On the Measurement Problem
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Divine Action and the World of Science: What Cosmology and Quantum Physics Teach Us About the Role of Providence in Nature 247 Bruce L
Journal of Biblical and Theological Studies JBTSVOLUME 2 | ISSUE 2 Christianity and the Philosophy of Science Divine Action and the World of Science: What Cosmology and Quantum Physics Teach Us about the Role of Providence in Nature 247 Bruce L. Gordon [JBTS 2.2 (2017): 247-298] Divine Action and the World of Science: What Cosmology and Quantum Physics Teach Us about the Role of Providence in Nature1 BRUCE L. GORDON Bruce L. Gordon is Associate Professor of the History and Philosophy of Science at Houston Baptist University and a Senior Fellow of Discovery Institute’s Center for Science and Culture Abstract: Modern science has revealed a world far more exotic and wonder- provoking than our wildest imaginings could have anticipated. It is the purpose of this essay to introduce the reader to the empirical discoveries and scientific concepts that limn our understanding of how reality is structured and interconnected—from the incomprehensibly large to the inconceivably small—and to draw out the metaphysical implications of this picture. What is unveiled is a universe in which Mind plays an indispensable role: from the uncanny life-giving precision inscribed in its initial conditions, mathematical regularities, and natural constants in the distant past, to its material insubstantiality and absolute dependence on transcendent causation for causal closure and phenomenological coherence in the present, the reality we inhabit is one in which divine action is before all things, in all things, and constitutes the very basis on which all things hold together (Colossians 1:17). §1. Introduction: The Intelligible Cosmos For science to be possible there has to be order present in nature and it has to be discoverable by the human mind. -
Effective Program Reasoning Using Bayesian Inference
EFFECTIVE PROGRAM REASONING USING BAYESIAN INFERENCE Sulekha Kulkarni A DISSERTATION in Computer and Information Science Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 2020 Supervisor of Dissertation Mayur Naik, Professor, Computer and Information Science Graduate Group Chairperson Mayur Naik, Professor, Computer and Information Science Dissertation Committee: Rajeev Alur, Zisman Family Professor of Computer and Information Science Val Tannen, Professor of Computer and Information Science Osbert Bastani, Research Assistant Professor of Computer and Information Science Suman Nath, Partner Research Manager, Microsoft Research, Redmond To my father, who set me on this path, to my mother, who leads by example, and to my husband, who is an infinite source of courage. ii Acknowledgments I want to thank my advisor Prof. Mayur Naik for giving me the invaluable opportunity to learn and experiment with different ideas at my own pace. He supported me through the ups and downs of research, and helped me make the Ph.D. a reality. I also want to thank Prof. Rajeev Alur, Prof. Val Tannen, Prof. Osbert Bastani, and Dr. Suman Nath for serving on my dissertation committee and for providing valuable feedback. I am deeply grateful to Prof. Alur and Prof. Tannen for their sound advice and support, and for going out of their way to help me through challenging times. I am also very grateful for Dr. Nath's able and inspiring mentorship during my internship at Microsoft Research, and during the collaboration that followed. Dr. Aditya Nori helped me start my Ph.D. -
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. -
1 Dependent and Independent Events 2 Complementary Events 3 Mutually Exclusive Events 4 Probability of Intersection of Events
1 Dependent and Independent Events Let A and B be events. We say that A is independent of B if P (AjB) = P (A). That is, the marginal probability of A is the same as the conditional probability of A, given B. This means that the probability of A occurring is not affected by B occurring. It turns out that, in this case, B is independent of A as well. So, we just say that A and B are independent. We say that A depends on B if P (AjB) 6= P (A). That is, the marginal probability of A is not the same as the conditional probability of A, given B. This means that the probability of A occurring is affected by B occurring. It turns out that, in this case, B depends on A as well. So, we just say that A and B are dependent. Consider these events from the card draw: A = drawing a king, B = drawing a spade, C = drawing a face card. Events A and B are independent. If you know that you have drawn a spade, this does not change the likelihood that you have actually drawn a king. Formally, the marginal probability of drawing a king is P (A) = 4=52. The conditional probability that your card is a king, given that it a spade, is P (AjB) = 1=13, which is the same as 4=52. Events A and C are dependent. If you know that you have drawn a face card, it is much more likely that you have actually drawn a king than it would be ordinarily. -
Introduction to Stochastic Processes - Lecture Notes (With 33 Illustrations)
Introduction to Stochastic Processes - Lecture Notes (with 33 illustrations) Gordan Žitković Department of Mathematics The University of Texas at Austin Contents 1 Probability review 4 1.1 Random variables . 4 1.2 Countable sets . 5 1.3 Discrete random variables . 5 1.4 Expectation . 7 1.5 Events and probability . 8 1.6 Dependence and independence . 9 1.7 Conditional probability . 10 1.8 Examples . 12 2 Mathematica in 15 min 15 2.1 Basic Syntax . 15 2.2 Numerical Approximation . 16 2.3 Expression Manipulation . 16 2.4 Lists and Functions . 17 2.5 Linear Algebra . 19 2.6 Predefined Constants . 20 2.7 Calculus . 20 2.8 Solving Equations . 22 2.9 Graphics . 22 2.10 Probability Distributions and Simulation . 23 2.11 Help Commands . 24 2.12 Common Mistakes . 25 3 Stochastic Processes 26 3.1 The canonical probability space . 27 3.2 Constructing the Random Walk . 28 3.3 Simulation . 29 3.3.1 Random number generation . 29 3.3.2 Simulation of Random Variables . 30 3.4 Monte Carlo Integration . 33 4 The Simple Random Walk 35 4.1 Construction . 35 4.2 The maximum . 36 1 CONTENTS 5 Generating functions 40 5.1 Definition and first properties . 40 5.2 Convolution and moments . 42 5.3 Random sums and Wald’s identity . 44 6 Random walks - advanced methods 48 6.1 Stopping times . 48 6.2 Wald’s identity II . 50 6.3 The distribution of the first hitting time T1 .......................... 52 6.3.1 A recursive formula . 52 6.3.2 Generating-function approach . -
Lecture 2: Modeling Random Experiments
Department of Mathematics Ma 3/103 KC Border Introduction to Probability and Statistics Winter 2021 Lecture 2: Modeling Random Experiments Relevant textbook passages: Pitman [5]: Sections 1.3–1.4., pp. 26–46. Larsen–Marx [4]: Sections 2.2–2.5, pp. 18–66. 2.1 Axioms for probability measures Recall from last time that a random experiment is an experiment that may be conducted under seemingly identical conditions, yet give different results. Coin tossing is everyone’s go-to example of a random experiment. The way we model random experiments is through the use of probabilities. We start with the sample space Ω, the set of possible outcomes of the experiment, and consider events, which are subsets E of the sample space. (We let F denote the collection of events.) 2.1.1 Definition A probability measure P or probability distribution attaches to each event E a number between 0 and 1 (inclusive) so as to obey the following axioms of probability: Normalization: P (?) = 0; and P (Ω) = 1. Nonnegativity: For each event E, we have P (E) > 0. Additivity: If EF = ?, then P (∪ F ) = P (E) + P (F ). Note that while the domain of P is technically F, the set of events, that is P : F → [0, 1], we may also refer to P as a probability (measure) on Ω, the set of realizations. 2.1.2 Remark To reduce the visual clutter created by layers of delimiters in our notation, we may omit some of them simply write something like P (f(ω) = 1) orP {ω ∈ Ω: f(ω) = 1} instead of P {ω ∈ Ω: f(ω) = 1} and we may write P (ω) instead of P {ω} . -
Probability and Counting Rules
blu03683_ch04.qxd 09/12/2005 12:45 PM Page 171 C HAPTER 44 Probability and Counting Rules Objectives Outline After completing this chapter, you should be able to 4–1 Introduction 1 Determine sample spaces and find the probability of an event, using classical 4–2 Sample Spaces and Probability probability or empirical probability. 4–3 The Addition Rules for Probability 2 Find the probability of compound events, using the addition rules. 4–4 The Multiplication Rules and Conditional 3 Find the probability of compound events, Probability using the multiplication rules. 4–5 Counting Rules 4 Find the conditional probability of an event. 5 Find the total number of outcomes in a 4–6 Probability and Counting Rules sequence of events, using the fundamental counting rule. 4–7 Summary 6 Find the number of ways that r objects can be selected from n objects, using the permutation rule. 7 Find the number of ways that r objects can be selected from n objects without regard to order, using the combination rule. 8 Find the probability of an event, using the counting rules. 4–1 blu03683_ch04.qxd 09/12/2005 12:45 PM Page 172 172 Chapter 4 Probability and Counting Rules Statistics Would You Bet Your Life? Today Humans not only bet money when they gamble, but also bet their lives by engaging in unhealthy activities such as smoking, drinking, using drugs, and exceeding the speed limit when driving. Many people don’t care about the risks involved in these activities since they do not understand the concepts of probability. -
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 . -
Negative Probability in the Framework of Combined Probability
Negative probability in the framework of combined probability Mark Burgin University of California, Los Angeles 405 Hilgard Ave. Los Angeles, CA 90095 Abstract Negative probability has found diverse applications in theoretical physics. Thus, construction of sound and rigorous mathematical foundations for negative probability is important for physics. There are different axiomatizations of conventional probability. So, it is natural that negative probability also has different axiomatic frameworks. In the previous publications (Burgin, 2009; 2010), negative probability was mathematically formalized and rigorously interpreted in the context of extended probability. In this work, axiomatic system that synthesizes conventional probability and negative probability is constructed in the form of combined probability. Both theoretical concepts – extended probability and combined probability – stretch conventional probability to negative values in a mathematically rigorous way. Here we obtain various properties of combined probability. In particular, relations between combined probability, extended probability and conventional probability are explicated. It is demonstrated (Theorems 3.1, 3.3 and 3.4) that extended probability and conventional probability are special cases of combined probability. 1 1. Introduction All students are taught that probability takes values only in the interval [0,1]. All conventional interpretations of probability support this assumption, while all popular formal descriptions, e.g., axioms for probability, such as Kolmogorov’s -
1 — a Single Random Variable
1 | A SINGLE RANDOM VARIABLE Questions involving probability abound in Computer Science: What is the probability of the PWF world falling over next week? • What is the probability of one packet colliding with another in a network? • What is the probability of an undergraduate not turning up for a lecture? • When addressing such questions there are often added complications: the question may be ill posed or the answer may vary with time. Which undergraduate? What lecture? Is the probability of turning up different on Saturdays? Let's start with something which appears easy to reason about: : : Introduction | Throwing a die Consider an experiment or trial which consists of throwing a mathematically ideal die. Such a die is often called a fair die or an unbiased die. Common sense suggests that: The outcome of a single throw cannot be predicted. • The outcome will necessarily be a random integer in the range 1 to 6. • The six possible outcomes are equiprobable, each having a probability of 1 . • 6 Without further qualification, serious probabilists would regard this collection of assertions, especially the second, as almost meaningless. Just what is a random integer? Giving proper mathematical rigour to the foundations of probability theory is quite a taxing task. To illustrate the difficulty, consider probability in a frequency sense. Thus a probability 1 of 6 means that, over a long run, one expects to throw a 5 (say) on one-sixth of the occasions that the die is thrown. If the actual proportion of 5s after n throws is p5(n) it would be nice to say: 1 lim p5(n) = n !1 6 Unfortunately this is utterly bogus mathematics! This is simply not a proper use of the idea of a limit. -
Probabilities, Random Variables and Distributions A
Probabilities, Random Variables and Distributions A Contents A.1 EventsandProbabilities................................ 318 A.1.1 Conditional Probabilities and Independence . ............. 318 A.1.2 Bayes’Theorem............................... 319 A.2 Random Variables . ................................. 319 A.2.1 Discrete Random Variables ......................... 319 A.2.2 Continuous Random Variables ....................... 320 A.2.3 TheChangeofVariablesFormula...................... 321 A.2.4 MultivariateNormalDistributions..................... 323 A.3 Expectation,VarianceandCovariance........................ 324 A.3.1 Expectation................................. 324 A.3.2 Variance................................... 325 A.3.3 Moments................................... 325 A.3.4 Conditional Expectation and Variance ................... 325 A.3.5 Covariance.................................. 326 A.3.6 Correlation.................................. 327 A.3.7 Jensen’sInequality............................. 328 A.3.8 Kullback–LeiblerDiscrepancyandInformationInequality......... 329 A.4 Convergence of Random Variables . 329 A.4.1 Modes of Convergence . 329 A.4.2 Continuous Mapping and Slutsky’s Theorem . 330 A.4.3 LawofLargeNumbers........................... 330 A.4.4 CentralLimitTheorem........................... 331 A.4.5 DeltaMethod................................ 331 A.5 ProbabilityDistributions............................... 332 A.5.1 UnivariateDiscreteDistributions...................... 333 A.5.2 Univariate Continuous Distributions . 335 -
Reformulating Bell's Theorem: the Search for a Truly Local Quantum
Reformulating Bell's Theorem: The Search for a Truly Local Quantum Theory∗ Mordecai Waegelly and Kelvin J. McQueenz March 2, 2020 Abstract The apparent nonlocality of quantum theory has been a persistent concern. Einstein et. al. (1935) and Bell (1964) emphasized the apparent nonlocality arising from entanglement correlations. While some interpretations embrace this nonlocality, modern variations of the Everett-inspired many worlds interpretation try to circumvent it. In this paper, we review Bell's \no-go" theorem and explain how it rests on three axioms, local causality, no superdeterminism, and one world. Although Bell is often taken to have shown that local causality is ruled out by the experimentally confirmed entanglement correlations, we make clear that it is the conjunction of the three axioms that is ruled out by these correlations. We then show that by assuming local causality and no superdeterminism, we can give a direct proof of many worlds. The remainder of the paper searches for a consistent, local, formulation of many worlds. We show that prominent formulations whose ontology is given by the wave function violate local causality, and we critically evaluate claims in the literature to the contrary. We ultimately identify a local many worlds interpretation that replaces the wave function with a separable Lorentz-invariant wave-field. We conclude with discussions of the Born rule, and other interpretations of quantum mechanics. Contents 1 Introduction 2 2 Local causality 3 3 Reformulating Bell's theorem 5 4 Proving Bell's theorem with many worlds 6 5 Global worlds 8 5.1 Global branching . .8 5.2 Global divergence .