Randomness and Degrees of Irregularity (Approximate Entropy/Computable/Maximally Random Sequences/Normal Numbers/Chaos) STEVE PINCUS*T and BURTON H
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Measure-Theoretic Probability I
Measure-Theoretic Probability I Steven P.Lalley Winter 2017 1 1 Measure Theory 1.1 Why Measure Theory? There are two different views – not necessarily exclusive – on what “probability” means: the subjectivist view and the frequentist view. To the subjectivist, probability is a system of laws that should govern a rational person’s behavior in situations where a bet must be placed (not necessarily just in a casino, but in situations where a decision must be made about how to proceed when only imperfect information about the outcome of the decision is available, for instance, should I allow Dr. Scissorhands to replace my arthritic knee by a plastic joint?). To the frequentist, the laws of probability describe the long- run relative frequencies of different events in “experiments” that can be repeated under roughly identical conditions, for instance, rolling a pair of dice. For the frequentist inter- pretation, it is imperative that probability spaces be large enough to allow a description of an experiment, like dice-rolling, that is repeated infinitely many times, and that the mathematical laws should permit easy handling of limits, so that one can make sense of things like “the probability that the long-run fraction of dice rolls where the two dice sum to 7 is 1/6”. But even for the subjectivist, the laws of probability should allow for description of situations where there might be a continuum of possible outcomes, or pos- sible actions to be taken. Once one is reconciled to the need for such flexibility, it soon becomes apparent that measure theory (the theory of countably additive, as opposed to merely finitely additive measures) is the only way to go. -
Probability and Statistics Lecture Notes
Probability and Statistics Lecture Notes Antonio Jiménez-Martínez Chapter 1 Probability spaces In this chapter we introduce the theoretical structures that will allow us to assign proba- bilities in a wide range of probability problems. 1.1. Examples of random phenomena Science attempts to formulate general laws on the basis of observation and experiment. The simplest and most used scheme of such laws is: if a set of conditions B is satisfied =) event A occurs. Examples of such laws are the law of gravity, the law of conservation of mass, and many other instances in chemistry, physics, biology... If event A occurs inevitably whenever the set of conditions B is satisfied, we say that A is certain or sure (under the set of conditions B). If A can never occur whenever B is satisfied, we say that A is impossible (under the set of conditions B). If A may or may not occur whenever B is satisfied, then A is said to be a random phenomenon. Random phenomena is our subject matter. Unlike certain and impossible events, the presence of randomness implies that the set of conditions B do not reflect all the necessary and sufficient conditions for the event A to occur. It might seem them impossible to make any worthwhile statements about random phenomena. However, experience has shown that many random phenomena exhibit a statistical regularity that makes them subject to study. For such random phenomena it is possible to estimate the chance of occurrence of the random event. This estimate can be obtained from laws, called probabilistic or stochastic, with the form: if a set of conditions B is satisfied event A occurs m times =) repeatedly n times out of the n repetitions. -
3 Autocorrelation
3 Autocorrelation Autocorrelation refers to the correlation of a time series with its own past and future values. Autocorrelation is also sometimes called “lagged correlation” or “serial correlation”, which refers to the correlation between members of a series of numbers arranged in time. Positive autocorrelation might be considered a specific form of “persistence”, a tendency for a system to remain in the same state from one observation to the next. For example, the likelihood of tomorrow being rainy is greater if today is rainy than if today is dry. Geophysical time series are frequently autocorrelated because of inertia or carryover processes in the physical system. For example, the slowly evolving and moving low pressure systems in the atmosphere might impart persistence to daily rainfall. Or the slow drainage of groundwater reserves might impart correlation to successive annual flows of a river. Or stored photosynthates might impart correlation to successive annual values of tree-ring indices. Autocorrelation complicates the application of statistical tests by reducing the effective sample size. Autocorrelation can also complicate the identification of significant covariance or correlation between time series (e.g., precipitation with a tree-ring series). Autocorrelation implies that a time series is predictable, probabilistically, as future values are correlated with current and past values. Three tools for assessing the autocorrelation of a time series are (1) the time series plot, (2) the lagged scatterplot, and (3) the autocorrelation function. 3.1 Time series plot Positively autocorrelated series are sometimes referred to as persistent because positive departures from the mean tend to be followed by positive depatures from the mean, and negative departures from the mean tend to be followed by negative departures (Figure 3.1). -
The Bayesian Approach to Statistics
THE BAYESIAN APPROACH TO STATISTICS ANTHONY O’HAGAN INTRODUCTION the true nature of scientific reasoning. The fi- nal section addresses various features of modern By far the most widely taught and used statisti- Bayesian methods that provide some explanation for the rapid increase in their adoption since the cal methods in practice are those of the frequen- 1980s. tist school. The ideas of frequentist inference, as set out in Chapter 5 of this book, rest on the frequency definition of probability (Chapter 2), BAYESIAN INFERENCE and were developed in the first half of the 20th century. This chapter concerns a radically differ- We first present the basic procedures of Bayesian ent approach to statistics, the Bayesian approach, inference. which depends instead on the subjective defini- tion of probability (Chapter 3). In some respects, Bayesian methods are older than frequentist ones, Bayes’s Theorem and the Nature of Learning having been the basis of very early statistical rea- Bayesian inference is a process of learning soning as far back as the 18th century. Bayesian from data. To give substance to this statement, statistics as it is now understood, however, dates we need to identify who is doing the learning and back to the 1950s, with subsequent development what they are learning about. in the second half of the 20th century. Over that time, the Bayesian approach has steadily gained Terms and Notation ground, and is now recognized as a legitimate al- ternative to the frequentist approach. The person doing the learning is an individual This chapter is organized into three sections. -
Determinism, Indeterminism and the Statistical Postulate
Tipicality, Explanation, The Statistical Postulate, and GRW Valia Allori Northern Illinois University [email protected] www.valiaallori.com Rutgers, October 24-26, 2019 1 Overview Context: Explanation of the macroscopic laws of thermodynamics in the Boltzmannian approach Among the ingredients: the statistical postulate (connected with the notion of probability) In this presentation: typicality Def: P is a typical property of X-type object/phenomena iff the vast majority of objects/phenomena of type X possesses P Part I: typicality is sufficient to explain macroscopic laws – explanatory schema based on typicality: you explain P if you explain that P is typical Part II: the statistical postulate as derivable from the dynamics Part III: if so, no preference for indeterministic theories in the quantum domain 2 Summary of Boltzmann- 1 Aim: ‘derive’ macroscopic laws of thermodynamics in terms of the microscopic Newtonian dynamics Problems: Technical ones: There are to many particles to do exact calculations Solution: statistical methods - if 푁 is big enough, one can use suitable mathematical technique to obtain information of macro systems even without having the exact solution Conceptual ones: Macro processes are irreversible, while micro processes are not Boltzmann 3 Summary of Boltzmann- 2 Postulate 1: the microscopic dynamics microstate 푋 = 푟1, … . 푟푁, 푣1, … . 푣푁 in phase space Partition of phase space into Macrostates Macrostate 푀(푋): set of macroscopically indistinguishable microstates Macroscopic view Many 푋 for a given 푀 given 푀, which 푋 is unknown Macro Properties (e.g. temperature): they slowly vary on the Macro scale There is a particular Macrostate which is incredibly bigger than the others There are many ways more to have, for instance, uniform temperature than not equilibrium (Macro)state 4 Summary of Boltzmann- 3 Entropy: (def) proportional to the size of the Macrostate in phase space (i.e. -
Topic 1: Basic Probability Definition of Sets
Topic 1: Basic probability ² Review of sets ² Sample space and probability measure ² Probability axioms ² Basic probability laws ² Conditional probability ² Bayes' rules ² Independence ² Counting ES150 { Harvard SEAS 1 De¯nition of Sets ² A set S is a collection of objects, which are the elements of the set. { The number of elements in a set S can be ¯nite S = fx1; x2; : : : ; xng or in¯nite but countable S = fx1; x2; : : :g or uncountably in¯nite. { S can also contain elements with a certain property S = fx j x satis¯es P g ² S is a subset of T if every element of S also belongs to T S ½ T or T S If S ½ T and T ½ S then S = T . ² The universal set is the set of all objects within a context. We then consider all sets S ½ . ES150 { Harvard SEAS 2 Set Operations and Properties ² Set operations { Complement Ac: set of all elements not in A { Union A \ B: set of all elements in A or B or both { Intersection A [ B: set of all elements common in both A and B { Di®erence A ¡ B: set containing all elements in A but not in B. ² Properties of set operations { Commutative: A \ B = B \ A and A [ B = B [ A. (But A ¡ B 6= B ¡ A). { Associative: (A \ B) \ C = A \ (B \ C) = A \ B \ C. (also for [) { Distributive: A \ (B [ C) = (A \ B) [ (A \ C) A [ (B \ C) = (A [ B) \ (A [ C) { DeMorgan's laws: (A \ B)c = Ac [ Bc (A [ B)c = Ac \ Bc ES150 { Harvard SEAS 3 Elements of probability theory A probabilistic model includes ² The sample space of an experiment { set of all possible outcomes { ¯nite or in¯nite { discrete or continuous { possibly multi-dimensional ² An event A is a set of outcomes { a subset of the sample space, A ½ . -
Measure Theory and Probability
Measure theory and probability Alexander Grigoryan University of Bielefeld Lecture Notes, October 2007 - February 2008 Contents 1 Construction of measures 3 1.1Introductionandexamples........................... 3 1.2 σ-additive measures ............................... 5 1.3 An example of using probability theory . .................. 7 1.4Extensionofmeasurefromsemi-ringtoaring................ 8 1.5 Extension of measure to a σ-algebra...................... 11 1.5.1 σ-rings and σ-algebras......................... 11 1.5.2 Outermeasure............................. 13 1.5.3 Symmetric difference.......................... 14 1.5.4 Measurable sets . ............................ 16 1.6 σ-finitemeasures................................ 20 1.7Nullsets..................................... 23 1.8 Lebesgue measure in Rn ............................ 25 1.8.1 Productmeasure............................ 25 1.8.2 Construction of measure in Rn. .................... 26 1.9 Probability spaces ................................ 28 1.10 Independence . ................................. 29 2 Integration 38 2.1 Measurable functions.............................. 38 2.2Sequencesofmeasurablefunctions....................... 42 2.3 The Lebesgue integral for finitemeasures................... 47 2.3.1 Simplefunctions............................ 47 2.3.2 Positivemeasurablefunctions..................... 49 2.3.3 Integrablefunctions........................... 52 2.4Integrationoversubsets............................ 56 2.5 The Lebesgue integral for σ-finitemeasure................. -
A Review of Graph and Network Complexity from an Algorithmic Information Perspective
entropy Review A Review of Graph and Network Complexity from an Algorithmic Information Perspective Hector Zenil 1,2,3,4,5,*, Narsis A. Kiani 1,2,3,4 and Jesper Tegnér 2,3,4,5 1 Algorithmic Dynamics Lab, Centre for Molecular Medicine, Karolinska Institute, 171 77 Stockholm, Sweden; [email protected] 2 Unit of Computational Medicine, Department of Medicine, Karolinska Institute, 171 77 Stockholm, Sweden; [email protected] 3 Science for Life Laboratory (SciLifeLab), 171 77 Stockholm, Sweden 4 Algorithmic Nature Group, Laboratoire de Recherche Scientifique (LABORES) for the Natural and Digital Sciences, 75005 Paris, France 5 Biological and Environmental Sciences and Engineering Division (BESE), King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia * Correspondence: [email protected] or [email protected] Received: 21 June 2018; Accepted: 20 July 2018; Published: 25 July 2018 Abstract: Information-theoretic-based measures have been useful in quantifying network complexity. Here we briefly survey and contrast (algorithmic) information-theoretic methods which have been used to characterize graphs and networks. We illustrate the strengths and limitations of Shannon’s entropy, lossless compressibility and algorithmic complexity when used to identify aspects and properties of complex networks. We review the fragility of computable measures on the one hand and the invariant properties of algorithmic measures on the other demonstrating how current approaches to algorithmic complexity are misguided and suffer of similar limitations than traditional statistical approaches such as Shannon entropy. Finally, we review some current definitions of algorithmic complexity which are used in analyzing labelled and unlabelled graphs. This analysis opens up several new opportunities to advance beyond traditional measures. -
Introduction to Bayesian Inference and Modeling Edps 590BAY
Introduction to Bayesian Inference and Modeling Edps 590BAY Carolyn J. Anderson Department of Educational Psychology c Board of Trustees, University of Illinois Fall 2019 Introduction What Why Probability Steps Example History Practice Overview ◮ What is Bayes theorem ◮ Why Bayesian analysis ◮ What is probability? ◮ Basic Steps ◮ An little example ◮ History (not all of the 705+ people that influenced development of Bayesian approach) ◮ In class work with probabilities Depending on the book that you select for this course, read either Gelman et al. Chapter 1 or Kruschke Chapters 1 & 2. C.J. Anderson (Illinois) Introduction Fall 2019 2.2/ 29 Introduction What Why Probability Steps Example History Practice Main References for Course Throughout the coures, I will take material from ◮ Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A., & Rubin, D.B. (20114). Bayesian Data Analysis, 3rd Edition. Boco Raton, FL, CRC/Taylor & Francis.** ◮ Hoff, P.D., (2009). A First Course in Bayesian Statistical Methods. NY: Sringer.** ◮ McElreath, R.M. (2016). Statistical Rethinking: A Bayesian Course with Examples in R and Stan. Boco Raton, FL, CRC/Taylor & Francis. ◮ Kruschke, J.K. (2015). Doing Bayesian Data Analysis: A Tutorial with JAGS and Stan. NY: Academic Press.** ** There are e-versions these of from the UofI library. There is a verson of McElreath, but I couldn’t get if from UofI e-collection. C.J. Anderson (Illinois) Introduction Fall 2019 3.3/ 29 Introduction What Why Probability Steps Example History Practice Bayes Theorem A whole semester on this? p(y|θ)p(θ) p(θ|y)= p(y) where ◮ y is data, sample from some population. -
1 Probability Measure and Random Variables
1 Probability measure and random variables 1.1 Probability spaces and measures We will use the term experiment in a very general way to refer to some process that produces a random outcome. Definition 1. The set of possible outcomes is called the sample space. We will typically denote an individual outcome by ω and the sample space by Ω. Set notation: A B, A is a subset of B, means that every element of A is also in B. The union⊂ A B of A and B is the of all elements that are in A or B, including those that∪ are in both. The intersection A B of A and B is the set of all elements that are in both of A and B. ∩ n j=1Aj is the set of elements that are in at least one of the Aj. ∪n j=1Aj is the set of elements that are in all of the Aj. ∩∞ ∞ j=1Aj, j=1Aj are ... Two∩ sets A∪ and B are disjoint if A B = . denotes the empty set, the set with no elements. ∩ ∅ ∅ Complements: The complement of an event A, denoted Ac, is the set of outcomes (in Ω) which are not in A. Note that the book writes it as Ω A. De Morgan’s laws: \ (A B)c = Ac Bc ∪ ∩ (A B)c = Ac Bc ∩ ∪ c c ( Aj) = Aj j j [ \ c c ( Aj) = Aj j j \ [ (1) Definition 2. Let Ω be a sample space. A collection of subsets of Ω is a σ-field if F 1. -
Autocorrelation
Autocorrelation David Gerbing School of Business Administration Portland State University January 30, 2016 Autocorrelation The difference between an actual data value and the forecasted data value from a model is the residual for that forecasted value. Residual: ei = Yi − Y^i One of the assumptions of the least squares estimation procedure for the coefficients of a regression model is that the residuals are purely random. One consequence of randomness is that the residuals would not correlate with anything else, including with each other at different time points. A value above the mean, that is, a value with a positive residual, would contain no information as to whether the next value in time would have a positive residual, or negative residual, with a data value below the mean. For example, flipping a fair coin yields random flips, with half of the flips resulting in a Head and the other half a Tail. If a Head is scored as a 1 and a Tail as a 0, and the probability of both Heads and Tails is 0.5, then calculate the value of the population mean as: Population Mean: µ = (0:5)(1) + (0:5)(0) = :5 The forecast of the outcome of the next flip of a fair coin is the mean of the process, 0.5, which is stable over time. What are the corresponding residuals? A residual value is the difference of the corresponding data value minus the mean. With this scoring system, a Head generates a positive residual from the mean, µ, Head: ei = 1 − µ = 1 − 0:5 = 0:5 A Tail generates a negative residual from the mean, Tail: ei = 0 − µ = 0 − 0:5 = −0:5 The error terms of the coin flips are independent of each other, so if the current flip is a Head, or if the last 5 flips are Heads, the forecast for the next flip is still µ = :5. -
Mild Vs. Wild Randomness: Focusing on Those Risks That Matter
with empirical sense1. Granted, it has been Mild vs. Wild Randomness: tinkered with using such methods as Focusing on those Risks that complementary “jumps”, stress testing, regime Matter switching or the elaborate methods known as GARCH, but while they represent a good effort, Benoit Mandelbrot & Nassim Nicholas Taleb they fail to remediate the bell curve’s irremediable flaws. Forthcoming in, The Known, the Unknown and the The problem is that measures of uncertainty Unknowable in Financial Institutions Frank using the bell curve simply disregard the possibility Diebold, Neil Doherty, and Richard Herring, of sharp jumps or discontinuities. Therefore they editors, Princeton: Princeton University Press. have no meaning or consequence. Using them is like focusing on the grass and missing out on the (gigantic) trees. Conventional studies of uncertainty, whether in statistics, economics, finance or social science, In fact, while the occasional and unpredictable have largely stayed close to the so-called “bell large deviations are rare, they cannot be dismissed curve”, a symmetrical graph that represents a as “outliers” because, cumulatively, their impact in probability distribution. Used to great effect to the long term is so dramatic. describe errors in astronomical measurement by The good news, especially for practitioners, is the 19th-century mathematician Carl Friedrich that the fractal model is both intuitively and Gauss, the bell curve, or Gaussian model, has computationally simpler than the Gaussian. It too since pervaded our business and scientific culture, has been around since the sixties, which makes us and terms like sigma, variance, standard deviation, wonder why it was not implemented before. correlation, R-square and Sharpe ratio are all The traditional Gaussian way of looking at the directly linked to it.