Are Deterministic Descriptions and Indeterministic Descriptions Observationally Equivalent?
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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. -
Observational Determinism for Concurrent Program Security
Observational Determinism for Concurrent Program Security Steve Zdancewic Andrew C. Myers Department of Computer and Information Science Computer Science Department University of Pennsylvania Cornell University [email protected] [email protected] Abstract This paper makes two contributions. First, it presents a definition of information-flow security that is appropriate for Noninterference is a property of sequential programs that concurrent systems. Second, it describes a simple but ex- is useful for expressing security policies for data confiden- pressive concurrent language with a type system that prov- tiality and integrity. However, extending noninterference to ably enforces security. concurrent programs has proved problematic. In this pa- Notions of secure information flow are usually based on per we present a relatively expressive secure concurrent lan- noninterference [15], a property only defined for determin- guage. This language, based on existing concurrent calculi, istic systems. Intuitively, noninterference requires that the provides first-class channels, higher-order functions, and an publicly visible results of a computation do not depend on unbounded number of threads. Well-typed programs obey a confidential (or secret) information. Generalizing noninter- generalization of noninterference that ensures immunity to ference to concurrent languages is problematic because these internal timing attacks and to attacks that exploit informa- languages are naturally nondeterministic: the order of execu- tion about the thread scheduler. Elimination of these refine- tion of concurrent threads is not specified by the language se- ment attacks is possible because the enforced security prop- mantics. Although this nondeterminism permits a variety of erty extends noninterference with observational determin- thread scheduler implementations, it also leads to refinement ism. -
An Introduction to Mathematical Modelling
An Introduction to Mathematical Modelling Glenn Marion, Bioinformatics and Statistics Scotland Given 2008 by Daniel Lawson and Glenn Marion 2008 Contents 1 Introduction 1 1.1 Whatismathematicalmodelling?. .......... 1 1.2 Whatobjectivescanmodellingachieve? . ............ 1 1.3 Classificationsofmodels . ......... 1 1.4 Stagesofmodelling............................... ....... 2 2 Building models 4 2.1 Gettingstarted .................................. ...... 4 2.2 Systemsanalysis ................................. ...... 4 2.2.1 Makingassumptions ............................. .... 4 2.2.2 Flowdiagrams .................................. 6 2.3 Choosingmathematicalequations. ........... 7 2.3.1 Equationsfromtheliterature . ........ 7 2.3.2 Analogiesfromphysics. ...... 8 2.3.3 Dataexploration ............................... .... 8 2.4 Solvingequations................................ ....... 9 2.4.1 Analytically.................................. .... 9 2.4.2 Numerically................................... 10 3 Studying models 12 3.1 Dimensionlessform............................... ....... 12 3.2 Asymptoticbehaviour ............................. ....... 12 3.3 Sensitivityanalysis . ......... 14 3.4 Modellingmodeloutput . ....... 16 4 Testing models 18 4.1 Testingtheassumptions . ........ 18 4.2 Modelstructure.................................. ...... 18 i 4.3 Predictionofpreviouslyunuseddata . ............ 18 4.3.1 Reasonsforpredictionerrors . ........ 20 4.4 Estimatingmodelparameters . ......... 20 4.5 Comparingtwomodelsforthesamesystem . ......... -
The Probability Set-Up.Pdf
CHAPTER 2 The probability set-up 2.1. Basic theory of probability We will have a sample space, denoted by S (sometimes Ω) that consists of all possible outcomes. For example, if we roll two dice, the sample space would be all possible pairs made up of the numbers one through six. An event is a subset of S. Another example is to toss a coin 2 times, and let S = fHH;HT;TH;TT g; or to let S be the possible orders in which 5 horses nish in a horse race; or S the possible prices of some stock at closing time today; or S = [0; 1); the age at which someone dies; or S the points in a circle, the possible places a dart can hit. We should also keep in mind that the same setting can be described using dierent sample set. For example, in two solutions in Example 1.30 we used two dierent sample sets. 2.1.1. Sets. We start by describing elementary operations on sets. By a set we mean a collection of distinct objects called elements of the set, and we consider a set as an object in its own right. Set operations Suppose S is a set. We say that A ⊂ S, that is, A is a subset of S if every element in A is contained in S; A [ B is the union of sets A ⊂ S and B ⊂ S and denotes the points of S that are in A or B or both; A \ B is the intersection of sets A ⊂ S and B ⊂ S and is the set of points that are in both A and B; ; denotes the empty set; Ac is the complement of A, that is, the points in S that are not in A. -
1.) What Is the Difference Between Observation and Interpretation? Write a Short Paragraph in Your Journal (5-6 Sentences) Defining Both, with One Example Each
Observation and Response: Creative Response Template can be adapted to fit a variety of classes and types of responses Goals: Students will: • Practice observation skills and develop their abilities to find multiple possibilities for interpretation and response to visual material; • Consider the difference between observation and interpretation; • Develop descriptive skills, including close reading and metaphorical description; • Develop research skills by considering a visual text or object as a primary source; • Engage in creative response to access and communicate intellectually or emotionally challenging material. Observation exercises Part I You will keep a journal as you go through this exercise. This can be a physical notebook that pleases you or a word document on your computer. At junctures during the exercise, you will be asked to comment, define, reflect, and create in your journal on what you just did. Journal responses will be posted to Canvas at the conclusion of the exercise, either as word documents or as Jpegs. 1.) What is the difference between observation and interpretation? Write a short paragraph in your journal (5-6 sentences) defining both, with one example each. 2.) Set a timer on your phone or computer (or oven…) and observe the image provided for 1 minute. Quickly write a preliminary research question about it. At first glance, what are you curious about? What do you want to know? 3.) Set your timer for 5 minutes and write as many observations as possible about it. Try to keep making observations, even when you feel stuck. Keep returning to the question, “What do I see?” No observation is too big or too small, too obvious or too obscure. -
1 Probabilities
1 Probabilities 1.1 Experiments with randomness We will use the term experiment in a very general way to refer to some process that produces a random outcome. Examples: (Ask class for some first) Here are some discrete examples: • roll a die • flip a coin • flip a coin until we get heads And here are some continuous examples: • height of a U of A student • random number in [0, 1] • the time it takes until a radioactive substance undergoes a decay These examples share the following common features: There is a proce- dure or natural phenomena called the experiment. It has a set of possible outcomes. There is a way to assign probabilities to sets of possible outcomes. We will call this a probability measure. 1.2 Outcomes and events Definition 1. An experiment is a well defined procedure or sequence of procedures that produces an outcome. 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. Definition 2. An event is a subset of the sample space. This definition will be changed when we come to the definition ofa σ-field. The next thing to define is a probability measure. Before we can do this properly we need some more structure, so for now we just make an informal definition. A probability measure is a function on the collection of events 1 that assign a number between 0 and 1 to each event and satisfies certain properties. NB: A probability measure is not a function on Ω. -
Propensities and Probabilities
ARTICLE IN PRESS Studies in History and Philosophy of Modern Physics 38 (2007) 593–625 www.elsevier.com/locate/shpsb Propensities and probabilities Nuel Belnap 1028-A Cathedral of Learning, University of Pittsburgh, Pittsburgh, PA 15260, USA Received 19 May 2006; accepted 6 September 2006 Abstract Popper’s introduction of ‘‘propensity’’ was intended to provide a solid conceptual foundation for objective single-case probabilities. By considering the partly opposed contributions of Humphreys and Miller and Salmon, it is argued that when properly understood, propensities can in fact be understood as objective single-case causal probabilities of transitions between concrete events. The chief claim is that propensities are well-explicated by describing how they fit into the existing formal theory of branching space-times, which is simultaneously indeterministic and causal. Several problematic examples, some commonsense and some quantum-mechanical, are used to make clear the advantages of invoking branching space-times theory in coming to understand propensities. r 2007 Elsevier Ltd. All rights reserved. Keywords: Propensities; Probabilities; Space-times; Originating causes; Indeterminism; Branching histories 1. Introduction You are flipping a fair coin fairly. You ascribe a probability to a single case by asserting The probability that heads will occur on this very next flip is about 50%. ð1Þ The rough idea of a single-case probability seems clear enough when one is told that the contrast is with either generalizations or frequencies attributed to populations asserted while you are flipping a fair coin fairly, such as In the long run; the probability of heads occurring among flips is about 50%. ð2Þ E-mail address: [email protected] 1355-2198/$ - see front matter r 2007 Elsevier Ltd. -
PDF Download Starting with Science Strategies for Introducing Young Children to Inquiry 1St Edition Ebook
STARTING WITH SCIENCE STRATEGIES FOR INTRODUCING YOUNG CHILDREN TO INQUIRY 1ST EDITION PDF, EPUB, EBOOK Marcia Talhelm Edson | 9781571108074 | | | | | Starting with Science Strategies for Introducing Young Children to Inquiry 1st edition PDF Book The presentation of the material is as good as the material utilizing star trek analogies, ancient wisdom and literature and so much more. Using Multivariate Statistics. Michael Gramling examines the impact of policy on practice in early childhood education. Part of a series on. Schauble and colleagues , for example, found that fifth grade students designed better experiments after instruction about the purpose of experimentation. For example, some suggest that learning about NoS enables children to understand the tentative and developmental NoS and science as a human activity, which makes science more interesting for children to learn Abd-El-Khalick a ; Driver et al. Research on teaching and learning of nature of science. The authors begin with theory in a cultural context as a foundation. What makes professional development effective? Frequently, the term NoS is utilised when considering matters about science. This book is a documentary account of a young intern who worked in the Reggio system in Italy and how she brought this pedagogy home to her school in St. Taking Science to School answers such questions as:. The content of the inquiries in science in the professional development programme was based on the different strands of the primary science curriculum, namely Living Things, Energy and Forces, Materials and Environmental Awareness and Care DES Exit interview. Begin to address the necessity of understanding other usually peer positions before they can discuss or comment on those positions. -
Indeterminism in Physics and Intuitionistic Mathematics
Indeterminism in Physics and Intuitionistic Mathematics Nicolas Gisin Group of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland (Dated: November 5, 2020) Most physics theories are deterministic, with the notable exception of quantum mechanics which, however, comes plagued by the so-called measurement problem. This state of affairs might well be due to the inability of standard mathematics to “speak” of indeterminism, its inability to present us a worldview in which new information is created as time passes. In such a case, scientific determinism would only be an illusion due to the timeless mathematical language scientists use. To investigate this possibility it is necessary to develop an alternative mathematical language that is both powerful enough to allow scientists to compute predictions and compatible with indeterminism and the passage of time. We argue that intuitionistic mathematics provides such a language and we illustrate it in simple terms. I. INTRODUCTION I have always been amazed by the huge difficulties that my fellow physicists seem to encounter when con- templating indeterminism and the highly sophisticated Physicists are not used to thinking of the world as in- circumlocutions they are ready to swallow to avoid the determinate and its evolution as indeterministic. New- straightforward conclusion that physics does not neces- ton’s equations, like Maxwell’s and Schr¨odinger’s equa- sarily present us a deterministic worldview. But even tions, are (partial) differential equations describing the more surprising to me is the attitude of most philoso- continuous evolution of the initial condition as a func- phers, especially philosophers of science. Indeed, most tion of a parameter identified with time. -
Wittgenstein on Freedom of the Will: Not Determinism, Yet Not Indeterminism
Wittgenstein on Freedom of the Will: Not Determinism, Yet Not Indeterminism Thomas Nadelhoffer This is a prepublication draft. This version is being revised for resubmission to a journal. Abstract Since the publication of Wittgenstein’s Lectures on Freedom of the Will, his remarks about free will and determinism have received very little attention. Insofar as these lectures give us an opportunity to see him at work on a traditional—and seemingly intractable—philosophical problem and given the voluminous secondary literature written about nearly every other facet of Wittgenstein’s life and philosophy, this neglect is both surprising and unfortunate. Perhaps these lectures have not attracted much attention because they are available to us only in the form of a single student’s notes (Yorick Smythies). Or perhaps it is because, as one Wittgenstein scholar put it, the lectures represent only “cursory reflections” that “are themselves uncompelling." (Glock 1996: 390) Either way, my goal is to show that Wittgenstein’s views about freedom of the will merit closer attention. All of these arguments might look as if I wanted to argue for the freedom of the will or against it. But I don't want to. --Ludwig Wittgenstein, Lectures on Freedom of the Will Since the publication of Wittgenstein’s Lectures on Freedom of the Will,1 his remarks from these lectures about free will and determinism have received very little attention.2 Insofar as these lectures give us an opportunity to see him at work on a traditional—and seemingly intractable— philosophical problem and given the voluminous secondary literature written about nearly every 1 Wittgenstein’s “Lectures on Freedom of the Will” will be abbreviated as LFW 1993 in this paper (see bibliography) since I am using the version reprinted in Philosophical Occasions (1993). -
Clustering and Classification of Multivariate Stochastic Time Series
Clemson University TigerPrints All Dissertations Dissertations 5-2012 Clustering and Classification of Multivariate Stochastic Time Series in the Time and Frequency Domains Ryan Schkoda Clemson University, [email protected] Follow this and additional works at: https://tigerprints.clemson.edu/all_dissertations Part of the Mechanical Engineering Commons Recommended Citation Schkoda, Ryan, "Clustering and Classification of Multivariate Stochastic Time Series in the Time and Frequency Domains" (2012). All Dissertations. 907. https://tigerprints.clemson.edu/all_dissertations/907 This Dissertation is brought to you for free and open access by the Dissertations at TigerPrints. It has been accepted for inclusion in All Dissertations by an authorized administrator of TigerPrints. For more information, please contact [email protected]. Clustering and Classification of Multivariate Stochastic Time Series in the Time and Frequency Domains A Dissertation Presented to the Graduate School of Clemson University In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Mechanical Engineering by Ryan F. Schkoda May 2012 Accepted by: Dr. John Wagner, Committee Chair Dr. Robert Lund Dr. Ardalan Vahidi Dr. Darren Dawson Abstract The dissertation primarily investigates the characterization and discrimina- tion of stochastic time series with an application to pattern recognition and fault detection. These techniques supplement traditional methodologies that make overly restrictive assumptions about the nature of a signal by accommodating stochastic behavior. The assumption that the signal under investigation is either deterministic or a deterministic signal polluted with white noise excludes an entire class of signals { stochastic time series. The research is concerned with this class of signals almost exclusively. The investigation considers signals in both the time and the frequency domains and makes use of both model-based and model-free techniques.