Analysis of low-degree polynomial algorithms is a powerful, newly-popular method for predicting computational thresholds in hypothesis testing problems. One limitation of current techniques for this analysis is their restriction to Bernoulli and Gaussian distributions. We expand this range of possibilities by performing the low-degree analysis of hypothesis testing for the Morris class of exponential families, giving a unified treatment of Gaussian, Poisson, gamma, binomial, negative binomial, and generalized hyperbolic secant distributions. We then give several algorithmic applications.
1. In models where a random signal is observed through an exponential family, the success or failure of low-degree polynomials is governed by the $z$-score overlap, the inner product of $z$-score vectors with respect to the null distribution of two independent copies of the signal.
2. In the same models, testing with low-degree polynomials exhibits channel monotonicity: the above distributions admit a total ordering by computational cost of hypothesis testing, according to a scalar parameter describing how the variance depends on the mean in an exponential family.
3. In a spiked matrix model with a particular non-Gaussian noise distribution, the low-degree prediction is incorrect unless polynomials with arbitrarily large degree in individual matrix entries are permitted. This shows that polynomials summing over self-avoiding walks and variants thereof, as proposed recently by Ding, Hopkins, and Steurer (2020) for spiked matrix models with heavy-tailed noise, are suboptimal for this model. Thus low-degree polynomials appear to offer a tradeoff between robustness and strong performance fine-tuned to specific models, and may struggle with problems requiring an algorithm to first examine the input and then use some intermediate computation to choose from one of several inference subroutines.
Analysis of Compositional Data with Positive Correlations among Components using a Nested Dirichlet Distribution with Application to a Morris Water Maze Experiment
Efficient automatic protein classification is of central importance in genomic annotation. As an independent way to check the reliability of the classification, we propose a statistical approach to test if two sets of protein domain sequences coming from two families of the Pfam database are significantly different. We model protein sequences as realizations of Variable Length Markov Chains (VLMC) and we use the context trees as a signature of each protein family. Our approach is based on a Kolmogorov--Smirnov-type goodness-of-fit test proposed by Balding et al. [Limit theorems for sequences o…
We treat quantum counterparts of testing problems whose optimal tests are given by chi-square, t and F tests. These quantum counterparts are formulated as quantum hypothesis testing problems concerning quantum Gaussian states families, and contain disturbance parameters, which have group symmetry. Quantum Hunt-Stein Theorem removes a part of these disturbance parameters, but other types of difficulty still remain. In order to remove them, combining quantum Hunt-Stein theorem and other reduction methods, we establish a general reduction theorem that reduces a complicated quantum hypothesis test…
We discuss an "operational" approach to testing convex composite hypotheses when the underlying distributions are heavy-tailed. It relies upon Euclidean separation of convex sets and can be seen as an extension of the approach to testing by convex optimization developed in [8, 12]. In particular, we show how one can construct quasi-optimal testing procedures for families of distributions which are majorated, in a certain precise sense, by a sub-spherical symmetric one and study the relationship between tests based on Euclidean separation and "potential-based tests." We apply the promoted metho…
Our purpose, is to put forward a change in the paradigm of testing by generalizing a very natural idea exposed by Morris DeGroot (1975) aiming to an approach that is attractive to all schools of statistics, in a procedure better suited for the needs of science. DeGroot's seminal idea is to base testing statistical hypothesis on minimizing the weighted sum of type I plus type II error instead of of the prevailing paradigm which is fixing type I error and minimizing type II error. DeGroot's result is that in simple vs simple hypothesis the optimal criterion is to reject, according to the likelih…
Equivalence on exponential families as Hessian manifolds and classification of exponential families of order 1 with constant Hessian sectional curvature
Tokai University（东海大学）；The University of Tokyo（东京大学）
We study the general problem of testing whether an unknown distribution belongs to a specified family of distributions. More specifically, given a distribution family $\mathcal{P}$ and sample access to an unknown discrete distribution $\mathbf{P}$, we want to distinguish (with high probability) between the case that $\mathbf{P} \in \mathcal{P}$ and the case that $\mathbf{P}$ is $ε$-far, in total variation distance, from every distribution in $\mathcal{P}$. This is the prototypical hypothesis testing problem that has received significant attention in statistics and, more recently, in theoretica…
Given a finite-valued sample $X_1,...,X_n$ we wish to test whether it was generated by a stationary ergodic process belonging to a family $H_0$, or it was generated by a stationary ergodic process outside $H_0$. We require the Type I error of the test to be uniformly bounded, while the type II error has to be mande not more than a finite number of times with probability 1. For this notion of consistency we provide necessary and sufficient conditions on the family $H_0$ for the existence of a consistent test. This criterion is illustrated with applications to testing for a membership to paramet…
Motivated by some cutting edge circular data such as from Smart Home technologies and roulette spins from online and casino, we construct some new rich classes of discrete distributions on the circle. We give four new general methods of construction, namely (i) maximum entropy, (ii) centered wrapping, (iii) marginalized and (iv) conditionalized methods. We motivate these methods on the line and then work on the circular case and provide some properties to gain insight into these constructions. We mainly focus on the last two methods (iii) and (iv) in the context of circular location families, …
When in a full exponential family the maximum likelihood estimate (MLE) does not exist, the MLE may exist in the Barndorff-Nielsen completion of the family. We propose a practical algorithm for finding the MLE in the completion based on repeated linear programming using the R contributed package rcdd and illustrate it with two generalized linear model examples. When the MLE for the null hypothesis lies in the completion, likelihood ratio tests of model comparison are almost unchanged from the usual case. Only the degrees of freedom need to be adjusted. When the MLE lies in the completion, conf…
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}}…
USC Information Sciences Institute（南加州大学信息科学研究所）；Sony CSL（索尼计算机科学实验室）；University of Toronto（多伦多大学）；Vector Institute（向量研究所）
The exponential family is well known in machine learning and statistical physics as the maximum entropy distribution subject to a set of observed constraints, while the geometric mixture path is common in MCMC methods such as annealed importance sampling. Linking these two ideas, recent work has interpreted the geometric mixture path as an exponential family of distributions to analyze the thermodynamic variational objective (TVO). We extend these likelihood ratio exponential families to include solutions to rate-distortion (RD) optimization, the information bottleneck (IB) method, and recent …
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 differe…
Goodness-of-fit tests based on the empirical Wasserstein distance are proposed for simple and composite null hypotheses involving general multivariate distributions. For group families, the procedure is to be implemented after preliminary reduction of the data via invariance.This property allows for calculation of exact critical values and p-values at finite sample sizes. Applications include testing for location--scale families and testing for families arising from affine transformations, such as elliptical distributions with given standard radial density and unspecified location vector and s…
In this paper we consider two statistical hypotheses for the families of Wishart type distributions. These distributions are analogs of the Wishart distributions defined and parametrized over a Lorentz cone. We test these hypotheses by means of maximal invariant statistics which are explicitly derived in the paper. The testing problems, respectively, concern the hypothesis that parameters are in a sub-Lorentz-cone, and the the hypothesis that two observations have the same parameter.
We consider the testing of mutual independence among all entries in a $d$-dimensional random vector based on $n$ independent observations. We study two families of distribution-free test statistics, which include Kendall's tau and Spearman's rho as important examples. We show that under the null hypothesis the test statistics of these two families converge weakly to Gumbel distributions, and propose tests that control the type I error in the high-dimensional setting where $d>n$. We further show that the two tests are rate-optimal in terms of power against sparse alternatives, and outperform co…
Consider a random sample from a continuous multivariate distribution function $F$ with copula $C$. In order to test the null hypothesis that $C$ belongs to a certain parametric family, we construct an empirical process on the unit hypercube that converges weakly to a standard Wiener process under the null hypothesis. This process can therefore serve as a `tests generator' for asymptotically distribution-free goodness-of-fit testing of copula families. We also prove maximal sensitivity of this process to contiguous alternatives. Finally, we demonstrate through a Monte Carlo simulation study tha…
