We study the fundamental tradeoffs between computational tractability and statistical accuracy for a general family of hypothesis testing problems with combinatorial structures. Based upon an oracle model of computation, which captures the interactions between algorithms and data, we establish a general lower bound that explicitly connects the minimum testing risk under computational budget constraints with the intrinsic probabilistic and combinatorial structures of statistical problems. This lower bound mirrors the classical statistical lower bound by Le Cam (1986) and allows us to quantify the optimal statistical performance achievable given limited computational budgets in a systematic fashion. Under this unified framework, we sharply characterize the statistical-computational phase transition for two testing problems, namely, normal mean detection and sparse principal component detection. For normal mean detection, we consider two combinatorial structures, namely, sparse set and perfect matching. For these problems we identify significant gaps between the optimal statistical accuracy that is achievable under computational tractability constraints and the classical statistical lower bounds. Compared with existing works on computational lower bounds for statistical problems, which consider general polynomial-time algorithms on Turing machines, and rely on computational hardness hypotheses on problems like planted clique detection, we focus on the oracle computational model, which covers a broad range of popular algorithms, and do not rely on unproven hypotheses. Moreover, our result provides an intuitive and concrete interpretation for the intrinsic computational intractability of high-dimensional statistical problems. One byproduct of our result is a lower bound for a strict generalization of the matrix permanent problem, which is of independent interest.
We introduce a new type of graphical model that we call a "memory factor network" (MFN). We show how to use MFNs to model the structure inherent in many types of data sets. We also introduce an associated message-passing style algorithm called "proactive message passing"' (PMP) that performs inference on MFNs. PMP comes with convergence guarantees and is efficient in comparison to competing algorithms such as variants of belief propagation. We specialize MFNs and PMP to a number of distinct types of data (discrete, continuous, labelled) and inference problems (interpolation, hypothesis testing), provide examples, and discuss approaches for efficient implementation.
Linear independence testing is a fundamental information-theoretic and statistical problem that can be posed as follows: given $n$ points $\{(X_i,Y_i)\}^n_{i=1}$ from a $p+q$ dimensional multivariate distribution where $X_i \in \mathbb{R}^p$ and $Y_i \in\mathbb{R}^q$, determine whether $a^T X$ and $b^T Y$ are uncorrelated for every $a \in \mathbb{R}^p, b\in \mathbb{R}^q$ or not. We give minimax lower bound for this problem (when $p+q,n \to \infty$, $(p+q)/n \leq \kappa < \infty$, without sparsity assumptions). In summary, our results imply that $n$ must be at least as large as $\sqrt {pq}/\|\Sigma_{XY}\|_F^2$ for any procedure (test) to have non-trivial power, where $\Sigma_{XY}$ is the cross-covariance matrix of $X,Y$. We also provide some evidence that the lower bound is tight, by connections to two-sample testing and regression in specific settings.
Automated feature selection is important for text categorization to reduce the feature size and to speed up the learning process of classifiers. In this paper, we present a novel and efficient feature selection framework based on the Information Theory, which aims to rank the features with their discriminative capacity for classification. We first revisit two information measures: Kullback-Leibler divergence and Jeffreys divergence for binary hypothesis testing, and analyze their asymptotic properties relating to type I and type II errors of a Bayesian classifier. We then introduce a new divergence measure, called Jeffreys-Multi-Hypothesis (JMH) divergence, to measure multi-distribution divergence for multi-class classification. Based on the JMH-divergence, we develop two efficient feature selection methods, termed maximum discrimination ($MD$) and $MD-\chi^2$ methods, for text categorization. The promising results of extensive experiments demonstrate the effectiveness of the proposed approaches.
When data analysts train a classifier and check if its accuracy is significantly different from chance, they are implicitly performing a two-sample test. We investigate the statistical properties of this flexible approach in the high-dimensional setting. We prove two results that hold for all classifiers in any dimensions: if its true error remains $\epsilon$-better than chance for some $\epsilon>0$ as $d,n \to \infty$, then (a) the permutation-based test is consistent (has power approaching to one), (b) a computationally efficient test based on a Gaussian approximation of the null distribution is also consistent. To get a finer understanding of the rates of consistency, we study a specialized setting of distinguishing Gaussians with mean-difference $\delta$ and common (known or unknown) covariance $\Sigma$, when $d/n \to c \in (0,\infty)$. We study variants of Fisher's linear discriminant analysis (LDA) such as "naive Bayes" in a nontrivial regime when $\epsilon \to 0$ (the Bayes classifier has true accuracy approaching 1/2), and contrast their power with corresponding variants of Hotelling's test. Surprisingly, the expressions for their power match exactly in terms of $n,d,\delta,\Sigma$, and the LDA approach is only worse by a constant factor, achieving an asymptotic relative efficiency (ARE) of $1/\sqrt{\pi}$ for balanced samples. We also extend our results to high-dimensional elliptical distributions with finite kurtosis. Other results of independent interest include minimax lower bounds, and the optimality of Hotelling's test when $d=o(n)$. Simulation results validate our theory, and we present practical takeaway messages along with natural open problems.
We propose a new algorithmic framework for sequential hypothesis testing with i.i.d. data, which includes A/B testing, nonparametric two-sample testing, and independence testing as special cases. It is novel in several ways: (a) it takes linear time and constant space to compute on the fly, (b) it has the same power guarantee as a non-sequential version of the test with the same computational constraints up to a small factor, and (c) it accesses only as many samples as are required - its stopping time adapts to the unknown difficulty of the problem. All our test statistics are constructed to be zero-mean martingales under the null hypothesis, and the rejection threshold is governed by a uniform non-asymptotic law of the iterated logarithm (LIL). For the case of nonparametric two-sample mean testing, we also provide a finite sample power analysis, and the first non-asymptotic stopping time calculations for this class of problems. We verify our predictions for type I and II errors and stopping times using simulations.
A great deal of effort has been devoted to reducing the risk of spurious scientific discoveries, from the use of sophisticated validation techniques, to deep statistical methods for controlling the false discovery rate in multiple hypothesis testing. However, there is a fundamental disconnect between the theoretical results and the practice of data analysis: the theory of statistical inference assumes a fixed collection of hypotheses to be tested, or learning algorithms to be applied, selected non-adaptively before the data are gathered, whereas in practice data is shared and reused with hypotheses and new analyses being generated on the basis of data exploration and the outcomes of previous analyses. In this work we initiate a principled study of how to guarantee the validity of statistical inference in adaptive data analysis. As an instance of this problem, we propose and investigate the question of estimating the expectations of $m$ adaptively chosen functions on an unknown distribution given $n$ random samples. We show that, surprisingly, there is a way to estimate an exponential in $n$ number of expectations accurately even if the functions are chosen adaptively. This gives an exponential improvement over standard empirical estimators that are limited to a linear number of estimates. Our result follows from a general technique that counter-intuitively involves actively perturbing and coordinating the estimates, using techniques developed for privacy preservation. We give additional applications of this technique to our question.
Discovering statistically significant patterns from databases is an important challenging problem. The main obstacle of this problem is in the difficulty of taking into account the selection bias, i.e., the bias arising from the fact that patterns are selected from extremely large number of candidates in databases. In this paper, we introduce a new approach for predictive pattern mining problems that can address the selection bias issue. Our approach is built on a recently popularized statistical inference framework called selective inference. In selective inference, statistical inferences (such as statistical hypothesis testing) are conducted based on sampling distributions conditional on a selection event. If the selection event is characterized in a tractable way, statistical inferences can be made without minding selection bias issue. However, in pattern mining problems, it is difficult to characterize the entire selection process of mining algorithms. Our main contribution in this paper is to solve this challenging problem for a class of predictive pattern mining problems by introducing a novel algorithmic framework. We demonstrate that our approach is useful for finding statistically significant patterns from databases.
Brain decoding is a popular multivariate approach for hypothesis testing in neuroimaging. It is well known that the brain maps derived from weights of linear classifiers are hard to interpret because of high correlations between predictors, low signal to noise ratios, and the high dimensionality of neuroimaging data. Therefore, improving the interpretability of brain decoding approaches is of primary interest in many neuroimaging studies. Despite extensive studies of this type, at present, there is no formal definition for interpretability of multivariate brain maps. As a consequence, there is no quantitative measure for evaluating the interpretability of different brain decoding methods. In this paper, first, we present a theoretical definition of interpretability in brain decoding; we show that the interpretability of multivariate brain maps can be decomposed into their reproducibility and representativeness. Second, as an application of the proposed theoretical definition, we formalize a heuristic method for approximating the interpretability of multivariate brain maps in a binary magnetoencephalography (MEG) decoding scenario. Third, we propose to combine the approximated interpretability and the performance of the brain decoding model into a new multi-objective criterion for model selection. Our results for the MEG data show that optimizing the hyper-parameters of the regularized linear classifier based on the proposed criterion results in more informative multivariate brain maps. More importantly, the presented definition provides the theoretical background for quantitative evaluation of interpretability, and hence, facilitates the development of more effective brain decoding algorithms in the future.
Structure discovery in graphical models is the determination of the topology of a graph that encodes conditional independence properties of the joint distribution of all variables in the model. For some class of probability distributions, an edge between two variables is present if and only if the corresponding entry in the precision matrix is non-zero. For a finite sample estimate of the precision matrix, entries close to zero may be due to low sample effects, or due to an actual association between variables; these two cases are not readily distinguishable. %Fisher provided a hypothesis test based on a parametric approximation to the distribution of an entry in the precision matrix of a Gaussian distribution, but this may not provide valid upper bounds on $p$-values for non-Gaussian distributions. Many related works on this topic consider potentially restrictive distributional or sparsity assumptions that may not apply to a data sample of interest, and direct estimation of the uncertainty of an estimate of the precision matrix for general distributions remains challenging. Consequently, we make use of results for $U$-statistics and apply them to the covariance matrix. By probabilistically bounding the distortion of the covariance matrix, we can apply Weyl's theorem to bound the distortion of the precision matrix, yielding a conservative, but sound test threshold for a much wider class of distributions than considered in previous works. The resulting test enables one to answer with statistical significance whether an edge is present in the graph, and convergence results are known for a wide range of distributions. The computational complexities is linear in the sample size enabling the application of the test to large data samples for which computation time becomes a limiting factor. We experimentally validate the correctness and scalability of the test on multivariate distributions for which the distributional assumptions of competing tests result in underestimates of the false positive ratio. By contrast, the proposed test remains sound, promising to be a useful tool for hypothesis testing for diverse real-world problems.
Neuroimaging research has predominantly drawn conclusions based on classical statistics, including null-hypothesis testing, t-tests, and ANOVA. Throughout recent years, statistical learning methods enjoy increasing popularity, including cross-validation, pattern classification, and sparsity-inducing regression. These two methodological families used for neuroimaging data analysis can be viewed as two extremes of a continuum. Yet, they originated from different historical contexts, build on different theories, rest on different assumptions, evaluate different outcome metrics, and permit different conclusions. This paper portrays commonalities and differences between classical statistics and statistical learning with their relation to neuroimaging research. The conceptual implications are illustrated in three common analysis scenarios. It is thus tried to resolve possible confusion between classical hypothesis testing and data-guided model estimation by discussing their ramifications for the neuroimaging access to neurobiology.
Consider a two-class clustering problem where we observe $X_i = \ell_i \mu + Z_i$, $Z_i \stackrel{iid}{\sim} N(0, I_p)$, $1 \leq i \leq n$. The feature vector $\mu\in R^p$ is unknown but is presumably sparse. The class labels $\ell_i\in\{-1, 1\}$ are also unknown and the main interest is to estimate them. We are interested in the statistical limits. In the two-dimensional phase space calibrating the rarity and strengths of useful features, we find the precise demarcation for the Region of Impossibility and Region of Possibility. In the former, useful features are too rare/weak for successful clustering. In the latter, useful features are strong enough to allow successful clustering. The results are extended to the case of colored noise using Le Cam's idea on comparison of experiments. We also extend the study on statistical limits for clustering to that for signal recovery and that for hypothesis testing. We compare the statistical limits for three problems and expose some interesting insight. We propose classical PCA and Important Features PCA (IF-PCA) for clustering. For a threshold $t > 0$, IF-PCA clusters by applying classical PCA to all columns of $X$ with an $L^2$-norm larger than $t$. We also propose two aggregation methods. For any parameter in the Region of Possibility, some of these methods yield successful clustering. We find an interesting phase transition for IF-PCA. Our results require delicate analysis, especially on post-selection Random Matrix Theory and on lower bound arguments.
Sobolev quantities (norms, inner products, and distances) of probability density functions are important in the theory of nonparametric statistics, but have rarely been used in practice, partly due to a lack of practical estimators. They also include, as special cases, $L^2$ quantities which are used in many applications. We propose and analyze a family of estimators for Sobolev quantities of unknown probability density functions. We bound the bias and variance of our estimators over finite samples, finding that they are generally minimax rate-optimal. Our estimators are significantly more computationally tractable than previous estimators, and exhibit a statistical/computational trade-off allowing them to adapt to computational constraints. We also draw theoretical connections to recent work on fast two-sample testing. Finally, we empirically validate our estimators on synthetic data.
An important task in network analysis is the detection of anomalous events in a network time series. These events could merely be times of interest in the network timeline or they could be examples of malicious activity or network malfunction. Hypothesis testing using network statistics to summarize the behavior of the network provides a robust framework for the anomaly detection decision process. Unfortunately, choosing network statistics that are dependent on confounding factors like the total number of nodes or edges can lead to incorrect conclusions (e.g., false positives and false negatives). In this dissertation we describe the challenges that face anomaly detection in dynamic network streams regarding confounding factors. We also provide two solutions to avoiding error due to confounding factors: the first is a randomization testing method that controls for confounding factors, and the second is a set of size-consistent network statistics which avoid confounding due to the most common factors, edge count and node count.
This article provides a thorough meta-analysis of the anomaly detection problem. To accomplish this we first identify approaches to benchmarking anomaly detection algorithms across the literature and produce a large corpus of anomaly detection benchmarks that vary in their construction across several dimensions we deem important to real-world applications: (a) point difficulty, (b) relative frequency of anomalies, (c) clusteredness of anomalies, and (d) relevance of features. We apply a representative set of anomaly detection algorithms to this corpus, yielding a very large collection of experimental results. We analyze these results to understand many phenomena observed in previous work. First we observe the effects of experimental design on experimental results. Second, results are evaluated with two metrics, ROC Area Under the Curve and Average Precision. We employ statistical hypothesis testing to demonstrate the value (or lack thereof) of our benchmarks. We then offer several approaches to summarizing our experimental results, drawing several conclusions about the impact of our methodology as well as the strengths and weaknesses of some algorithms. Last, we compare results against a trivial solution as an alternate means of normalizing the reported performance of algorithms. The intended contributions of this article are many; in addition to providing a large publicly-available corpus of anomaly detection benchmarks, we provide an ontology for describing anomaly detection contexts, a methodology for controlling various aspects of benchmark creation, guidelines for future experimental design and a discussion of the many potential pitfalls of trying to measure success in this field.
In this paper, we initiate a principled study of how the generalization properties of approximate differential privacy can be used to perform adaptive hypothesis testing, while giving statistically valid $p$-value corrections. We do this by observing that the guarantees of algorithms with bounded approximate max-information are sufficient to correct the $p$-values of adaptively chosen hypotheses, and then by proving that algorithms that satisfy $(\epsilon,\delta)$-differential privacy have bounded approximate max information when their inputs are drawn from a product distribution. This substantially extends the known connection between differential privacy and max-information, which previously was only known to hold for (pure) $(\epsilon,0)$-differential privacy. It also extends our understanding of max-information as a partially unifying measure controlling the generalization properties of adaptive data analyses. We also show a lower bound, proving that (despite the strong composition properties of max-information), when data is drawn from a product distribution, $(\epsilon,\delta)$-differentially private algorithms can come first in a composition with other algorithms satisfying max-information bounds, but not necessarily second if the composition is required to itself satisfy a nontrivial max-information bound. This, in particular, implies that the connection between $(\epsilon,\delta)$-differential privacy and max-information holds only for inputs drawn from product distributions, unlike the connection between $(\epsilon,0)$-differential privacy and max-information.
Information theoretic measures (e.g. the Kullback Liebler divergence and Shannon mutual information) have been used for exploring possibly nonlinear multivariate dependencies in high dimension. If these dependencies are assumed to follow a Markov factor graph model, this exploration process is called structure discovery. For discrete-valued samples, estimates of the information divergence over the parametric class of multinomial models lead to structure discovery methods whose mean squared error achieves parametric convergence rates as the sample size grows. However, a naive application of this method to continuous nonparametric multivariate models converges much more slowly. In this paper we introduce a new method for nonparametric structure discovery that uses weighted ensemble divergence estimators that achieve parametric convergence rates and obey an asymptotic central limit theorem that facilitates hypothesis testing and other types of statistical validation.
Detection rules have traditionally been designed for rational agents that minimize the Bayes risk (average decision cost). With the advent of crowd-sensing systems, there is a need to redesign binary hypothesis testing rules for behavioral agents, whose cognitive behavior is not captured by traditional utility functions such as Bayes risk. In this paper, we adopt prospect theory based models for decision makers. We consider special agent models namely optimists and pessimists in this paper, and derive optimal detection rules under different scenarios. Using an illustrative example, we also show how the decision rule of a human agent deviates from the Bayesian decision rule under various behavioral models, considered in this paper.
In analyzing high-dimensional models, sparsity of the model parameter is a common but often undesirable assumption. In this paper, we study the following two-sample testing problem: given two samples generated by two high-dimensional linear models, we aim to test whether the regression coefficients of the two linear models are identical. We propose a framework named TIERS (short for TestIng Equality of Regression Slopes), which solves the two-sample testing problem without making any assumptions on the sparsity of the regression parameters. TIERS builds a new model by convolving the two samples in such a way that the original hypothesis translates into a new moment condition. A self-normalization construction is then developed to form a moment test. We provide rigorous theory for the developed framework. Under very weak conditions of the feature covariance, we show that the accuracy of the proposed test in controlling Type I errors is robust both to the lack of sparsity in the features and to the heavy tails in the error distribution, even when the sample size is much smaller than the feature dimension. Moreover, we discuss minimax optimality and efficiency properties of the proposed test. Simulation analysis demonstrates excellent finite-sample performance of our test. In deriving the test, we also develop tools that are of independent interest. The test is built upon a novel estimator, called Auto-aDaptive Dantzig Selector (ADDS), which not only automatically chooses an appropriate scale of the error term but also incorporates prior information. To effectively approximate the critical value of the test statistic, we develop a novel high-dimensional plug-in approach that complements the recent advances in Gaussian approximation theory.
In many practical applications of multiple hypothesis testing using the False Discovery Rate (FDR), the given hypotheses can be naturally partitioned into groups, and one may not only want to control the number of false discoveries (wrongly rejected null hypotheses), but also the number of falsely discovered groups of hypotheses (we say a group is falsely discovered if at least one hypothesis within that group is rejected, when in reality the group contains only nulls). In this paper, we introduce the p-filter, a procedure which unifies and generalizes the standard FDR procedure by Benjamini and Hochberg and global null testing procedure by Simes. We first prove that our proposed method can simultaneously control the overall FDR at the finest level (individual hypotheses treated separately) and the group FDR at coarser levels (when such groups are user-specified). We then generalize the p-filter procedure even further to handle multiple partitions of hypotheses, since that might be natural in many applications. For example, in neuroscience experiments, we may have a hypothesis for every (discretized) location in the brain, and at every (discretized) timepoint: does the stimulus correlate with activity in location x at time t after the stimulus was presented? In this setting, one might want to group hypotheses by location and by time. Importantly, our procedure can handle multiple partitions which are nonhierarchical (i.e. one partition may arrange p-values by voxel, and another partition arranges them by time point; neither one is nested inside the other). We prove that our procedure controls FDR simultaneously across these multiple lay- ers, under assumptions that are standard in the literature: we do not need the hypotheses to be independent, but require a nonnegative dependence condition known as PRDS.