DOCSLIB.ORG

Digitizing Omics Profiles by Divergence from a Baseline

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Digitizing Omics Profiles by Divergence from a Baseline Digitizing omics profiles by divergence from a baseline INAUGURAL ARTICLE Wikum Dinalankaraa,1, Qian Keb,1, Yiran Xub, Lanlan Jib, Nicole Paganea, Anching Liena, Tejasvi Matama, Elana J. Fertiga, Nathan D. Pricec, Laurent Younesb, Luigi Marchionnia,2, and Donald Gemanb,2 aDepartment of Oncology, Johns Hopkins University School of Medicine, Baltimore, MD 21205; bDepartment of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD 21218; and cInstitute for Systems Biology, Seattle, WA 98109 This contribution is part of the special series of Inaugural Articles by members of the National Academy of Sciences elected in 2015. Contributed by Donald Geman, March 19, 2018 (sent for review December 13, 2017; reviewed by Nicholas P. Tatonetti and Bin Yu) Data collected from omics technologies have revealed perva- relative to a baseline, may be essential to fully realize the poten- sive heterogeneity and stochasticity of molecular states within tial of high-dimensional omics assays; in principle, they pro- and between phenotypes. A prominent example of such het- vide unprecedented quantification of normal and disease-specific erogeneity occurs between genome-wide mRNA, microRNA, and variation, which can inform clinical approaches for diagnosis and methylation profiles from one individual tumor to another, even prevention of complex diseases. within a cancer subtype. However, current methods in bioinfor- To develop this high-dimensional analogue to low-dimensional matics, such as detecting differentially expressed genes or CpG clinical tests, we present a unified framework to characterize sites, are population-based and therefore do not effectively model normal and nonnormal variation by quantizing high-resolution intersample diversity. Here we introduce a unified theory to quan- measurements into a few discrete states based on divergence tify sample-level heterogeneity that is applicable to a single omics from a baseline. This simplified data representation is intended profile. Specifically, we simplify an omics profile to a digital repre- to capture most of the important biological information neces- sentation based on the omics profiles from a set of samples from sary to characterize phenotypes of interest. Briefly, suppose we a reference or baseline population (e.g., normal tissues). The state are given a particular subset of omics features, for example the of any subprofile (e.g., expression vector for a subset of genes) expression levels of any set of genes; important special cases are is said to be “divergent” if it lies outside the estimated support a single gene or a functional get set. Suppose also that sam- of the baseline distribution and is consequently interpreted as ple observations of this subset are available from a population STATISTICS “dysregulated” relative to that baseline. We focus on two cases: declared as the baseline. Then the realization of this subset of single features (e.g., individual genes) and distinguished subsets features in any sample profile is said to be “divergent” if it lies (e.g., regulatory pathways). Notably, since the divergence analy- outside the support of the baseline probability distribution. For sis is at the individual sample level, dysregulation can be analyzed instance, if the phenotypes of interest are cancers of a particular probabilistically; for example, one can estimate the probability tissue, then a natural “baseline” population is normal samples that a gene or pathway is divergent in some population. Finally, from that tissue. However, if the goal is to determine the unique the reduction in complexity facilitates a more “personalized” and features of gene regulation in particular (normal) tissue, the biologically interpretable analysis of variation, as illustrated by BIOPHYSICS AND experiments involving tissue characterization, disease detection Significance and progression, and disease–pathway associations. COMPUTATIONAL BIOLOGY Technological advances enable increasingly comprehensive stochasticity j digitization j dysregulation j cancer j precision medicine profiling of the molecular landscapes of cells, and these data can inform the personalized treatment of complex diseases. n recent decades, technological advances have enabled global Two major obstacles are the complexity of these data and the Iprofiling of genetic variants, RNA species, epigenetic marks, high degree of person-to-person heterogeneity. We develop proteins, metabolites, and other previously unknown molecular a highly simplified, personalized data representation by com- features, enabling the characterization of complex biological sys- paring the profile of an individual to the range of landscapes tems over distinct molecular domains. These high-dimensional in a baseline population, thereby mimicking basic clinical diag- measurements have been made on thousands of samples and are nostic testing for departures of selected variables from normal collectively referred to as omics data. Through complex com- levels. Moreover, our method can be applied to any data putational and statistical analyses of these data from different modality and at any level of granularity, from single fea- cell types and tissues, across individuals and model organisms, tures to any subset of features treated as a single entity, and from diseased and healthy conditions, we have substan- for example the gene expression levels in a pathway. Experi- tially enhanced our understanding of molecular mechanisms ments involve both healthy human tissues and various cancer underlying cell functioning, tissue organization, and organism subtypes. development. These data are also beginning to impact clinical Author contributions: L.M. and D.G. initiated the project; W.D., Q.K., E.J.F., N.D.P., L.Y., practice for complex diseases (1–3). L.M., and D.G. designed the experiments and interpreted the results; W.D., Q.K., Y.X., Nevertheless, omics data have yet to significantly inform the and L.J. performed research; W.D., N.P., A.L., T.M., and L.M. collected and curated data; standard of care, as was expected when introduced. In part, and W.D., E.J.F., L.Y., L.M., and D.G. wrote the paper. this limitation arises from insufficient genome-wide characteriza- Reviewers: N.P.T., Columbia University; and B.Y., University of California, Berkeley. tion of normal variation, impeding progress toward determining The authors declare no conflict of interest. whether a single omics profile reflects molecular dysregulation This open access article is distributed under Creative Commons Attribution- that is indicative of a complex disease. The use of deviation NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND). from a normal range is already widely used clinically for low- See QnAs on page 4528. dimensional laboratory tests and biomarkers, risk assessment, 1 W.D. and Q.K. contributed equally to this work. disease diagnosis and prognosis, and therapy selection. Similar 2 To whom correspondence may be addressed. Email: [email protected] or marchion@jhu. approaches are now needed for omics data modalities. Notably, edu. efforts to monitor large-scale omics data in the population are This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. now underway (4). Methods to quantify genome-wide variation, 1073/pnas.1721628115/-/DCSupplemental. especially techniques to determine where an omics profile falls Published online April 16, 2018. www.pnas.org/cgi/doi/10.1073/pnas.1721628115 PNAS j May 1, 2018 j vol. 115 j no. 18 j 4545–4552 Downloaded by guest on September 28, 2021 baseline could be all other tissues types combined. The novelty Divergence. Suppose we are given a reference joint probability distribution of our framework is that it applies to any subset of features and P0 of (Xj, j 2 J ) associated with some baseline phenotype. Of course, in prac- to any new profile by itself. tice, we only observe samples from this distribution where each sample is Divergence is a sample property. In contrast, putative discrim- an omics profile from a single data modality, and the choice of reference inating sets of features, such as “differentially expressed genes,” phenotype depends on the particular study and data modality. As will be seen in Results, the baseline phenotype need not be a single tissue type or a are defined in terms of population-level distributions. For exam- disease-free phenotype and can be any population that is meaningful under ple, given two classes or phenotypes, a gene is differentially the analysis carried out. expressed if its distribution differs in the two classes. However, Given an omics profile X and a subset S ⊂ J , we will apply the defi- S S knowing the status of a molecular feature in the context of pop- nition of divergence to the random vector U = Q (X) = (Qj(X), j 2 S). We ulations may say very little about the status of that feature for will focus on single features S = fjg (e.g., single genes) and on sets S a particular individual. Divergence then enables the prospec- with 1 jSj jJ j (e.g., gene pathways). Let m = jSj, so that US takes m S m S tive analysis of individual samples, as required for applications values in [0, 1] . We describe below how the support U0 ⊂ [0, 1] of U can be estimated from the data, resulting in a set U^S that is explic- to precision and personalized medicine. It also enables a proba- 0 bilistic analysis at the phenotype level, where the probability that itly defined. When presented with a new sample X, we define a binary variable by a specific feature (say gene), or set of features (say pathway), is ( S S 1 if Q (X) 2= U^ , divergent is a well-defined concept. Phenotypes can then be char- Z = 0 acterized based on the corresponding divergence probabilities 0 otherwise : and predicted for unlabeled samples by examining the diver- We will then say that the set S is divergent (for the considered sample) if gent set. This approach can be applied to any high-dimensional and only if Z = 1. omics dataset, yielding simplified matrices of digitized values that can be analyzed statistically at the feature, sample, or Support Estimation.
Load more

Recommended publications
  • IR-IITBHU at TREC 2016 Open Search Track: Retrieving Documents Using Divergence from Randomness Model in Terrier
    IR-IITBHU at TREC 2016 Open Search Track: Retrieving documents using Divergence From Randomness model in Terrier Mitodru Niyogi1 and Sukomal Pal2 1Department of Information Technology, Government College of Engineering & Ceramic Technology, Kolkata 2Department of Computer Science & Engineering, Indian Institute of Technology(BHU), Varanasi Abstract In our participation at TREC 2016 Open Search Track which focuses on ad-hoc scientic literature search, we used Terrier, a modular and a scalable Information Retrieval framework as a tool to rank documents. The organizers provided live data as documents, queries and user interac- tions from real search engine that were available through Living Lab API framework. The data was then converted into TREC format to be used in Terrier. We used Divergence from Randomness (DFR) model, specically, the Inverse expected document frequency model for randomness, the ratio of two Bernoulli's processes for rst normalisation, and normalisation 2 for term frequency normalization with natural logarithm, i.e., In_expC2 model to rank the available documents in response to a set of queries. Al- together we submit 391 runs for sites CiteSeerX and SSOAR to the Open Search Track via the Living Lab API framework. We received an `out- come' of 0.72 for test queries and 0.62 for train queries of site CiteSeerX at the end of Round 3 Test Period where, the outcome is computed as: #wins / (#wins + #losses). A `win' occurs when the participant achieves more clicks on his documents than those of the site and `loss' otherwise. Complete relevance judgments is awaited at the moment. We look forward to getting the users' feedback and work further with them.
    [Show full text]
  • Subnational Inequality Divergence
    Subnational Inequality Divergence Tom VanHeuvelen1 University of Minnesota Department of Sociology Abstract How have inequality levels across local labor markets in the subnational United States changed over the past eight decades? In this study, I examine inequality divergence, or the inequality of inequalities. While divergence trends of central tendencies such as per capita income have been well documented, less is known about the descriptive trends or contributing mechanisms for inequality. In this study, I construct wage inequality measures in 722 local labor markets covering the entire contiguous United States across 22 waves of Census and American Community Survey data from 1940-2019 to assess the historical trends of inequality divergence. I apply variance decomposition and counterfactual techniques to develop main conclusions. Inequality divergence follows a u-shaped pattern, declining through 1990 but with contemporary divergence at as high a level as any time in the past 80 years. Early era convergence occurred broadly and primarily worked to reduce interregional differences, whereas modern inequality divergence operates through a combination of novel mechanisms, most notably through highly unequal urban areas separating from other labor markets. Overall, results show geographical fragmentation of inequality underneath overall inequality growth in recent years, highlighting the fundamental importance of spatial trends for broader stratification outcomes. 1 Correspondence: [email protected]. A previous version of this manuscript was presented at the 2021 Population Association of American annual conference. Thank you to Jane VanHeuvelen and Peter Catron for their helpful comments. Recent changes in the United States have situated geographical residence as a key pillar of the unequal distribution of economic resources (Austin et al.
    [Show full text]
  • A Generalized Divergence for Statistical Inference
    A GENERALIZED DIVERGENCE FOR STATISTICAL INFERENCE Abhik Ghosh, Ian R. Harris, Avijit Maji Ayanendranath Basu and Leandro Pardo TECHNICAL REPORT NO. BIRU/2013/3 2013 BAYESIAN AND INTERDISCIPLINARY RESEARCH UNIT INDIAN STATISTICAL INSTITUTE 203, Barrackpore Trunk Road Kolkata – 700 108 INDIA A Generalized Divergence for Statistical Inference Abhik Ghosh Indian Statistical Institute, Kolkata, India. Ian R. Harris Southern Methodist University, Dallas, USA. Avijit Maji Indian Statistical Institute, Kolkata, India. Ayanendranath Basu Indian Statistical Institute, Kolkata, India. Leandro Pardo Complutense University, Madrid, Spain. Summary. The power divergence (PD) and the density power divergence (DPD) families have proved to be useful tools in the area of robust inference. The families have striking similarities, but also have fundamental differences; yet both families are extremely useful in their own ways. In this paper we provide a comprehensive description of the two families and tie in their role in statistical theory and practice. At the end, the families are seen to be a part of a superfamily which contains both of these families as special cases. In the process, the limitation of the influence function as an effective descriptor of the robustness of the estimator is also demonstrated. Keywords: Robust Estimation, Divergence, Influence Function 2 Ghosh et al. 1. Introduction The density-based minimum divergence approach is an useful technique in para- metric inference. Here the closeness of the data and the model is quantified by a suitable measure of density-based divergence between the data density and the model density. Many of these methods have been particularly useful because of the strong robustness properties that they inherently possess.
    [Show full text]
  • Divergence Measures for Statistical Data Processing—An Annotated Bibliography
    Signal Processing 93 (2013) 621–633 Contents lists available at SciVerse ScienceDirect Signal Processing journal homepage: www.elsevier.com/locate/sigpro Review Divergence measures for statistical data processing—An annotated bibliography Michele Basseville n,1 IRISA, Campus de Beaulieu, 35042 Rennes Cedex, France article info abstract Article history: This paper provides an annotated bibliography for investigations based on or related to Received 19 April 2012 divergence measures for statistical data processing and inference problems. Received in revised form & 2012 Elsevier B.V. All rights reserved. 17 July 2012 Accepted 5 September 2012 Available online 14 September 2012 Keywords: Divergence Distance Information f-Divergence Bregman divergence Learning Estimation Detection Classification Recognition Compression Indexing Contents 1. Introduction ......................................................................................... 621 2. f-Divergences ........................................................................................ 622 3. Bregman divergences.................................................................................. 623 4. a-Divergences ....................................................................................... 624 5. Handling more than two distributions .................................................................... 625 6. Inference based on entropy and divergence criteria.......................................................... 626 7. Spectral divergence measures . .......................................................................
    [Show full text]
  • Robust Hypothesis Testing with Α-Divergence Gokhan¨ Gul,¨ Student Member, IEEE, Abdelhak M
    1 Robust Hypothesis Testing with α-Divergence Gokhan¨ Gul,¨ Student Member, IEEE, Abdelhak M. Zoubir, Fellow, IEEE, Abstract—A robust minimax test for two composite hypotheses, effects that go unmodeled [2]. which are determined by the neighborhoods of two nominal dis- Imprecise knowledge of F0 or F1 leads, in general, to per- tributions with respect to a set of distances - called α−divergence formance degradation and a useful approach is to extend the distances, is proposed. Sion’s minimax theorem is adopted to characterize the saddle value condition. Least favorable distri- known model by accepting a set of distributions Fi, under each butions, the robust decision rule and the robust likelihood ratio hypothesis Hi, that are populated by probability distributions test are derived. If the nominal probability distributions satisfy Gi, which are at the neighborhood of the nominal distribution a symmetry condition, the design procedure is shown to be Fi based on some distance D [1]. Under some mild conditions simplified considerably. The parameters controlling the degree of on D, it can be shown that the best (error minimizing) decision robustness are bounded from above and the bounds are shown to ^ be resulting from a solution of a set of equations. The simulations rule δ for the worst case (error maximizing) pair of probability ^ ^ performed evaluate and exemplify the theoretical derivations. distributions (G0; G1) 2 F0 × F1 accepts a saddle value. Therefore, such a test design guarantees a certain level of Index Terms—Detection, hypothesis testing, robustness, least favorable distributions, minimax optimization, likelihood ratio detection at all times. This type of optimization is known test.
    [Show full text]
  • Divergence Tendencies in the European Integration Process: a Danger for the Sustainability of the E(M)U?
    Journal of Risk and Financial Management Article Divergence Tendencies in the European Integration Process: A Danger for the Sustainability of the E(M)U? Linda Glawe and Helmut Wagner * Faculty of Economics, University of Hagen, 58084 Hagen, Germany; [email protected] * Correspondence: [email protected] Abstract: The European integration process started with the aim of reducing the differences in income and/or living standards between the participating countries over time. To achieve this, a certain alignment of institutions and structures was seen as a necessary precondition. While the goal of this income and institutional convergence was successfully achieved over a long period of time, this convergence development has weakened or even turned into divergence in the last one to two decades. This paper provides an overview of the empirical evidence for these convergence and divergence developments and develops policy implications (the challenges and possible ways out). Keywords: European integration process; convergence dynamics; divergence; institutional quality; economic growth and economic development; European Union JEL Classification: F55; O10; O11; O43; O52 Citation: Glawe, Linda, and Helmut Wagner. 2021. Divergence Tendencies 1. Introduction in the European Integration Process: It is conventional wisdom that the larger and more complex a project is, the greater the A Danger for the Sustainability of the risk that it will fail. This also applies to the project of European integration. This project was E(M)U? Journal of Risk and Financial initially small and therefore manageable and also successful. Over time, however, more and Management 14: 104. https:// more extensions were made (in terms of scope, and in terms of areas of application/depth), doi.org/10.3390/jrfm14030104 so that it has become increasingly difficult and complex to keep the situation/project stable/sustainable.
    [Show full text]
  • GDP PER CAPITA VERSUS MEDIAN HOUSEHOLD INCOME: WHAT GIVES RISE to DIVERGENCE OVER TIME? Brian Nolan, Max Roser, and Stefan
    GDP PER CAPITA VERSUS MEDIAN HOUSEHOLD INCOME: WHAT GIVES RISE TO DIVERGENCE OVER TIME? Brian Nolan, Max Roser, and Stefan Thewissen INET Oxford WorkinG Paper no. 2016-03 Employment, Equity and Growth proGramme GDP per capita versus median household income: What gives rise to divergence over time?⇤ § Brian Nolan,† Max Roser,‡ and Stefan Thewissen May 2016 Abstract Divergence between the evolution of GDP per capita and the income of a‘typical’householdasmeasuredinhouseholdsurveysisgivingrisetoa range of serious concerns, especially in the USA. This paper investigates the extent of that divergence and the factors that contribute to it across 27 OECD countries, using data from OECD National Accounts and the Luxembourg Income Study. While GDP per capita has risen faster than median household income in most of these countries over the period these data cover, the size of that divergence varied very substantially, with the USA a clear outlier. The paper distinguishes a number of factors contributing to such a divergence, and finds wide variation across countries in the impact of the various factors. Further, both the extent of that divergence and the role of the various contributory factors vary widely over time for most of the countries studied. These findings have serious implications for the monitoring and assessment of changes in household incomes and living standards over time. keywords: Economic growth, inequality, household incomes ⇤The authors are grateful for comments and advice from Tony Atkinson (University of Oxford) and Facundo Alvaredo (Paris School of Economics). †Institute for New Economic Thinking at the Oxford Martin School, Department of Social Policy and Intervention, and Nuffield College, University of Oxford.
    [Show full text]
  • On the Use of Entropy to Improve Model Selection Criteria
    entropy Article On the Use of Entropy to Improve Model Selection Criteria Andrea Murari 1, Emmanuele Peluso 2, Francesco Cianfrani 2, Pasquale Gaudio 2 and Michele Lungaroni 2,* 1 Consorzio RFX (CNR, ENEA, INFN, Universita’ di Padova, Acciaierie Venete SpA), 35127 Padova, Italy; [email protected] 2 Department of Industrial Engineering, University of Rome “Tor Vergata”, 00133 Roma, Italy; [email protected] (E.P.); [email protected] (F.C.); [email protected] (P.G.) * Correspondence: [email protected]; Tel.: +39-(0)6-7259-7196 Received: 21 January 2019; Accepted: 10 April 2019; Published: 12 April 2019 Abstract: The most widely used forms of model selection criteria, the Bayesian Information Criterion (BIC) and the Akaike Information Criterion (AIC), are expressed in terms of synthetic indicators of the residual distribution: the variance and the mean-squared error of the residuals respectively. In many applications in science, the noise affecting the data can be expected to have a Gaussian distribution. Therefore, at the same level of variance and mean-squared error, models, whose residuals are more uniformly distributed, should be favoured. The degree of uniformity of the residuals can be quantified by the Shannon entropy. Including the Shannon entropy in the BIC and AIC expressions improves significantly these criteria. The better performances have been demonstrated empirically with a series of simulations for various classes of functions and for different levels and statistics of the noise. In presence of outliers, a better treatment of the errors, using the Geodesic Distance, has proved essential. Keywords: Model Selection Criteria; Bayesian Information Criterion (BIC); Akaike Information Criterion (AIC); Shannon Entropy; Geodesic Distance 1.
    [Show full text]
  • Stagnating Median Incomes Despite Economic Growth: Explaining the Divergence in 27 OECD Countries
    12/3/2018 Economic growth with stagnating median incomes: New analysis | VOX, CEPR Policy Portal VOX CEPR Policy Portal Create account | Login | Subscribe Research-based policy analysis and commentary from leading economists Search Columns Video Vox Vox Talks Publications Blogs&Reviews People Debates Events About By Topic By Date By Reads By Tag Stagnating median incomes despite economic growth: Explaining the divergence in 27 OECD countries Brian Nolan, Max Roser, Stefan Thewissen 27 August 2016 Brian Nolan With inequality rising and household incomes across developed countries stagnating, accurate monitoring of Professor of Social Policy and living standards cannot be achieved by relying on GDP per capita alone. This column analyses the path of Director of Employment, Equity, divergence between household income and GDP per capita for 27 OECD countries. It finds several reasons why and Growth Programme, Institute GDP per capita has outpaced median incomes, and recommends assigning median income a central place in for New Economic Thinking; official monitoring and assessment of living standards over time. Senior Research Fellow, Nuffield College, University of Oxford 255 A A Increasing inequality and stagnating incomes for Related ordinary workers and their families are now at the centre Growth quality: A new index of economic and political debate across the rich Montfort Mlachila, René Tapsoba, Sampawende Tapsoba countries. Against that backdrop, the fact that median Max Roser household income has lagged so far behind growth in The turtle’s progress: Secular stagnation meets the headwinds Programme Director, Oxford GDP per capita in the US – widely attributed to growing Robert J. Gordon Martin School, University of inequality – has reinforced concerns about relying on Global income distribution since 1988 Oxford GDP as the core measure of economic prosperity Christoph Lakner, Branko Milanovic (Stiglitz et al.
    [Show full text]
  • Lecture 7: Hypothesis Testing and KL Divergence 1 Introducing The
    ECE 830 Spring 2015 Statistical Signal Processing instructor: R. Nowak Lecture 7: Hypothesis Testing and KL Divergence 1 Introducing the Kullback-Leibler Divergence iid Suppose X1;X2;:::;Xn ∼ q(x) and we have two models for q(x), p0(x) and p1(x). In past lectures we have seen that the likelihood ratio test (LRT) is optimal, assuming that q is p0 or p1. The error probabilities can be computed numerically in many cases. The error probabilities converge to 0 as the number of samples n grows, but numerical calculations do not always yield insight into rate of convergence. In this lecture we will see that the rate is exponential in n and parameterized the Kullback-Leibler (KL) divergence, which quantifies the differences between the distributions p0 and p1. Our analysis will also give insight into the performance of the LRT when q is neither p0 nor p1. This is important since in practice p0 and p1 may be imperfect models for reality, q in this context. The LRT acts as one would expect in such cases, it picks the model that is closest (in the sense of KL divergence) to q. To begin our discusion, recall the likelihood ratio is n Y p1(xi) Λ = p (x ) i=1 0 i The log likelihood ratio, normalized by dividing by n, is then n 1 X p1(xi) Λ^ = log n n p (x ) i=1 0 i p1(xi) Note that Λ^ n is itself a random variable, and is in fact a sum of iid random variables Li = log which p0(xi) are independent because the xi are.
    [Show full text]
  • Improved Security Proofs in Lattice-Based Cryptography: Using the Rényi Divergence Rather Than the Statistical Distance
    Improved security proofs in lattice-based cryptography: using the Rényi divergence rather than the statistical distance Shi Bai1, Tancrède Lepoint3, Adeline Roux-Langlois4, Amin Sakzad5, Damien Stehlé2, and Ron Steinfeld5 1 Department of Mathematical Sciences, Florida Atlantic University, USA. [email protected] – http://cosweb1.fau.edu/~sbai 2 ENS de Lyon, Laboratoire LIP (U. Lyon, CNRS, ENSL, INRIA, UCBL), France. [email protected] – http://perso.ens-lyon.fr/damien.stehle 3 SRI International, USA. [email protected] – https://tlepoint.github.io/ 4 CNRS/IRISA, Rennes, France. [email protected] – http://people.irisa.fr/Adeline.Roux-Langlois/ 5 Faculty of Information Technology, Monash University, Clayton, Australia. [email protected], [email protected] – http://users.monash.edu.au/~rste/ Abstract. The Rényi divergence is a measure of closeness of two prob- ability distributions. We show that it can often be used as an alternative to the statistical distance in security proofs for lattice-based cryptogra- phy. Using the Rényi divergence is particularly suited for security proofs of primitives in which the attacker is required to solve a search problem (e.g., forging a signature). We show that it may also be used in the case of distinguishing problems (e.g., semantic security of encryption schemes), when they enjoy a public sampleability property. The techniques lead to security proofs for schemes with smaller parameters, and sometimes to simpler security proofs than the existing ones. 1 Introduction Let D1 and D2 be two non-vanishing probability distributions over a common measurable support X. Let a ∈ (1, +∞). The Rényi divergence [Rén61,vEH14] (RD for short) Ra(D1kD2) of order a between D1 and D2 is defined as the a−1 ((a − 1)th root of the) expected value of (D1(x)/D2(x)) over the randomness of x sampled from D1.
    [Show full text]
  • Minimum Description Length Model Selection
    Minimum Description Length Model Selection Problems and Extensions Steven de Rooij Minimum Description Length Model Selection Problems and Extensions ILLC Dissertation Series DS-2008-07 For further information about ILLC-publications, please contact Institute for Logic, Language and Computation Universiteit van Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam phone: +31-20-525 6051 fax: +31-20-525 5206 e-mail: [email protected] homepage: http://www.illc.uva.nl/ Minimum Description Length Model Selection Problems and Extensions Academisch Proefschrift ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magnificus prof.dr. D. C. van den Boom ten overstaan van een door het college voor promoties ingestelde commissie, in het openbaar te verdedigen in de Agnietenkapel op woensdag 10 september 2008, te 12.00 uur door Steven de Rooij geboren te Amersfoort. Promotiecommissie: Promotor: prof.dr.ir. P.M.B. Vit´anyi Co-promotor: dr. P.D. Grunwald¨ Overige leden: prof.dr. P.W. Adriaans prof.dr. A.P. Dawid prof.dr. C.A.J. Klaassen prof.dr.ir. R.J.H. Scha dr. M.W. van Someren prof.dr. V. Vovk dr.ir. F.M.J. Willems Faculteit der Natuurwetenschappen, Wiskunde en Informatica The investigations were funded by the Netherlands Organization for Scientific Research (NWO), project 612.052.004 on Universal Learning, and were supported in part by the IST Programme of the European Community, under the PASCAL Network of Excellence, IST-2002-506778. Copyright c 2008 by Steven de Rooij Cover inspired by Randall Munroe’s xkcd webcomic (www.xkcd.org) Printed and bound by PrintPartners Ipskamp (www.ppi.nl) ISBN: 978-90-5776-181-2 Parts of this thesis are based on material contained in the following papers: An Empirical Study of Minimum Description Length Model Selection with Infinite • Parametric Complexity Steven de Rooij and Peter Grunwald¨ In: Journal of Mathematical Psychology, special issue on model selection.
    [Show full text]