# Review. Bayesianism and Reliability. Today s Class

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Review Bayesianism and Reliability Models and Simulations in Philosophy April 14th, 2014 Last Class: Difference between individual and social epistemology Why simulations are particularly useful for social epistemology and philosophy of science Intro. to Agent-Based Models (abms) Today s Class Question: Why discuss individual epistemology given that the focus of this course is social epistemology and philosophy of science? Today: Two theories of what is rational for an individual to believe Bayesianism Logical Reliability Answer: To Model how group members do or ought to change their beliefs. Develop criteria of group rationality.

2 Outline Bayesianism 1 Bayesianism Probability Theory Probabilistic Beliefs? Measuring Belief Dutch Book Theorem Conditionalization 2 Logical Reliability Bayesianism is the conjunction of two theses: Beliefs = Probabilities Beliefs updated by conditionalization 3 Course Structure 4 NetLogo 5 References The Sample Space Events In probability, there is a set Ω called the sample space that represents all the possible outcomes of an experiment. For instance: Experiment 1: Roll a die. Ω = {1, 2, 3, 4, 5, 6}. Experiment 2: Flip a coin twice. Ω = { H, H, T, T, H, T, T, H } Subsets of the sample space are called events.

3 Events Events Experiment 1: Roll a die Example Event: The event the die lands on an odd number is A = {1, 3, 5}. Experiment 2: Flip a coin twice Example Event: The event the coin lands heads exactly once is: A = { H, T, T, H } Impossible Event Probability Axioms In both experiments, represents the impossible event: Experiment 1: The die does not land on any face. Experiment 2: The coin lands on neither heads nor tails. A probability measure is a function that assigns every event A some number P(A) between 0 and 1 (inclusive) such that P(Ω) = 1, where Ω is the entire sample space, and Finite Additivity: If A B =, then P(A B) = P(A) + P(B).

4 Probability Axioms Scientists and statisticians often assume Countable Additivity: If E 1, E 2,... is a countable sequence of events and E i E j = whenever i j, then P( n E n ) = n P(E n ). Belief and Probability Properties of Probability The first thesis of Bayesianism is that degrees of belief are representable by probabilities. Probabilities are always comparable. Numbers can be compared: either x > y, or y > x, or x = y. Are degrees of belief always comparable? E.g., What s more likely: (i) aliens invade Earth before 2200 or (ii) monkeys increase in intelligence and become our overlords within the next 3000 years?

5 Properties of Probability Quantitative Comparisons Probabilities are always quantitatively comparable. For example,.642 is 32.1 times greater than.02. Tremendously Awful Really awful (but better?) Properties of Probability Properties of Probability Probabilities are always quantitatively comparable. Comparisons of numbers can be quantified. E.g..642 is 32.1 times greater than.02. Compare: Clearly, Nickelback is worse than Creed. Are there 32.1 times as bad? Can assessments of degrees of belief be made so precise? Probabilities are bounded (between 0 and 1) Compare: The length of objects can be assigned a number, but there is no maximum length. Why is it that there is some maximum degree of certainty? And some minimum degree of confidence?

6 Properties of Probability Properties of Probability Probabilities are additive. Compare: Ice cream is delicious. So is sausage. Is the deliciousness of sausage with a side of ice cream the sum of the degrees to which others are delicious? Moreover, why didn t you multiply the two numbers? What s so special about addition? To answer these questions, let s consider how we might measure beliefs... Measuring Probability Odds How can degrees of belief be measured? Idea: Betting behavior (with small sums) On April 16th, the football team FC Bayern will play Kaiserlautern. Several bookies are offering around 3 : 2 odds on Kaiserlautern.

7 Odds Odds That is, you pay \$2 for a bet such that You win \$3 if Kaiserlautern wins. You get nothing otherwise. A bookie who offers you such a bet clearly thinks that FC Bayern will likely beat Kaiserlautern. Can we quantify how much so? Fair Odds Odds and Degrees of Belief Define the bookie s degree of belief P(E) in an event E to be where a : b are her fair betting odds for E. b b+a Notice that her degrees of belief are numbers between 0 and 1. Say the bookie considers her odds a : b on an event E to be fair if she is willing to both sell and buy the bet a : b on E.

8 Odds and Degrees of Belief Dutch Book Question: Would it be prudent for the bookie s degrees of beliefs to satisfy the remaining probability axioms? Let E be the event that FC Bayern wins its next match Let E c the event they don t. Suppose the bookie posts 2 : 1 odds on E and 2 : 1 odds on E c. Dutch Book Dutch Book Notice the bookies degrees of belief are not probabilities as and hence, P(E) = P(E c ) = = 1 3 You are very wise and decide to buy both bets from the bookie. So you pay \$1 for a bet that pays \$2 if FC Bayern wins, and You pay \$1 for a bet that pays \$2 if FC Bayern does not win. Regardless of what happens, you win \$1, i.e., you are guaranteed to take money from the bookie. P(Ω) = P(E E c ) = P(E) + P(E c ) = 2 3.

9 Dutch Book Theorem Dutch Book Theorem and Bayesianism Theorem (Dutch Book Theorem) There is no series of bets against the bookie that ensures that she loses money for sure if and only if her degrees of belief obey the probability axioms. The Dutch book theorem is one (of several) arguments for the Bayesian s first thesis: Beliefs = Probabilities Bayesianism Updating Degrees of Belief Bayesianism is the conjunction of two theses: Beliefs = Probabilities Update by conditionalization Suppose your degree of belief at time t that my sibling is male is P t (M). You learn at time t + 1 that my sibling s name is Evan. What should your degrees of belief be then?

10 Conditionalization Diachronic Dutch Books Conditionalization is the thesis that upon learning E, you ought to revise your degrees of belief in M as follows: P t+1 (M) = P t (M E) := P t(m&e) P t (E) There are Dutch Book theorems for conditionalization as well, but we ll skip them. Conditionalization seems most plausible when P t are frequencies. Normative Arguments Descriptive Models Together, these theorems seem to provide a normative argument for the theses: Your degrees of belief ought to act like probabilities, and Your ought to update your degrees of belief by conditionalization. This is what Strevens [2006] calls the a priori justification for Bayesianism. Sometimes Bayesianism is also descriptive of how humans in fact reason [Gopnik and Wellman, 2012].

11 Bayesianism in this Course Outline Question: Why will we care about Bayesianism in this course? Answer: Many of the models we consider will assume group members have probabilistic beliefs. Some assume they update those beliefs by conditionalization. 1 Bayesianism Probability Theory Probabilistic Beliefs? Measuring Belief Dutch Book Theorem Conditionalization 2 Logical Reliability 3 Course Structure 4 NetLogo 5 References Bayesianism and Truth Truth and Rationality Notice the arguments for Bayesianism had the following structure: If you want to avoid sure loss, then Beliefs = Probabilities Update = Conditionalization In particular, neither argument showed a relationship between your beliefs and what is true. I argue that epistemic states and changes of such states as well as the rationality criteria governing epistemic dynamics can be, and should be, formulated independently of the factual connections between the epistemic inputs and the outer world. One consequence of this position is that the concept of truth is irrelevant... Gaerdenfors [1988], page 9.

12 Bayesianism and Truth Logical Reliability Defined There are theorems that connect Bayesian updating with developing true beliefs, but they may not be as widely applicable as many imagine [Belot, 2013]. Kelly [1996] argues that the method we use to update our beliefs ought be logically reliable, i.e., it is guaranteed to converge in every possible circumstance consistent with... background assumptions. Logical Reliability Defined Logical Reliability in this Course The logical reliabilist conceives of inductive problems the way a computer scientist conceives of computational problems. A solution to a computational problem (an algorithm) is supposed to be guaranteed by its mathematical structure to output a correct answer on every possible input. The logical perspective stands in sharp contrast with the received view among inductive methodologists, who are often more interested in whether a belief is justified by evidence... Logical reliabilism simply demands of inductive methods what is routinely required of algorithms. Question: Why will we care about logical reliability in this course? Answer: Many of the models we consider will evaluate the rationality of an organizational structure by asking whether all group members beliefs tend towards the truth. Kelly [1996], page 4.

13 Bounded Rationality Bounded Rationality Kelly [1996] also discusses another important issue in this course: bounded rationality. For Kelly, one should not prescribe methods for updating beliefs that cannot not, in principle, be carried out by a computer. Traditional methodological analysis has centered on quixotic ideal agents who have divine cognitive powers... This idealization is harmless only so long as what is good for ideal agents is good for their more benighted brethren. It will be seen, however, that rules the seem harmless and inevitable for ideal agents are in fact disastrous for computationally bounded agents. A prime example is the simple requirement that a hypothesis be dropped as soon as it is inconsistent with the data, a rule endorse by almost all inductive methodologies [e.g., Bayesianism]. It turns out the there are problems for which computable methods can be consistent in this sense, and some computable method is a reliable solution, but no computable, consistent method is a reliable solution. Kelly [1996], page 6. Logical Reliability in this Course Structure of the Course Question: Why will we care about bounded rationality in this course? Answer: Some of the models we consider will evaluate ideal agents. We should carefully consider whether the models apply to non-ideal ones. Other models consider boundedly rational agents. We should consider whether agents limitations are realistic and normatively defensible. Upcoming Weeks: Three Units: Disagreement, Diversity, and Testimony Each unit has two parts: 1 An introduction to a traditional problem in epistemology of philosophy of science 2 Analysis of computer models aimed at answering said question.

14 Structure of the Course NetLogo Course Website: Printable slides Sample code from class Today we will discuss: Data types in NetLogo Declaring and modifying variables Arithmetic, Boolean, and List Operations Global vs. Local Variables References I Belot, G. (2013). Bayesian orgulity. Philosophy of Science, 80(4): Gaerdenfors, P. (1988). Knowledge in flux: Modeling the dynamics of epistemic states. MIT press. Gopnik, A. and Wellman, H. M. (2012). Reconstructing constructivism: Causal models, bayesian learning mechanisms, and the theory theory. Psychological bulletin, 138(6): Kelly, K. T. (1996). The logic of reliable inquiry. Oxford University Press, New York. Strevens, M. (2006). Notes on bayesian confirmation theory.

### When Betting Odds and Credences Come Apart: More Worries for Dutch Book Arguments

When Betting Odds and Credences Come Apart: More Worries for Dutch Book Arguments Darren BRADLEY and Hannes LEITGEB If an agent believes that the probability of E being true is 1/2, should she accept a

### Lecture 8 The Subjective Theory of Betting on Theories

Lecture 8 The Subjective Theory of Betting on Theories Patrick Maher Philosophy 517 Spring 2007 Introduction The subjective theory of probability holds that the laws of probability are laws that rational

### Economics 1011a: Intermediate Microeconomics

Lecture 11: Choice Under Uncertainty Economics 1011a: Intermediate Microeconomics Lecture 11: Choice Under Uncertainty Tuesday, October 21, 2008 Last class we wrapped up consumption over time. Today we

### Pascal is here expressing a kind of skepticism about the ability of human reason to deliver an answer to this question.

Pascal s wager So far we have discussed a number of arguments for or against the existence of God. In the reading for today, Pascal asks not Does God exist? but Should we believe in God? What is distinctive

### The "Dutch Book" argument, tracing back to independent work by. F.Ramsey (1926) and B.deFinetti (1937), offers prudential grounds for

The "Dutch Book" argument, tracing back to independent work by F.Ramsey (1926) and B.deFinetti (1937), offers prudential grounds for action in conformity with personal probability. Under several structural

The result of the bayesian analysis is the probability distribution of every possible hypothesis H, given one real data set D. This prestatistical approach to our problem was the standard approach of Laplace

### The Bayesian Approach to the Philosophy of Science

The Bayesian Approach to the Philosophy of Science Michael Strevens For the Macmillan Encyclopedia of Philosophy, second edition The posthumous publication, in 1763, of Thomas Bayes Essay Towards Solving

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According

### Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22

CS 70 Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice, roulette

### Basic Probability. Probability: The part of Mathematics devoted to quantify uncertainty

AMS 5 PROBABILITY Basic Probability Probability: The part of Mathematics devoted to quantify uncertainty Frequency Theory Bayesian Theory Game: Playing Backgammon. The chance of getting (6,6) is 1/36.

### arxiv:1112.0829v1 [math.pr] 5 Dec 2011

How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly

### Handout #1: Mathematical Reasoning

Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or

### Betting systems: how not to lose your money gambling

Betting systems: how not to lose your money gambling G. Berkolaiko Department of Mathematics Texas A&M University 28 April 2007 / Mini Fair, Math Awareness Month 2007 Gambling and Games of Chance Simple

### 1. Probabilities lie between 0 and 1 inclusive. 2. Certainty implies unity probability. 3. Probabilities of exclusive events add. 4. Bayes s rule.

Probabilities as betting odds and the Dutch book Carlton M. Caves 2000 June 24 Substantially modified 2001 September 6 Minor modifications 2001 November 30 Material on the lottery-ticket approach added

### A Few Basics of Probability

A Few Basics of Probability Philosophy 57 Spring, 2004 1 Introduction This handout distinguishes between inductive and deductive logic, and then introduces probability, a concept essential to the study

### Probability Models.S1 Introduction to Probability

Probability Models.S1 Introduction to Probability Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard The stochastic chapters of this book involve random variability. Decisions are

### Unit 19: Probability Models

Unit 19: Probability Models Summary of Video Probability is the language of uncertainty. Using statistics, we can better predict the outcomes of random phenomena over the long term from the very complex,

### Lecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett

Lecture Note 1 Set and Probability Theory MIT 14.30 Spring 2006 Herman Bennett 1 Set Theory 1.1 Definitions and Theorems 1. Experiment: any action or process whose outcome is subject to uncertainty. 2.

### Using game-theoretic probability for probability judgment. Glenn Shafer

Workshop on Game-Theoretic Probability and Related Topics Tokyo University February 28, 2008 Using game-theoretic probability for probability judgment Glenn Shafer For 170 years, people have asked whether

### V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPECTED VALUE

V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPETED VALUE A game of chance featured at an amusement park is played as follows: You pay \$ to play. A penny and a nickel are flipped. You win \$ if either

### Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice,

### Iterated Dynamic Belief Revision. Sonja Smets, University of Groningen. website: http://www.vub.ac.be/clwf/ss

LSE-Groningen Workshop I 1 Iterated Dynamic Belief Revision Sonja Smets, University of Groningen website: http://www.vub.ac.be/clwf/ss Joint work with Alexandru Baltag, COMLAB, Oxford University LSE-Groningen

### 6.042/18.062J Mathematics for Computer Science. Expected Value I

6.42/8.62J Mathematics for Computer Science Srini Devadas and Eric Lehman May 3, 25 Lecture otes Expected Value I The expectation or expected value of a random variable is a single number that tells you

### Fundamentals of Mathematics Lecture 6: Propositional Logic

Fundamentals of Mathematics Lecture 6: Propositional Logic Guan-Shieng Huang National Chi Nan University, Taiwan Spring, 2008 1 / 39 Connectives Propositional Connectives I 1 Negation: (not A) A A T F

### AMS 5 CHANCE VARIABILITY

AMS 5 CHANCE VARIABILITY The Law of Averages When tossing a fair coin the chances of tails and heads are the same: 50% and 50%. So if the coin is tossed a large number of times, the number of heads and

### Last time we had arrived at the following provisional interpretation of Aquinas second way:

Aquinas Third Way Last time we had arrived at the following provisional interpretation of Aquinas second way: 1. 2. 3. 4. At least one thing has an efficient cause. Every causal chain must either be circular,

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### Section 7C: The Law of Large Numbers

Section 7C: The Law of Large Numbers Example. You flip a coin 00 times. Suppose the coin is fair. How many times would you expect to get heads? tails? One would expect a fair coin to come up heads half

### Self-locating belief and the Sleeping Beauty problem

Self-locating belief and the Sleeping Beauty problem Adam Elga Analysis, 60(2): 143-147, 2000. In addition to being uncertain about what the world is like, one can also be uncertain about one s own spatial

### 5544 = 2 2772 = 2 2 1386 = 2 2 2 693. Now we have to find a divisor of 693. We can try 3, and 693 = 3 231,and we keep dividing by 3 to get: 1

MATH 13150: Freshman Seminar Unit 8 1. Prime numbers 1.1. Primes. A number bigger than 1 is called prime if its only divisors are 1 and itself. For example, 3 is prime because the only numbers dividing

### Solutions: Problems for Chapter 3. Solutions: Problems for Chapter 3

Problem A: You are dealt five cards from a standard deck. Are you more likely to be dealt two pairs or three of a kind? experiment: choose 5 cards at random from a standard deck Ω = {5-combinations of

### P(A) = P - denotes a probability. A, B, and C - denote specific events. P (A) - denotes the probability of event A occurring. Chapter 4 Probability

4-1 Overview 4-2 Fundamentals 4-3 Addition Rule Chapter 4 Probability 4-4 Multiplication Rule: Basics 4-5 Multiplication Rule: Complements and Conditional Probability 4-6 Probabilities Through Simulations

### Prediction Markets, Fair Games and Martingales

Chapter 3 Prediction Markets, Fair Games and Martingales Prediction markets...... are speculative markets created for the purpose of making predictions. The current market prices can then be interpreted

### Elementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.

Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole

### In the situations that we will encounter, we may generally calculate the probability of an event

What does it mean for something to be random? An event is called random if the process which produces the outcome is sufficiently complicated that we are unable to predict the precise result and are instead

### Probability and Expected Value

Probability and Expected Value This handout provides an introduction to probability and expected value. Some of you may already be familiar with some of these topics. Probability and expected value are

### Expected Value. 24 February 2014. Expected Value 24 February 2014 1/19

Expected Value 24 February 2014 Expected Value 24 February 2014 1/19 This week we discuss the notion of expected value and how it applies to probability situations, including the various New Mexico Lottery

### How to bet and win: and why not to trust a winner. Niall MacKay. Department of Mathematics

How to bet and win: and why not to trust a winner Niall MacKay Department of Mathematics Two ways to win Arbitrage: exploit market imperfections to make a profit, with certainty Two ways to win Arbitrage:

### Probability in Philosophy

Probability in Philosophy Brian Weatherson Rutgers and Arché June, 2008 Brian Weatherson (Rutgers and Arché) Probability in Philosophy June, 2008 1 / 87 Orthodox Bayesianism Statics Credences are Probabilities

### Theorem 2. If x Q and y R \ Q, then. (a) x + y R \ Q, and. (b) xy Q.

Math 305 Fall 011 The Density of Q in R The following two theorems tell us what happens when we add and multiply by rational numbers. For the first one, we see that if we add or multiply two rational numbers

### Probabilistic Strategies: Solutions

Probability Victor Xu Probabilistic Strategies: Solutions Western PA ARML Practice April 3, 2016 1 Problems 1. You roll two 6-sided dice. What s the probability of rolling at least one 6? There is a 1

### Math 317 HW #5 Solutions

Math 317 HW #5 Solutions 1. Exercise 2.4.2. (a) Prove that the sequence defined by x 1 = 3 and converges. x n+1 = 1 4 x n Proof. I intend to use the Monotone Convergence Theorem, so my goal is to show

### Oh Yeah? Well, Prove It.

Oh Yeah? Well, Prove It. MT 43A - Abstract Algebra Fall 009 A large part of mathematics consists of building up a theoretical framework that allows us to solve problems. This theoretical framework is built

### Betting interpretations of probability

Betting interpretations of probability Glenn Shafer June 21, 2010 Third Workshop on Game-Theoretic Probability and Related Topics Royal Holloway, University of London 1 Outline 1. Probability began with

### Chapter 4 Lecture Notes

Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a real-valued function defined on the sample space of some experiment. For instance,

### Proof: A logical argument establishing the truth of the theorem given the truth of the axioms and any previously proven theorems.

Math 232 - Discrete Math 2.1 Direct Proofs and Counterexamples Notes Axiom: Proposition that is assumed to be true. Proof: A logical argument establishing the truth of the theorem given the truth of the

### MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

### Expected Value and the Game of Craps

Expected Value and the Game of Craps Blake Thornton Craps is a gambling game found in most casinos based on rolling two six sided dice. Most players who walk into a casino and try to play craps for the

### What Is Probability?

1 What Is Probability? The idea: Uncertainty can often be "quantified" i.e., we can talk about degrees of certainty or uncertainty. This is the idea of probability: a higher probability expresses a higher

### Probability Review. ICPSR Applied Bayesian Modeling

Probability Review ICPSR Applied Bayesian Modeling Random Variables Flip a coin. Will it be heads or tails? The outcome of a single event is random, or unpredictable What if we flip a coin 10 times? How

### MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 1 9/3/2008 PROBABILISTIC MODELS AND PROBABILITY MEASURES

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 1 9/3/2008 PROBABILISTIC MODELS AND PROBABILITY MEASURES Contents 1. Probabilistic experiments 2. Sample space 3. Discrete probability

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us

### IS HUME S ACCOUNT SCEPTICAL?

Michael Lacewing Hume on knowledge HUME S FORK Empiricism denies that we can know anything about how the world outside our own minds is without relying on sense experience. In IV, Hume argues that we can

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 13. Random Variables: Distribution and Expectation

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 3 Random Variables: Distribution and Expectation Random Variables Question: The homeworks of 20 students are collected

### Examples of infinite sample spaces. Math 425 Introduction to Probability Lecture 12. Example of coin tosses. Axiom 3 Strong form

Infinite Discrete Sample Spaces s of infinite sample spaces Math 425 Introduction to Probability Lecture 2 Kenneth Harris kaharri@umich.edu Department of Mathematics University of Michigan February 4,

### 1. (First passage/hitting times/gambler s ruin problem:) Suppose that X has a discrete state space and let i be a fixed state. Let

Copyright c 2009 by Karl Sigman 1 Stopping Times 1.1 Stopping Times: Definition Given a stochastic process X = {X n : n 0}, a random time τ is a discrete random variable on the same probability space as

### WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly

### Section 1. Statements and Truth Tables. Definition 1.1: A mathematical statement is a declarative sentence that is true or false, but not both.

M3210 Supplemental Notes: Basic Logic Concepts In this course we will examine statements about mathematical concepts and relationships between these concepts (definitions, theorems). We will also consider

### Physics 509: Intro to Bayesian Analysis

Physics 509: Intro to Bayesian Analysis Scott Oser Lecture #5 Thomas Bayes Physics 509 1 Outline Previously: we now have introduced all of the basic probability distributions and some useful computing

### Interpretations of Probability

Interpretations of Probability a. The Classical View: Subsets of a set of equally likely, mutually exclusive possibilities Problem of infinite or indefinite sets Problem of "equally likely" b. The Relative

### 6.3 Probabilities with Large Numbers

6.3 Probabilities with Large Numbers In general, we can t perfectly predict any single outcome when there are numerous things that could happen. But, when we repeatedly observe many observations, we expect

### PROBABILITY PROBABILITY

PROBABILITY PROBABILITY Documents prepared for use in course B0.305, New York University, Stern School of Business Types of probability page 3 The catchall term probability refers to several distinct ideas.

### Mathematical Induction

Mathematical Induction In logic, we often want to prove that every member of an infinite set has some feature. E.g., we would like to show: N 1 : is a number 1 : has the feature Φ ( x)(n 1 x! 1 x) How

### Math 317 HW #7 Solutions

Math 17 HW #7 Solutions 1. Exercise..5. Decide which of the following sets are compact. For those that are not compact, show how Definition..1 breaks down. In other words, give an example of a sequence

### Lecture 11: Probability models

Lecture 11: Probability models Probability is the mathematical toolbox to describe phenomena or experiments where randomness occur. To have a probability model we need the following ingredients A sample

### White Paper. Probability

Probability Probability is central to project management. We need to understand the probability of achieving cost or time outcomes and the probability of a risk event occurring. Probability is also closely

### Phil 420: Metaphysics Spring 2008. [Handout 4] Hilary Putnam: Why There Isn t A Ready-Made World

1 Putnam s Main Theses: 1. There is no ready-made world. Phil 420: Metaphysics Spring 2008 [Handout 4] Hilary Putnam: Why There Isn t A Ready-Made World * [A ready-made world]: The world itself has to

### Introductory Probability. MATH 107: Finite Mathematics University of Louisville. March 5, 2014

Introductory Probability MATH 07: Finite Mathematics University of Louisville March 5, 204 What is probability? Counting and probability 2 / 3 Probability in our daily lives We see chances, odds, and probabilities

### Combinatorics: The Fine Art of Counting

Combinatorics: The Fine Art of Counting Week 7 Lecture Notes Discrete Probability Continued Note Binomial coefficients are written horizontally. The symbol ~ is used to mean approximately equal. The Bernoulli

### Belief and Degrees of Belief

Belief and Degrees of Belief Franz Huber, California Institute of Technology Draft June 12, 2006 Contents 1 Introduction 2 2 The Objects of Belief 3 3 Theories of Degrees of Belief 4 3.1 Subjective Probabilities.......................

### Self-Referential Probabilities

Self-Referential Probabilities And a Kripkean semantics Catrin Campbell-Moore Munich Center for Mathematical Philosophy Bridges 2 September 2015 Supported by Introduction Languages that can talk about

### Full and Complete Binary Trees

Full and Complete Binary Trees Binary Tree Theorems 1 Here are two important types of binary trees. Note that the definitions, while similar, are logically independent. Definition: a binary tree T is full

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 1

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 1 Course Outline CS70 is a course on Discrete Mathematics and Probability for EECS Students. The purpose of the course

### General Philosophy. Dr Peter Millican, Hertford College. Lecture 3: Induction

General Philosophy Dr Peter Millican, Hertford College Lecture 3: Induction Hume s s Fork 2 Enquiry IV starts with a vital distinction between types of proposition: Relations of ideas can be known a priori

### Chapter 6 Finite sets and infinite sets. Copyright 2013, 2005, 2001 Pearson Education, Inc. Section 3.1, Slide 1

Chapter 6 Finite sets and infinite sets Copyright 013, 005, 001 Pearson Education, Inc. Section 3.1, Slide 1 Section 6. PROPERTIES OF THE NATURE NUMBERS 013 Pearson Education, Inc.1 Slide Recall that denotes

### Lecture 3. Mathematical Induction

Lecture 3 Mathematical Induction Induction is a fundamental reasoning process in which general conclusion is based on particular cases It contrasts with deduction, the reasoning process in which conclusion

### 36 Odds, Expected Value, and Conditional Probability

36 Odds, Expected Value, and Conditional Probability What s the difference between probabilities and odds? To answer this question, let s consider a game that involves rolling a die. If one gets the face

### Lecture 11 Uncertainty

Lecture 11 Uncertainty 1. Contingent Claims and the State-Preference Model 1) Contingent Commodities and Contingent Claims Using the simple two-good model we have developed throughout this course, think

### CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE

CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify

### This chapter is all about cardinality of sets. At first this looks like a

CHAPTER Cardinality of Sets This chapter is all about cardinality of sets At first this looks like a very simple concept To find the cardinality of a set, just count its elements If A = { a, b, c, d },

### Economics 1011a: Intermediate Microeconomics

Lecture 12: More Uncertainty Economics 1011a: Intermediate Microeconomics Lecture 12: More on Uncertainty Thursday, October 23, 2008 Last class we introduced choice under uncertainty. Today we will explore

### Events. Independence. Coin Tossing. Random Phenomena

Random Phenomena Events A random phenomenon is a situation in which we know what outcomes could happen, but we don t know which particular outcome did or will happen For any random phenomenon, each attempt,

### PHLA The Problem of Induction

Knowledge versus mere justified belief Knowledge implies truth Justified belief does not imply truth Knowledge implies the impossibility of error Justified belief does not imply impossibility of error

### Comparison of frequentist and Bayesian inference. Class 20, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Comparison of frequentist and Bayesian inference. Class 20, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Be able to explain the difference between the p-value and a posterior

### MATHEMATICAL INDUCTION. Mathematical Induction. This is a powerful method to prove properties of positive integers.

MATHEMATICAL INDUCTION MIGUEL A LERMA (Last updated: February 8, 003) Mathematical Induction This is a powerful method to prove properties of positive integers Principle of Mathematical Induction Let P

### Descartes Handout #2. Meditation II and III

Descartes Handout #2 Meditation II and III I. Meditation II: The Cogito and Certainty A. I think, therefore I am cogito ergo sum In Meditation II Descartes proposes a truth that cannot be undermined by

### Decision Theory. 36.1 Rational prospecting

36 Decision Theory Decision theory is trivial, apart from computational details (just like playing chess!). You have a choice of various actions, a. The world may be in one of many states x; which one

### 1/9. Locke 1: Critique of Innate Ideas

1/9 Locke 1: Critique of Innate Ideas This week we are going to begin looking at a new area by turning our attention to the work of John Locke, who is probably the most famous English philosopher of all

Chapter 3 Probability 2012 Pearson Education, Inc. All rights reserved. 1 of 20 Chapter Outline 3.1 Basic Concepts of Probability 3.2 Conditional Probability and the Multiplication Rule 3.3 The Addition

### Math 421: Probability and Statistics I Note Set 2

Math 421: Probability and Statistics I Note Set 2 Marcus Pendergrass September 13, 2013 4 Discrete Probability Discrete probability is concerned with situations in which you can essentially list all the

### CHAPTER 2 Estimating Probabilities

CHAPTER 2 Estimating Probabilities Machine Learning Copyright c 2016. Tom M. Mitchell. All rights reserved. *DRAFT OF January 24, 2016* *PLEASE DO NOT DISTRIBUTE WITHOUT AUTHOR S PERMISSION* This is a

### What Is Probability? 1

What Is Probability? 1 Glenn Shafer 1 Introduction What is probability? What does it mean to say that the probability of an event is 75%? Is this the frequency with which the event happens? Is it the degree

### MA 1125 Lecture 14 - Expected Values. Friday, February 28, 2014. Objectives: Introduce expected values.

MA 5 Lecture 4 - Expected Values Friday, February 2, 24. Objectives: Introduce expected values.. Means, Variances, and Standard Deviations of Probability Distributions Two classes ago, we computed the

### An Innocent Investigation

An Innocent Investigation D. Joyce, Clark University January 2006 The beginning. Have you ever wondered why every number is either even or odd? I don t mean to ask if you ever wondered whether every number

### Software Verification and System Assurance

Software Verification and System Assurance John Rushby Based on joint work with Bev Littlewood (City University UK) Computer Science Laboratory SRI International Menlo Park CA USA John Rushby, SR I Verification

### Probability OPRE 6301

Probability OPRE 6301 Random Experiment... Recall that our eventual goal in this course is to go from the random sample to the population. The theory that allows for this transition is the theory of probability.