sample tests.
Determining the extent to which different cognitive modalities (understood here as the set of cognitive processes underlying the elaboration of a stimulus by the brain) rely on overlapping neural representations is a fundamental issue in cognitive neuroscience. In the last decade, the identification of shared activity patterns has been mostly framed as a supervised learning problem. For instance, a classifier is trained to discriminate categories (e.g. faces vs. houses) in modality I (e.g. perception) and tested on the same categories in modality II (e.g. imagery). This type of analysis is often referred to as cross-modal decoding. In this paper we take a different approach and instead formulate the problem of assessing shared patterns across modalities within the framework of statistical hypothesis testing. We propose both an appropriate test statistic and a scheme based on permutation testing to compute the significance of this test while making only minimal distributional assumption. We denote this test cross-modal permutation test (CMPT). We also provide empirical evidence on synthetic datasets that our approach has greater statistical power than the cross-modal decoding method while maintaining low Type I errors (rejecting a true null hypothesis). We compare both approaches on an fMRI dataset with three different cognitive modalities (perception, imagery, visual search). Finally, we show how CMPT can be combined with Searchlight analysis to explore spatial distribution of shared activity patterns.
One important partition of algorithms for controlling the false discovery rate (FDR) in multiple testing is into offline and online algorithms. The first generally achieve significantly higher power of discovery, while the latter allow making decisions sequentially as well as adaptively formulating hypotheses based on past observations. Using existing methodology, it is unclear how one could trade off the benefits of these two broad families of algorithms, all the while preserving their formal FDR guarantees. To this end, we introduce $\text{Batch}_{\text{BH}}$ and $\text{Batch}_{\text{St-BH}}$, algorithms for controlling the FDR when a possibly infinite sequence of batches of hypotheses is tested by repeated application of one of the most widely used offline algorithms, the Benjamini-Hochberg (BH) method or Storey's improvement of the BH method. We show that our algorithms interpolate between existing online and offline methodology, thus trading off the best of both worlds.
When learning behavior, training data is often generated by the learner itself; this can result in unstable training dynamics, and this problem has particularly important applications in safety-sensitive real-world control tasks such as robotics. In this work, we propose a principled and model-agnostic approach to mitigate the issue of unstable learning dynamics by maintaining a history of a reinforcement learning agent over the course of training, and reverting to the parameters of a previous agent whenever performance significantly decreases. We develop techniques for evaluating this performance through statistical hypothesis testing of continued improvement, and evaluate them on a standard suite of challenging benchmark tasks involving continuous control of simulated robots. We show improvements over state-of-the-art reinforcement learning algorithms in performance and robustness to hyperparameters, outperforming DDPG in 5 out of 6 evaluation environments and showing no decrease in performance with TD3, which is known to be relatively stable. In this way, our approach takes an important step towards increasing data efficiency and stability in training for real-world robotic applications.
We study nonzero-sum hypothesis testing games that arise in the context of adversarial classification, in both the Bayesian as well as the Neyman-Pearson frameworks. We first show that these games admit mixed strategy Nash equilibria, and then we examine some interesting concentration phenomena of these equilibria. Our main results are on the exponential rates of convergence of classification errors at equilibrium, which are analogous to the well-known Chernoff-Stein lemma and Chernoff information that describe the error exponents in the classical binary hypothesis testing problem, but with parameters derived from the adversarial model. The results are validated through numerical experiments.
The paper presents new metrics to quantify and test for (i) the equality of distributions and (ii) the independence between two high-dimensional random vectors. We show that the energy distance based on the usual Euclidean distance cannot completely characterize the homogeneity of two high-dimensional distributions in the sense that it only detects the equality of means and the traces of covariance matrices in the high-dimensional setup. We propose a new class of metrics which inherits the desirable properties of the energy distance and maximum mean discrepancy/(generalized) distance covariance and the Hilbert-Schmidt Independence Criterion in the low-dimensional setting and is capable of detecting the homogeneity of/completely characterizing independence between the low-dimensional marginal distributions in the high dimensional setup. We further propose t-tests based on the new metrics to perform high-dimensional two-sample testing/independence testing and study their asymptotic behavior under both high dimension low sample size (HDLSS) and high dimension medium sample size (HDMSS) setups. The computational complexity of the t-tests only grows linearly with the dimension and thus is scalable to very high dimensional data. We demonstrate the superior power behavior of the proposed tests for homogeneity of distributions and independence via both simulated and real datasets.
We propose a novel supervised learning method to optimize the kernel in the maximum mean discrepancy generative adversarial networks (MMD GANs), and the kernel support vector machines (SVMs). Specifically, we characterize a distributionally robust optimization problem to compute a good distribution for the random feature model of Rahimi and Recht. Due to the fact that the distributional optimization is infinite dimensional, we consider a Monte-Carlo sample average approximation (SAA) to obtain a more tractable finite dimensional optimization problem. We subsequently leverage a particle stochastic gradient descent (SGD) method to solve the derived finite dimensional optimization problem. Based on a mean-field analysis, we then prove that the empirical distribution of the interactive particles system at each iteration of the SGD follows the path of the gradient descent flow on the Wasserstein manifold. We also establish the non-asymptotic consistency of the finite sample estimator. We evaluate our kernel learning method for the hypothesis testing problem by evaluating the kernel MMD statistics, and show that our learning method indeed attains better power of the test for larger threshold values compared to an untrained kernel. Moreover, our empirical evaluation on benchmark data-sets shows the advantage of our kernel learning approach compared to alternative kernel learning methods.
The recent success of generative adversarial networks and variational learning suggests training a classifier network may work well in addressing the classical two-sample problem. Network-based tests have the computational advantage that the algorithm scales to large samples. This paper proposes a two-sample statistic which is the difference of the logit function, provided by a trained classification neural network, evaluated on the testing set split of the two datasets. Theoretically, we prove the testing power to differentiate two sub-exponential densities given that the network is sufficiently parametrized. When the two densities lie on or near to low-dimensional manifolds embedded in possibly high-dimensional space, the needed network complexity is reduced to only scale with the intrinsic dimensionality. Both the approximation and estimation error analysis are based on a new result of near-manifold integral approximation. In experiments, the proposed method demonstrates better performance than previous network-based tests using classification accuracy as the two-sample statistic, and compares favorably to certain kernel maximum mean discrepancy tests on synthetic datasets and hand-written digit datasets.
Are two sets of observations drawn from the same distribution? This problem is a two-sample test. Kernel methods lead to many appealing properties. Indeed state-of-the-art approaches use the $L^2$ distance between kernel-based distribution representatives to derive their test statistics. Here, we show that $L^p$ distances (with $p\geq 1$) between these distribution representatives give metrics on the space of distributions that are well-behaved to detect differences between distributions as they metrize the weak convergence. Moreover, for analytic kernels, we show that the $L^1$ geometry gives improved testing power for scalable computational procedures. Specifically, we derive a finite dimensional approximation of the metric given as the $\ell_1$ norm of a vector which captures differences of expectations of analytic functions evaluated at spatial locations or frequencies (i.e, features). The features can be chosen to maximize the differences of the distributions and give interpretable indications of how they differs. Using an $\ell_1$ norm gives better detection because differences between representatives are dense as we use analytic kernels (non-zero almost everywhere). The tests are consistent, while much faster than state-of-the-art quadratic-time kernel-based tests. Experiments on artificial and real-world problems demonstrate improved power/time tradeoff than the state of the art, based on $\ell_2$ norms, and in some cases, better outright power than even the most expensive quadratic-time tests.
Given two distinct datasets, an important question is if they have arisen from the the same data generating function or alternatively how their data generating functions diverge from one another. In this paper, we introduce an approach for measuring the distance between two datasets with high dimensionality using variational autoencoders. This approach is augmented by a permutation hypothesis test in order to check the hypothesis that the data generating distributions are the same within a significance level. We evaluate both the distance measurement and hypothesis testing approaches on generated and on public datasets. According to the results the proposed approach can be used for data exploration (e.g. by quantifying the discrepancy/separability between categories of images), which can be particularly useful in the early phases of the pipeline of most machine learning projects.
The combined algorithm selection and hyperparameter tuning (CASH) problem is characterized by large hierarchical hyperparameter spaces. Model-free hyperparameter tuning methods can explore such large spaces efficiently since they are highly parallelizable across multiple machines. When no prior knowledge or meta-data exists to boost their performance, these methods commonly sample random configurations following a uniform distribution. In this work, we propose a novel sampling distribution as an alternative to uniform sampling and prove theoretically that it has a better chance of finding the best configuration in a worst-case setting. In order to compare competing methods rigorously in an experimental setting, one must perform statistical hypothesis testing. We show that there is little-to-no agreement in the automated machine learning literature regarding which methods should be used. We contrast this disparity with the methods recommended by the broader statistics literature, and identify a suitable approach. We then select three popular model-free solutions to CASH and evaluate their performance, with uniform sampling as well as the proposed sampling scheme, across 67 datasets from the OpenML platform. We investigate the trade-off between exploration and exploitation across the three algorithms, and verify empirically that the proposed sampling distribution improves performance in all cases.
Whenever a social media user decides to share a story, she is typically pleased to receive likes, comments, shares, or, more generally, feedback from her followers. As a result, she may feel compelled to use the feedback she receives to (re-)estimate her followers' preferences and decides which stories to share next to receive more (positive) feedback. Under which conditions can she succeed? In this work, we first look into this problem from a theoretical perspective and then provide a set of practical algorithms to identify and characterize such behavior in social media. More specifically, we address the above problem from the viewpoint of sequential decision making and utility maximization. For a wide variety of utility functions, we first show that, to succeed, a user needs to actively trade off exploitation-- sharing stories which lead to more (positive) feedback--and exploration-- sharing stories to learn about her followers' preferences. However, exploration is not necessary if a user utilizes the feedback her followers provide to other users in addition to the feedback she receives. Then, we develop a utility estimation framework for observation data, which relies on statistical hypothesis testing to determine whether a user utilizes the feedback she receives from each of her followers to decide what to post next. Experiments on synthetic data illustrate our theoretical findings and show that our estimation framework is able to accurately recover users' underlying utility functions. Experiments on several real datasets gathered from Twitter and Reddit reveal that up to 82% (43%) of the Twitter (Reddit) users in our datasets do use the feedback they receive to decide what to post next.
In this work we use Recurrent Neural Networks and Multilayer Perceptrons to predict NYSE, NASDAQ and AMEX stock prices from historical data. We experiment with different architectures and compare data normalization techniques. Then, we leverage those findings to question the efficient-market hypothesis through a formal statistical test.
We characterize the asymptotic performance of nonparametric one- and two-sample testing. The exponential decay rate or error exponent of the type-II error probability is used as the asymptotic performance metric, and an optimal test achieves the maximum rate subject to a constant level constraint on the type-I error probability. With Sanov's theorem, we derive a sufficient condition for one-sample tests to achieve the optimal error exponent in the universal setting, i.e., for any distribution defining the alternative hypothesis. We then show that two classes of Maximum Mean Discrepancy (MMD) based tests attain the optimal type-II error exponent on $\mathbb R^d$, while the quadratic-time Kernel Stein Discrepancy (KSD) based tests achieve this optimality with an asymptotic level constraint. For general two-sample testing, however, Sanov's theorem is insufficient to obtain a similar sufficient condition. We proceed to establish an extended version of Sanov's theorem and derive an exact error exponent for the quadratic-time MMD based two-sample tests. The obtained error exponent is further shown to be optimal among all two-sample tests satisfying a given level constraint. Our work hence provides an achievability result for optimal nonparametric one- and two-sample testing in the universal setting. Application to off-line change detection and related issues are also discussed.
This is an up-to-date introduction to and overview of the Minimum Description Length (MDL) Principle, a theory of inductive inference that can be applied to general problems in statistics, machine learning and pattern recognition. While MDL was originally based on data compression ideas, this introduction can be read without any knowledge thereof. It takes into account all major developments since 2007, the last time an extensive overview was written. These include new methods for model selection and averaging and hypothesis testing, as well as the first completely general definition of {\em MDL estimators}. Incorporating these developments, MDL can be seen as a powerful extension of both penalized likelihood and Bayesian approaches, in which penalization functions and prior distributions are replaced by more general luckiness functions, average-case methodology is replaced by a more robust worst-case approach, and in which methods classically viewed as highly distinct, such as AIC vs BIC and cross-validation vs Bayes can, to a large extent, be viewed from a unified perspective.
These notes survey and explore an emerging method, which we call the low-degree method, for predicting and understanding statistical-versus-computational tradeoffs in high-dimensional inference problems. In short, the method posits that a certain quantity -- the second moment of the low-degree likelihood ratio -- gives insight into how much computational time is required to solve a given hypothesis testing problem, which can in turn be used to predict the computational hardness of a variety of statistical inference tasks. While this method originated in the study of the sum-of-squares (SoS) hierarchy of convex programs, we present a self-contained introduction that does not require knowledge of SoS. In addition to showing how to carry out predictions using the method, we include a discussion investigating both rigorous and conjectural consequences of these predictions. These notes include some new results, simplified proofs, and refined conjectures. For instance, we point out a formal connection between spectral methods and the low-degree likelihood ratio, and we give a sharp low-degree lower bound against subexponential-time algorithms for tensor PCA.
Surrogate testing techniques have been used widely to investigate the presence of dynamical nonlinearities, an essential ingredient of deterministic chaotic processes. Traditional surrogate testing subscribes to statistical hypothesis testing and investigates potential differences in discriminant statistics between the given empirical sample and its surrogate counterparts. The choice and estimation of the discriminant statistics can be challenging across short time series. Also, conclusion based on a single empirical sample is an inherent limitation. The present study proposes a recurrent neural network classification framework that uses the raw time series obviating the need for discriminant statistic while accommodating multiple time series realizations for enhanced generalizability of the findings. The results are demonstrated on short time series with lengths (L = 32, 64, 128) from continuous and discrete dynamical systems in chaotic regimes, nonlinear transform of linearly correlated noise and experimental data. Accuracy of the classifier is shown to be markedly higher than >> 50% for the processes in chaotic regimes whereas those of nonlinearly correlated noise were around ~50% similar to that of random guess from a one-sample binomial test. These results are promising and elucidate the usefulness of the proposed framework in identifying potential dynamical nonlinearities from short experimental time series.
We present a general framework for hypothesis testing on distributions of sets of individual examples. Sets may represent many common data sources such as groups of observations in time series, collections of words in text or a batch of images of a given phenomenon. This observation pattern, however, differs from the common assumptions required for hypothesis testing: each set differs in size, may have differing levels of noise, and also may incorporate nuisance variability, irrelevant for the analysis of the phenomenon of interest; all features that bias test decisions if not accounted for. In this paper, we propose to interpret sets as independent samples from a collection of latent probability distributions, and introduce kernel two-sample and independence tests in this latent space of distributions. We prove the consistency of tests and observe them to outperform in a wide range of synthetic experiments. Finally, we showcase their use in practice with experiments of healthcare and climate data, where previously heuristics were needed for feature extraction and testing.
We consider the hypothesis testing problem of detecting conditional dependence, with a focus on high-dimensional feature spaces. Our contribution is a new test statistic based on samples from a generative adversarial network designed to approximate directly a conditional distribution that encodes the null hypothesis, in a manner that maximizes power (the rate of true negatives). We show that such an approach requires only that density approximation be viable in order to ensure that we control type I error (the rate of false positives); in particular, no assumptions need to be made on the form of the distributions or feature dependencies. Using synthetic simulations with high-dimensional data we demonstrate significant gains in power over competing methods. In addition, we illustrate the use of our test to discover causal markers of disease in genetic data.