Hypothesis Testing Framework Research Articles

Quantum hypothesis testing in many-body systems

One of the key tasks in physics is to perform measurements in order to determine the state of a system. Often, measurements are aimed at determining the values of physical parameters, but one can also ask simpler questions, such as ``is the system in state A or state B?''. In quantum mechanics, the latter type of measurements can be studied and optimized using the framework of quantum hypothesis testing. In many cases one can explicitly find the optimal measurement in the limit where one has simultaneous access to a large number n of identical copies of the system, and estimate the expected error as n becomes large. Interestingly, error estimates turn out to involve various quantum information theoretic quantities such as relative entropy, thereby giving these quantities operational meaning. In this paper we consider the application of quantum hypothesis testing to quantum many-body systems and quantum field theory. We review some of the necessary background material, and study in some detail the situation where the two states one wants to distinguish are parametrically close. The relevant error estimates involve quantities such as the variance of relative entropy, for which we prove a new inequality. We explore the optimal measurement strategy for spin chains and two-dimensional conformal field theory, focusing on the task of distinguishing reduced density matrices of subsystems. The optimal strategy turns out to be somewhat cumbersome to implement in practice, and we discuss a possible alternative strategy and the corresponding errors.

Analysis of similarities (ANOSIM) for 2‐way layouts using a generalised ANOSIM statistic, with comparative notes on Permutational Multivariate Analysis of Variance (PERMANOVA)

AbstractANOSIM (Analysis of Similarities) is a robust non‐parametric hypothesis testing framework for differences in resemblances among groups of samples. The generalised ANOSIM statistic RO is defined as the slope of the linear regression of ranked resemblances from observations against ranked distances in a model describing the unordered or ordered distances among samples under an alternative to the null hypothesis. In the absence of ordering, this becomes the standard ANOSIM R statistic. The construction of 2‐way tests using the generalised statistic in various nested and crossed designs, with and without ordered factors, and with or without replication, is described. Examples are given of 2‐way tests with ordered factors in marine ecological studies: 1. phytal meiofaunal communities in species of macroalgae with increasing physical complexity, among islands in the Isles of Scilly; 2. coral community composition across intertidal flats in Thailand, sampled in different years; 3. macrofauna inhabiting kelp holdfasts from different places in response to an oil spill; 4. experimental effects of salinity stress and food restriction on nematode communities. ANOSIM is fully non‐parametric and thus cannot, for two‐way crossed designs, decompose factors into (metric‐based) main effects and interactions; this requires at least semi‐parametric modelling, such as provided by PERMANOVA. The two approaches therefore test very different hypotheses: ANOSIM gives a robust, comparable and globally interpretable measure of magnitude of overall community change associated with each factor, having excised any possible effect from the factor(s) it is crossed with, irrespective of whether the factors interact or not. PERMANOVA cannot do this because the presence of interactions will compromise (sometimes totally) any overall measures of the main effects of each factor. Conversely, PERMANOVA can test for interactions involving directional (but non‐magnitudinal) community change, which are entirely invisible to ANOSIM. The two methods are therefore seen as complementary, rather than as alternatives.

Stochastic resonance benefits in signal detection under MAP criterion

In this paper, a noise enhanced hypothesis testing framework for a general nonlinear system is established under the Maximum A Posteriori (MAP) criterion, where an additive noise is added to the nonlinear system input and then the optimal MAP test for the discrimination between two hypotheses is made based on the noise modified nonlinear system output. The improvability of the noise modified MAP detection system is presented first, and then the noise enhanced decision solutions for the performance optimization are explored under three kinds of different cases. In opt SR case, the minimization of the average error probability is studied, and the optimal additive noise is proved as a constant vector, i.e. its probability distribution function is expressed by Dirac type δ(n-no). Furthermore, constraints on detection and false-alarm probabilities are considered in the minimization problem in UCSR and CSR cases. The difference is that no randomization between different detectors is allowed in UCSR case, while the randomization between detectors is feasible in CSR case. Finally, numerical results for sine transform and amplitude limit systems are given to confirm the theoretical results. The performance comparisons of no SR (i.e., in the absence of additive noise), opt SR, CSR and UCSR cases demonstrate that the SR phenomenon could occur to improve the optimal MAP decision under certain conditions.

A novel Mendelian randomization method with binary risk factor and outcome.

Mendelian randomization (MR) applies instrumental variable (IV) methods to observational data using a genetic variant as an IV. Several Monte-Carlo studies investigate the performance of MR methods with binary outcomes, but few consider them in conjunction with binary risk factors. To develop a novel MR estimator for scenarios with a binary risk factor and outcome; and compare to existing MR estimators via simulations and real data analysis. A bivariate Bernoulli distribution is adapted to the IV setting. Empirical bias and asymptotic coverage probabilities are estimated via simulations. The proposed method is compared to the Wald method, two-stage predictor substitution (2SPS), two-stage residual inclusion (2SRI), and the generalized method of moments (GMM). An analysis is performed using existing data from the CLEAR study to estimate the potential causal effect of smoking on rheumatoid arthritis risk in African Americans. Bias was low for the proposed method and comparable to 2SPS. The Wald method was often biased towards the null. Coverage was adequate for the proposed method, 2SPS, and 2SRI. Coverage for the Wald and GMM methods was poor in several scenarios. The causal effect of ever smoking on rheumatoid arthritis risk was not statistically significant using a variety of genetic instruments. Simulations suggest the proposed MR method is sound with binary risk factors and outcomes, and comparable to 2SPS and 2SRI in terms of bias. The proposed method also provides more natural framework for hypothesis testing compared to 2SPS or 2SRI, which require ad-hoc variance adjustments.

Assessment of significance of conditionally independent GWAS signals.

Multiple independently associated SNPs within a linkage disequilibrium region are a common phenomenon. Conditional analysis has been successful in identifying secondary signals. While conditional association tests are limited to specific genomic regions, they are benchmarked with genome-wide scale criterion, a conservative strategy. Within the weighted hypothesis testing framework, we developed a 'quasi-adaptive' method that uses the pairwise correlation (r2) and physical distance (d) from the index association to construct priority functions G =G(r2, d), which assign an SNP-specific α-threshold to each SNP. Family-wise error rate (FWER) and power of the approach were evaluated via simulations based on real GWAS data. We compared a series of different G-functions. Simulations under the null hypothesis on 1,100 primary SNPs confirmed appropriate empirical FWER for all G-functions. A G-function with optimal r2 = 0.3 between index and secondary SNP which down-weighted SNPs at higher distance step-wise-strong and gave more emphasis on d than on r2 had overall best power. It also gave the best results in application to the real datasets. As a proof of concept, 'quasi-adaptive' method was applied to GWAS on free thyroxine (FT4), inflammatory bowel disease (IBD) and human height. Application of the algorithm revealed 5 secondary signals in our example GWAS on FT4, 5 secondary signals in case of the IBD and 19 secondary signals on human height, that would have gone undetected with the established genome-wide threshold (α=5×10-8). https://github.com/sghasemi64/Secondary-Signal. Supplementary data are available at Bioinformatics online.

Bioverse: A Simulation Framework to Assess the Statistical Power of Future Biosignature Surveys

Abstract Next-generation space observatories will conduct the first systematic surveys of terrestrial exoplanet atmospheres and search for evidence of life beyond Earth. While in-depth observations of the nearest habitable worlds may yield enticing results, there are fundamental questions about planetary habitability and evolution that can only be answered through population-level studies of dozens to hundreds of terrestrial planets. To determine the requirements for next-generation observatories to address these questions, we have developed Bioverse. Bioverse combines existing knowledge of exoplanet statistics with a survey simulation and hypothesis testing framework to determine whether proposed space-based direct imaging and transit-spectroscopy surveys will be capable of detecting various hypothetical statistical relationships between the properties of terrestrial exoplanets. Following a description of the code, we apply Bioverse to determine whether an ambitious direct imaging or transit survey would be able to determine the extent of the circumstellar habitable zone and study the evolution of Earth-like planets. Given recent evidence that Earth-sized habitable zone planets are likely much rarer than previously believed, we find that space missions with large search volumes will be necessary to study the population of terrestrial and habitable worlds. Moving forward, Bioverse provides a methodology for performing trade studies of future observatory concepts to maximize their ability to address population-level questions, including and beyond the specific examples explored here.

Hypothesis-based Acceptance Sampling for Modules F and F1 of the European Measuring Instruments Directive

Millions of measuring instruments are verified each year before being placed on the markets worldwide. In the EU, such initial conformity assessments are regulated by the Measuring Instruments Directive (MID). The MID modules F and F1 on product verification allow for statistical acceptance sampling, whereby only random subsets of instruments need to be inspected. This article re-interprets the acceptance sampling conditions formulated by the MID. The new interpretation is contrasted with the one advanced in WELMEC guide 8.10, and three advantages have become apparent. First, an economic advantage of the new interpretation is a producers’ risk bounded from above, such that measuring instruments with sufficient quality are accepted with a guaranteed probability of no less than 95%. Second, a conceptual advantage is that the new MID interpretation fits into the well known, formal framework of statistical hypothesis testing. Thirdly, the new interpretation applies unambiguously to finite-sized lots, even very small ones. We conclude that the new interpretation is to be preferred and suggest re-formulating the statistical sampling conditions in the MID. Re-interpreting the MID conditions implies that currently available sampling plans are either not admissible or not optimal. We derive a new acceptance sampling scheme and recommend its application. Supplementary materials for this article are available online.

Pengaruh Pengampunan Pajak dan Kemudahan Administrasi Pajak Terhadap Kepatuhan Wajib Pajak Orang Pribadi pada KPP Pratama Jakarta Penjaringan

Tax amnesty is a government policy program that provides an opportunity for taxpayers to pay their tax arrears without administrative sanctions in order to improve taxpayer compliance and state revenue. Given the importance of tax forgiveness programs in the implementation of government policies, especially in the implementation of self assessment system, it is interesting to examine how the implementation of tax capability is done by the Directorate General of Taxes in this case in the Tax Office Pratama Jakarta Penjaringan. In the preparation of this study, the authors use a quantitative approach, namely research that emphasizes the analysis on numerical data or numbers obtained by statistical methods and conducted on inferential research or in the framework of hypothesis testing so obtained the significance of the relationship between variables studied. The purpose of this study is to determine the implementation of tax pardons conducted by KPP Pratama Jakarta Penjaringan, ease of tax administration and changes in taxpayer compliance increase. The conclusion of research, that the implementation of tax amnesty and the ease of tax administration conducted by KPP Penjaringan in the period 2015-2017 have a strong influence on taxpayer compliance. The coefficient of determination (adjusted R2) shows the variable of Tax Amnesty (X1) and Ease of Tax Administration (X2) can explain or explain Personal Taxpayer Compliance KPP Pratama Jakarta Penjaringan (Y) of 28%. KPP is advised to improve tax administration system, service to Taxpayer, tax law enforcement and tax rate and intensive socialization to taxpayer about good tax administration knowledge.

Does trait-based joint species distribution modelling reveal the signature of competition in stream macroinvertebrate communities?

The occupancy and abundance of species are jointly driven by local factors, such as environmental characteristics and biotic interactions, and regional-scale factors, such as dispersal and climate. Recently, it has been shown that biotic interactions shape species occupancies and abundances beyond local extents. However, for small ectothermic animals, particularly for those occurring in freshwater environments, the importance of biotic interactions remains understudied. Species-to-species associations from joint species distribution models (i.e. species associations while controlling for environmental characteristics) are increasingly used to draw hypotheses of which species possibly show biotic interactions. We studied whether species-to-species associations from joint species distribution models show signs of competition using a hypothesis testing framework in stream macroinvertebrate communities at regional extent. We sampled aquatic macroinvertebrates from 105 stream sites in western Finland encompassing a latitudinal gradient of c. 500 kilometres. We hypothesized that if competition drives these associations (H1) functionally, similar species are mostly negatively associated, whereas functionally dissimilar species show random associations. We further hypothesized that the relationship between functional dissimilarity and the strength of association is more pronounced (H2) for abundances rather than occupancies, (H3) at small grain (i.e. stream site) rather than at large grain (i.e. river basin), and (H4) among species having weak dispersal ability than among species with high dispersal ability. Stream macroinvertebrates showed both negative and positive species-to-species associations while controlling for habitat characteristics. However, the negative associations were mostly at large grain (river basin) rather than at small grain (stream site), in occupancy rather than abundance, and not related to species functional dissimilarity or to their dispersal ability. Thus, all our hypotheses considering possible competition (H1-H4) were rejected. Competition does not appear to be a major driving force of stream macroinvertebrate communities at the spatial grain sizes considered. The observed positive associations in occupancy at small grain (stream site) may be attributed to species' similar microhabitat preferences, whereas at large grain (river basin), they may stem from metacommunity dynamics. Our results highlight that species traits were necessary to interpret whether or not species-to-species associations from joint species distribution models resulted from biotic interactions.

Sepal: identifying transcript profiles with spatial patterns by diffusion-based modeling.

MotivationCollection of spatial signals in large numbers has become a routine task in multiple omics-fields, but parsing of these rich datasets still pose certain challenges. In whole or near-full transcriptome spatial techniques, spurious expression profiles are intermixed with those exhibiting an organized structure. To distinguish profiles with spatial patterns from the background noise, a metric that enables quantification of spatial structure is desirable. Current methods designed for similar purposes tend to be built around a framework of statistical hypothesis testing, hence we were compelled to explore a fundamentally different strategy.ResultsWe propose an unexplored approach to analyze spatial transcriptomics data, simulating diffusion of individual transcripts to extract genes with spatial patterns. The method performed as expected when presented with synthetic data. When applied to real data, it identified genes with distinct spatial profiles, involved in key biological processes or characteristic for certain cell types. Compared to existing methods, ours seemed to be less informed by the genes’ expression levels and showed better time performance when run with multiple cores.Availabilityand implementationOpen-source Python package with a command line interface (CLI), freely available at https://github.com/almaan/sepal under an MIT licence. A mirror of the GitHub repository can be found at Zenodo, doi: 10.5281/zenodo.4573237.Supplementary informationSupplementary data are available at Bioinformatics online.

Revealing posturographic profile of patients with Parkinsonian syndromes through a novel hypothesis testing framework based on machine learning.

Falling in Parkinsonian syndromes (PS) is associated with postural instability and consists a common cause of disability among PS patients. Current posturographic practices record the body’s center-of-pressure displacement (statokinesigram) while the patient stands on a force platform. Statokinesigrams, after appropriate processing, can offer numerous posturographic features. This fact, although beneficial, challenges the efforts for valid statistics via standard univariate approaches. In this work, 123 PS patients were classified into fallers (PSF) or non-faller (PSNF) based on the clinical assessment, and underwent simple Romberg Test (eyes open/eyes closed). We developed a non-parametric multivariate two-sample test (ts-AUC) based on machine learning, in order to examine statokinesigrams’ differences between PSF and PSNF. We analyzed posturographic features using both multiple testing with p-value adjustment and ts-AUC. While ts-AUC showed significant difference between groups (p-value = 0.01), multiple testing did not agree with this result (eyes open). PSF showed significantly increased antero-posterior movements as well as increased posturographic area compared to PSNF. Our study highlights the superiority of ts-AUC compared to standard statistical tools in distinguishing PSF and PSNF in multidimensional space. Machine learning-based statistical tests can be seen as a natural extension of classical statistics and should be considered, especially when dealing with multifactorial assessments.

A hypothesis-driven statistical approach for identifying ecosystem indicators of coho and Chinook salmon marine survival

Efforts to understand causes of declines in productivity of species of concern often involve retrospective evaluation of multiple possible causes based on trends in relevant ecological indicators. We describe a hypothesis testing framework for examining declines in marine survival for coho and Chinook salmon in the Salish Sea. Independent populations of both anadromous species have declined over the last 50 years, prompting extensive examination of mortality in different life stages. Previous studies have identified declining trends in marine survival, and we re-evaluated these trends in light of a number of possible hypotheses for declines. We laid out seven potential explanations for declines: changes in predator buffering related to abundance and timing, density-dependent or -independent food availability, water quality, timing of freshwater delivery to Puget Sound, and anthropogenic impacts. We compiled ecosystem indicators relevant to these hypotheses and used generalized additive models (GAMs) to examine multivariate relationships with survival from multiple coho and Chinook salmon stocks. We also developed additional models using the most informative indicators based on variable importance weighting (VIW) from the seven hypothesis groups. We examined how these models explained overall trends in marine survival, as well as survival in three temporal stanzas (before, during, and after a major decline, based on statistical breakpoint analysis). Across the entire time series, best fitting models explained 30–40% of the variation in the survival data. Best fitting models were from multiple hypotheses, including predation (abundance and timing), competition, water quality, and anthropogenic impacts; the freshwater delivery hypothesis was the least supported. Different models performed best (lowest error) during different stanzas of the coho salmon marine survival time series and the two VIW models were generally the top performing models, but performance varied in different years. Indicators with the strongest support included seal abundance, herring abundance, timing of hatchery salmon releases, and indicators related to water properties like stratification and temperature. These findings suggest that multiple processes embedded in several of our hypotheses influence marine survival but that an ecological “smoking gun” for Salish Sea salmon declines will remain elusive.

Determining sample size for progression criteria for pragmatic pilot RCTs: the hypothesis test strikes back!

BackgroundThe current CONSORT guidelines for reporting pilot trials do not recommend hypothesis testing of clinical outcomes on the basis that a pilot trial is under-powered to detect such differences and this is the aim of the main trial. It states that primary evaluation should focus on descriptive analysis of feasibility/process outcomes (e.g. recruitment, adherence, treatment fidelity). Whilst the argument for not testing clinical outcomes is justifiable, the same does not necessarily apply to feasibility/process outcomes, where differences may be large and detectable with small samples. Moreover, there remains much ambiguity around sample size for pilot trials.MethodsMany pilot trials adopt a ‘traffic light’ system for evaluating progression to the main trial determined by a set of criteria set up a priori. We construct a hypothesis testing approach for binary feasibility outcomes focused around this system that tests against being in the RED zone (unacceptable outcome) based on an expectation of being in the GREEN zone (acceptable outcome) and choose the sample size to give high power to reject being in the RED zone if the GREEN zone holds true. Pilot point estimates falling in the RED zone will be statistically non-significant and in the GREEN zone will be significant; the AMBER zone designates potentially acceptable outcome and statistical tests may be significant or non-significant.ResultsFor example, in relation to treatment fidelity, if we assume the upper boundary of the RED zone is 50% and the lower boundary of the GREEN zone is 75% (designating unacceptable and acceptable treatment fidelity, respectively), the sample size required for analysis given 90% power and one-sided 5% alpha would be around n = 34 (intervention group alone). Observed treatment fidelity in the range of 0–17 participants (0–50%) will fall into the RED zone and be statistically non-significant, 18–25 (51–74%) fall into AMBER and may or may not be significant and 26–34 (75–100%) fall into GREEN and will be significant indicating acceptable fidelity.DiscussionIn general, several key process outcomes are assessed for progression to a main trial; a composite approach would require appraising the rules of progression across all these outcomes. This methodology provides a formal framework for hypothesis testing and sample size indication around process outcome evaluation for pilot RCTs.

A General Framework for Distributed Inference With Uncertain Models

This paper studies the problem of distributed classification with a network of heterogeneous agents. The agents seek to jointly identify the underlying target class that best describes a sequence of observations. The problem is first abstracted to a hypothesis-testing framework, where we assume that the agents seek to agree on the hypothesis (target class) that best matches the distribution of observations. Non-Bayesian social learning theory provides a framework that solves this problem in an efficient manner by allowing the agents to sequentially communicate and update their beliefs for each hypothesis over the network. Most existing approaches assume that agents have access to exact statistical models for each hypothesis. However, in many practical applications, agents learn the likelihood models based on limited data, which induces uncertainty in the likelihood function parameters. In this work, we build upon the concept of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">uncertain models</i> to incorporate the agents’ uncertainty in the likelihoods by identifying a broad set of parametric distribution that allows the agents’ beliefs to converge to the same result as a centralized approach. Furthermore, we empirically explore extensions to non-parametric models to provide a generalized framework of uncertain models in non-Bayesian social learning.

Copy Move Source-Target Disambiguation Through Multi-Branch CNNs

We propose a method to identify the source and target regions of a copy-move forgery so allow a correct localisation of the tampered area. First, we cast the problem into a hypothesis testing framework whose goal is to decide which region between the two nearly-duplicate regions detected by a generic copy-move detector is the original one. Then we design a multi-branch CNN architecture that solves the hypothesis testing problem by learning a set of features capable to reveal the presence of interpolation artefacts and boundary inconsistencies in the copy-moved area. The proposed architecture, trained on a synthetic dataset explicitly built for this purpose, achieves good results on copy-move forgeries from both synthetic and realistic datasets. Based on our tests, the proposed disambiguation method can reliably reveal the target region even in realistic cases where an approximate version of the copy-move localization mask is provided by a state-of-the-art copy-move detection algorithm.

Toward catchment hydro‐biogeochemical theories

AbstractHeadwater catchments are the fundamental units that connect the land to the ocean. Hydrological flow and biogeochemical processes are intricately coupled, yet their respective sciences have progressed without much integration. Reaction kinetic theories that prescribe rate dependence on environmental variables (e.g., temperature and water content) have advanced substantially, mostly in well‐mixed reactors, columns, and warming experiments without considering the characteristics of hydrological flow at the catchment scale. These theories have shown significant divergence from observations in natural systems. On the other hand, hydrological theories, including transit time theory, have progressed substantially yet have not been incorporated into understanding reactions at the catchment scale. Here we advocate for the development of integrated hydro‐biogeochemical theories across gradients of climate, vegetation, and geology conditions. The lack of such theories presents barriers for understanding mechanisms and forecasting the future of the Critical Zone under human‐ and climate‐induced perturbations. Although integration has started and co‐located measurements are well under way, tremendous challenges remain. In particular, even in this era of “big data,” we are still limited by data and will need to (1) intensify measurements beyond river channels and characterize the vertical connectivity and broadly the shallow and deep subsurface; (2) expand to older water dating beyond the time scales reflected in stable water isotopes; (3) combine the use of reactive solutes, nonreactive tracers, and isotopes; and (4) augment measurements in environments that are undergoing rapid changes. To develop integrated theories, it is essential to (1) engage models at all stages to develop model‐informed data collection strategies and to maximize data usage; (2) adopt a “simple but not simplistic,” or fit‐for‐purpose approach to include essential processes in process‐based models; (3) blend the use of process‐based and data‐driven models in the framework of “theory‐guided data science.” Within the framework of hypothesis testing, model‐data fusion can advance integrated theories that mechanistically link catchments' internal structures and external drivers to their functioning. It can not only advance the field of hydro‐biogeochemistry, but also enable hind‐ and fore‐casting and serve the society at large. Broadly, future education will need to cultivate thinkers at the intersections of traditional disciplines with hollistic approaches for understanding interacting processes in complex earth systems.This article is categorized under: Engineering Water &gt; Methods

A False Discovery Rate Approach to Optimal Volatility Forecasting Model Selection

Forecasting volatility as a proxy to risk is critical to investment decisions and the study of investment behaviour. Toward this, a growing literature has attempted but not yet been able to identify an optimal volatility forecasting model. In this paper, we show that this is because the optimal model is dynamic. The choice of the optimal model depends on three factors: the asset in question; the period being studied; and the fitted innovation distribution. Therefore, we propose a novel empirical strategy that enables the researcher to select the optimal model as a function of these three factors. Using our Multiple Hypothesis Testing framework with False Discovery Rate approach, we are able to consistently identify significantly more accurate models relative to the literature’s benchmark volatility forecasting models of ARCH(1) and GARCH(1,1).

The Black Lives Matter Movement: A Call to Action for Couple and Family Therapists.

The frequent police killings during the COVID-19 pandemic forced a reckoning among Americans from all backgrounds and propelled the Black Lives Matter movement into a global force. This manuscript addresses major issues to aid practitioners in the effective treatment of African Americans via the lens of Critical Race Theory and the Bioecological Model. We place the impacts of racism on Black families in historical context and outline the sources of Black family resilience. We critique structural racism embedded in all aspects of psychology and allied fields. We provide an overview of racial socialization and related issues affecting the parenting decisions in Black families, as well as a detailed overview of impacts of structural racism on couple dynamics. Recommendations are made for engaging racial issues in therapy, providing emotional support and validation to couples and families experiencing discrimination and racial trauma, and using Black cultural strengths as therapeutic resources.

Recovering Data Permutations From Noisy Observations: The Linear Regime

This article considers a noisy data structure recovery problem. The goal is to investigate the following question: given a noisy observation of a permuted data set, according to which permutation was the original data sorted? The focus is on scenarios where data is generated according to an isotropic Gaussian distribution, and the noise is additive Gaussian with an arbitrary covariance matrix. This problem is posed within a hypothesis testing framework. The objective is to study the linear regime in which the optimal decoder has a polynomial complexity in the data size, and it declares the permutation by simply computing a permutation-independent linear function of the noisy observations. The main result of this article is a complete characterization of the linear regime in terms of the noise covariance matrix. Specifically, it is shown that this matrix must have a very flat spectrum with at most three distinct eigenvalues to induce the linear regime. Several practically relevant implications of this result are discussed, and the error probability incurred by the decision criterion in the linear regime is also characterized. A core technical component consists of using linear algebraic and geometric tools, such as Steiner symmetrization.

Non‐stochastic hypothesis testing for privacy

In this study, I consider privacy against hypothesis testing adversaries within a non-stochastic framework. He developed a theory of non-stochastic hypothesis testing by borrowing the notion of uncertain variables from non-stochastic information theory. I define tests as binary-valued mappings on uncertain variables and proved a fundamental bound on the best performance of the tests in non-stochastic hypothesis testing. I provide parallels between stochastic and non-stochastic hypothesis-testing frameworks. I use the performance bound in non-stochastic hypothesis testing to develop a measure of privacy. I then construct the reporting policies with the prescribed privacy and utility guarantees. The utility of a reporting policy is measured by the distance between the reported and original values. Finally, I present the notion of indistinguishability as a measure of privacy by extending the identifiability from the privacy literature to the non-stochastic framework. I prove that the linear quantisers can indeed achieve identifiability for responding to linear queries on private datasets.