CS340 Machine learning Bayesian statistics 3
|
|
- Dylan Rodgers
- 5 years ago
- Views:
Transcription
1 CS340 Machine learning Bayesian statistics 3 1
2 Outline Conjugate analysis of µ and σ 2 Bayesian model selection Summarizing the posterior 2
3 Unknown mean and precision The likelihood function is p(d µ,λ) = = 1 (2π) n/2λn/2 exp 1 (2π) n/2λn/2 exp The natural conjugate prior is normal gamma ( ( λ 2 λ 2 ) n (x i µ) 2 i=1 p(µ,λ) = NG(µ,λ µ 0,κ 0,α 0,β 0 ) def [ n(µ x) 2 + ]) n (x i x) 2 i=1 = N(µ µ 0,(κ 0 λ) 1 )Ga(λ α 0,rate=β 0 ) ( 1 = λ α 1 0 2exp λ [ κ0 (µ µ 0 ) 2 ] ) +2β 0 Z NG 2 3
4 Used for positive reals Gamma distribution Ga(x shape=a,rate=b) = Γ(a) xa 1 e xb, x,a,b>0 1 Ga(x shape=α,scale=β) = β α Γ(α) xα 1 e x/β b a Bishop Matlab 4
5 Posterior is also NG Just update the hyper-parameters p(µ,λ D) = NG(µ,λ µ n,κ n,α n,β n ) µ n = κ 0µ 0 +nx κ 0 +n κ n = κ 0 +n α n = α 0 +n/2 n β n = β (x i x) 2 + κ 0n(x µ 0 ) 2 2(κ 0 +n) i=1 Derivation of this result not on exam 5
6 Posterior marginals Variance p(λ D) = Ga(λ α n,β n ) Mean p(µ D) = T 2αn (µ µ n, β n α n κ n ) Student t distribution Derivation of this result not on exam 6
7 Student t distribution Approaches Gaussian as ν t ν (x µ,σ 2 ) [ 1+ 1 ] ( ν+1 ν (x µ 2 ) σ )2 7
8 Robustness of t distribution Student t less affected by outliers Bishop 2.16 Gaussian 8
9 Posterior predictive distribution Also a t distribution (fatter tails than Gaussian due to uncertainty in λ) p(x D) = t 2αn (x µ n, β n(κ n +1) α n κ n ) Derivation of this result not on exam 9
10 Uninformative prior It can be shown (see handout) that an uninformative prior has the form p(µ,λ) 1 λ This can be emulated using the following hyper-parameters κ 0 = 0 a 0 = 1 2 b 0 = 0 This prior is improper (does not integrate to 1), but the posterior is proper if n 1 Derivation of this result not on exam 10
11 Outline Conjugate analysis of µ and σ 2 Bayesian model selection Summarizing the posterior 11
12 Bayesian model selection Suppose we have K possible models, each with parameters θ i. The posterior over models is defined using the marginal likelihood ( evidence ) p(d M=i), which is the normalizing constant from the posterior over parameters p(m =i D) = p(d M =i) = p(m =i)p(d M =i) p(d) p(d θ,m =i)p(θ M =i)dθ p(θ D,M =i) = p(d θ,m =i)p(θ M=i) p(d M =i) 12
13 Bayes factors To compare two models, use posterior odds O ij = p(m i D) p(m j D) = p(d M i)p(m i ) p(d M j )p(m j ) Bayes factor Prior odds The Bayes factor BF(i,j) is a Bayesian version of a likelihood ratio test, that can be used to compare models of different complexity 13
14 Marginal likelihood for Beta-Bernoulli Since we know p(θ D) = Be(α 1,α 0 ) p(θ D) = p(θ)p(d θ) p(d) [ ] 1 1 = p(d) B(α 1,α 0 ) θα 1 1 (1 θ) α 0 1 [θ N 1 (1 θ) N ] 0 = θα 1 1 (1 θ) α 0 1 B(α 1,α 0 ) Hence the marginal likelihood is a ratio of normalizing constants p(d)= p(d θ)p(θ)dθ= B(α 1,α 0) B(α 1,α 0 ) 14
15 Example: is the Eurocoin biased? Suppose we toss a coin N=250 times and observe N 1 =141 heads and N 0 =109 tails. Consider two hypotheses: H 0 that θ=0.5 and H 1 that θ 0.5. Actually, we can let H 1 be p(θ) = U(0,1), since p(θ=0.5 H 1 ) = 0 (pdf). For H 0, marginal likelihood is p(d H 0 )=0.5 N For H 1, marginal likelihood is P(D H 1 )= 1 Hence the Bayes factor is 0 P(D θ,h 1 )P(θ H 1 )dθ= B(α 1+N 1,α 0 +N 0 ) B(α 1,α 0 ) BF(1,0)= P(D H 1) P(D H 0 ) = B(α 1+N 1,α 0 +N 0 ) B(α 1,α 0 ) N 15
16 Bayes factor vs prior strength Let α 1 =α 0 range from 0 to The largest BF in favor of H1 (biased coin) is only 2.0, which is very weak evidence of bias. BF(1,0) α 16
17 Bayesian Occam s razor The use of the marginal likelihood p(d M) automatically penalizes overly complex models, since they spread their probability mass very widely (predict that everything is possible), so the probability of the actual data is small. Too simple, cannot predict D Just right Too complex, can predict everything Bishop
18 Bayesian Occam s razor for biased coin Blue line = p(d H 0 ) = 0.5 N Red curve = p(d H 1 ) = p(d θ) Beta(θ 1,1) d θ If we have already observed 4 heads, it is much more likely to observe a 5 th head than Marginal a likelihood tail, since of biased θ gets coin updated sequentially If we observe 2 or 3 heads out of 5, the simpler model is more likely num heads 18
19 Bayesian Information Criterion (BIC) If we make a Gaussian approximation to p(θ D) (Laplace approximation), and approximate H N d, the log marginal likelihood becomes logp(d) logp(d θ ML ) 1 2 dlogn Here d is the dimension/ number of free parameters. AIC (Akaike Info criterion) is defined as logp(d) logp(d θ ML ) d Can use penalized log-likelihood for model selection instead of cross-validation. 19
20 Outline Conjugate analysis of µ and σ 2 Bayesian model selection Summarizing the posterior 20
21 Summarizing the posterior If p(θ D) is too complex to plot, we can compute various summary statistics, such as posterior mean, mode and median ˆθ mean = E[θ D] ˆθ MAP = argmaxp(θ D) θ ˆθ median = t:p(θ>t D)=0.5 21
22 Bayesian credible intervals We can represent our uncertainty using a posterior credible interval We set p(l θ u D) 1 α l=f 1 (α/2),u=f 1 (1 α/2) 22
23 Example We see 47 heads out of 100 trials. Using a Beta(1,1) prior, what is the 95% credible interval for probability of heads? S = 47; N = 100; a = S+1; b = (N-S)+1; alpha = 0.05; l = betainv(alpha/2, a, b); u = betainv(1-alpha/2, a, b); CI = [l,u]
24 Posterior sampling If θ is high-dimensional, it is hard to visualize p(θ D). A common strategy is to draw typical values θ s p(θ D) and analyze the resulting samples. Eg we can generate fake data p(x s θ s ) to see if it looks like the real data (a simple kind of posterior predictive check of model adequacy). See handout for some examples. 24
CS340 Machine learning Bayesian model selection
CS340 Machine learning Bayesian model selection Bayesian model selection Suppose we have several models, each with potentially different numbers of parameters. Example: M0 = constant, M1 = straight line,
More informationChapter 7: Estimation Sections
1 / 31 : Estimation Sections 7.1 Statistical Inference Bayesian Methods: 7.2 Prior and Posterior Distributions 7.3 Conjugate Prior Distributions 7.4 Bayes Estimators Frequentist Methods: 7.5 Maximum Likelihood
More informationChapter 7: Estimation Sections
1 / 40 Chapter 7: Estimation Sections 7.1 Statistical Inference Bayesian Methods: Chapter 7 7.2 Prior and Posterior Distributions 7.3 Conjugate Prior Distributions 7.4 Bayes Estimators Frequentist Methods:
More informationST440/550: Applied Bayesian Analysis. (5) Multi-parameter models - Summarizing the posterior
(5) Multi-parameter models - Summarizing the posterior Models with more than one parameter Thus far we have studied single-parameter models, but most analyses have several parameters For example, consider
More information(5) Multi-parameter models - Summarizing the posterior
(5) Multi-parameter models - Summarizing the posterior Spring, 2017 Models with more than one parameter Thus far we have studied single-parameter models, but most analyses have several parameters For example,
More informationConjugate priors: Beta and normal Class 15, Jeremy Orloff and Jonathan Bloom
1 Learning Goals Conjugate s: Beta and normal Class 15, 18.05 Jeremy Orloff and Jonathan Bloom 1. Understand the benefits of conjugate s.. Be able to update a beta given a Bernoulli, binomial, or geometric
More informationConjugate Models. Patrick Lam
Conjugate Models Patrick Lam Outline Conjugate Models What is Conjugacy? The Beta-Binomial Model The Normal Model Normal Model with Unknown Mean, Known Variance Normal Model with Known Mean, Unknown Variance
More informationObjective Bayesian Analysis for Heteroscedastic Regression
Analysis for Heteroscedastic Regression & Esther Salazar Universidade Federal do Rio de Janeiro Colóquio Inter-institucional: Modelos Estocásticos e Aplicações 2009 Collaborators: Marco Ferreira and Thais
More informationChapter 7: Estimation Sections
Chapter 7: Estimation Sections 7.1 Statistical Inference Bayesian Methods: 7.2 Prior and Posterior Distributions 7.3 Conjugate Prior Distributions Frequentist Methods: 7.5 Maximum Likelihood Estimators
More informationChapter 8: Sampling distributions of estimators Sections
Chapter 8 continued Chapter 8: Sampling distributions of estimators Sections 8.1 Sampling distribution of a statistic 8.2 The Chi-square distributions 8.3 Joint Distribution of the sample mean and sample
More informationPosterior Inference. , where should we start? Consider the following computational procedure: 1. draw samples. 2. convert. 3. compute properties
Posterior Inference Example. Consider a binomial model where we have a posterior distribution for the probability term, θ. Suppose we want to make inferences about the log-odds γ = log ( θ 1 θ), where
More informationSTART HERE: Instructions. 1 Exponential Family [Zhou, Manzil]
START HERE: Instructions Thanks a lot to John A.W.B. Constanzo and Shi Zong for providing and allowing to use the latex source files for quick preparation of the HW solution. The homework was due at 9:00am
More informationTwo hours. To be supplied by the Examinations Office: Mathematical Formula Tables and Statistical Tables THE UNIVERSITY OF MANCHESTER
Two hours MATH20802 To be supplied by the Examinations Office: Mathematical Formula Tables and Statistical Tables THE UNIVERSITY OF MANCHESTER STATISTICAL METHODS Answer any FOUR of the SIX questions.
More informationSTAT 425: Introduction to Bayesian Analysis
STAT 45: Introduction to Bayesian Analysis Marina Vannucci Rice University, USA Fall 018 Marina Vannucci (Rice University, USA) Bayesian Analysis (Part 1) Fall 018 1 / 37 Lectures 9-11: Multi-parameter
More informationNon-informative Priors Multiparameter Models
Non-informative Priors Multiparameter Models Statistics 220 Spring 2005 Copyright c 2005 by Mark E. Irwin Prior Types Informative vs Non-informative There has been a desire for a prior distributions that
More informationcontinuous rv Note for a legitimate pdf, we have f (x) 0 and f (x)dx = 1. For a continuous rv, P(X = c) = c f (x)dx = 0, hence
continuous rv Let X be a continuous rv. Then a probability distribution or probability density function (pdf) of X is a function f(x) such that for any two numbers a and b with a b, P(a X b) = b a f (x)dx.
More informationChapter 4: Commonly Used Distributions. Statistics for Engineers and Scientists Fourth Edition William Navidi
Chapter 4: Commonly Used Distributions Statistics for Engineers and Scientists Fourth Edition William Navidi 2014 by Education. This is proprietary material solely for authorized instructor use. Not authorized
More informationOutline. Review Continuation of exercises from last time
Bayesian Models II Outline Review Continuation of exercises from last time 2 Review of terms from last time Probability density function aka pdf or density Likelihood function aka likelihood Conditional
More informationThis is a open-book exam. Assigned: Friday November 27th 2009 at 16:00. Due: Monday November 30th 2009 before 10:00.
University of Iceland School of Engineering and Sciences Department of Industrial Engineering, Mechanical Engineering and Computer Science IÐN106F Industrial Statistics II - Bayesian Data Analysis Fall
More informationCS 361: Probability & Statistics
March 12, 2018 CS 361: Probability & Statistics Inference Binomial likelihood: Example Suppose we have a coin with an unknown probability of heads. We flip the coin 10 times and observe 2 heads. What can
More informationME3620. Theory of Engineering Experimentation. Spring Chapter III. Random Variables and Probability Distributions.
ME3620 Theory of Engineering Experimentation Chapter III. Random Variables and Probability Distributions Chapter III 1 3.2 Random Variables In an experiment, a measurement is usually denoted by a variable
More informationReview for Final Exam Spring 2014 Jeremy Orloff and Jonathan Bloom
Review for Final Exam 18.05 Spring 2014 Jeremy Orloff and Jonathan Bloom THANK YOU!!!! JON!! PETER!! RUTHI!! ERIKA!! ALL OF YOU!!!! Probability Counting Sets Inclusion-exclusion principle Rule of product
More informationBayesian Hierarchical/ Multilevel and Latent-Variable (Random-Effects) Modeling
Bayesian Hierarchical/ Multilevel and Latent-Variable (Random-Effects) Modeling 1: Formulation of Bayesian models and fitting them with MCMC in WinBUGS David Draper Department of Applied Mathematics and
More informationCommon one-parameter models
Common one-parameter models In this section we will explore common one-parameter models, including: 1. Binomial data with beta prior on the probability 2. Poisson data with gamma prior on the rate 3. Gaussian
More information4-1. Chapter 4. Commonly Used Distributions by The McGraw-Hill Companies, Inc. All rights reserved.
4-1 Chapter 4 Commonly Used Distributions 2014 by The Companies, Inc. All rights reserved. Section 4.1: The Bernoulli Distribution 4-2 We use the Bernoulli distribution when we have an experiment which
More informationCommonly Used Distributions
Chapter 4: Commonly Used Distributions 1 Introduction Statistical inference involves drawing a sample from a population and analyzing the sample data to learn about the population. We often have some knowledge
More informationThe Bernoulli distribution
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this
More informationBusiness Statistics 41000: Probability 3
Business Statistics 41000: Probability 3 Drew D. Creal University of Chicago, Booth School of Business February 7 and 8, 2014 1 Class information Drew D. Creal Email: dcreal@chicagobooth.edu Office: 404
More informationدرس هفتم یادگیري ماشین. (Machine Learning) دانشگاه فردوسی مشهد دانشکده مهندسی رضا منصفی
یادگیري ماشین توزیع هاي نمونه و تخمین نقطه اي پارامترها Sampling Distributions and Point Estimation of Parameter (Machine Learning) دانشگاه فردوسی مشهد دانشکده مهندسی رضا منصفی درس هفتم 1 Outline Introduction
More informationBayesian course - problem set 3 (lecture 4)
Bayesian course - problem set 3 (lecture 4) Ben Lambert November 14, 2016 1 Ticked off Imagine once again that you are investigating the occurrence of Lyme disease in the UK. This is a vector-borne disease
More informationDefinition 9.1 A point estimate is any function T (X 1,..., X n ) of a random sample. We often write an estimator of the parameter θ as ˆθ.
9 Point estimation 9.1 Rationale behind point estimation When sampling from a population described by a pdf f(x θ) or probability function P [X = x θ] knowledge of θ gives knowledge of the entire population.
More informationExam 2 Spring 2015 Statistics for Applications 4/9/2015
18.443 Exam 2 Spring 2015 Statistics for Applications 4/9/2015 1. True or False (and state why). (a). The significance level of a statistical test is not equal to the probability that the null hypothesis
More informationThe Normal Distribution
The Normal Distribution The normal distribution plays a central role in probability theory and in statistics. It is often used as a model for the distribution of continuous random variables. Like all models,
More information8.1 Estimation of the Mean and Proportion
8.1 Estimation of the Mean and Proportion Statistical inference enables us to make judgments about a population on the basis of sample information. The mean, standard deviation, and proportions of a population
More informationChapter 5. Statistical inference for Parametric Models
Chapter 5. Statistical inference for Parametric Models Outline Overview Parameter estimation Method of moments How good are method of moments estimates? Interval estimation Statistical Inference for Parametric
More informationBayesian Normal Stuff
Bayesian Normal Stuff - Set-up of the basic model of a normally distributed random variable with unknown mean and variance (a two-parameter model). - Discuss philosophies of prior selection - Implementation
More informationSYSM 6304 Risk and Decision Analysis Lecture 2: Fitting Distributions to Data
SYSM 6304 Risk and Decision Analysis Lecture 2: Fitting Distributions to Data M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu September 5, 2015
More informationBack to estimators...
Back to estimators... So far, we have: Identified estimators for common parameters Discussed the sampling distributions of estimators Introduced ways to judge the goodness of an estimator (bias, MSE, etc.)
More informationModeling skewness and kurtosis in Stochastic Volatility Models
Modeling skewness and kurtosis in Stochastic Volatility Models Georgios Tsiotas University of Crete, Department of Economics, GR December 19, 2006 Abstract Stochastic volatility models have been seen as
More informationHigh Dimensional Bayesian Optimisation and Bandits via Additive Models
1/20 High Dimensional Bayesian Optimisation and Bandits via Additive Models Kirthevasan Kandasamy, Jeff Schneider, Barnabás Póczos ICML 15 July 8 2015 2/20 Bandits & Optimisation Maximum Likelihood inference
More informationCSC 411: Lecture 08: Generative Models for Classification
CSC 411: Lecture 08: Generative Models for Classification Richard Zemel, Raquel Urtasun and Sanja Fidler University of Toronto Zemel, Urtasun, Fidler (UofT) CSC 411: 08-Generative Models 1 / 23 Today Classification
More informationMVE051/MSG Lecture 7
MVE051/MSG810 2017 Lecture 7 Petter Mostad Chalmers November 20, 2017 The purpose of collecting and analyzing data Purpose: To build and select models for parts of the real world (which can be used for
More informationECE 340 Probabilistic Methods in Engineering M/W 3-4:15. Lecture 10: Continuous RV Families. Prof. Vince Calhoun
ECE 340 Probabilistic Methods in Engineering M/W 3-4:15 Lecture 10: Continuous RV Families Prof. Vince Calhoun 1 Reading This class: Section 4.4-4.5 Next class: Section 4.6-4.7 2 Homework 3.9, 3.49, 4.5,
More informationRandom Variables Handout. Xavier Vilà
Random Variables Handout Xavier Vilà Course 2004-2005 1 Discrete Random Variables. 1.1 Introduction 1.1.1 Definition of Random Variable A random variable X is a function that maps each possible outcome
More informationMATH 3200 Exam 3 Dr. Syring
. Suppose n eligible voters are polled (randomly sampled) from a population of size N. The poll asks voters whether they support or do not support increasing local taxes to fund public parks. Let M be
More informationLecture Notes 6. Assume F belongs to a family of distributions, (e.g. F is Normal), indexed by some parameter θ.
Sufficient Statistics Lecture Notes 6 Sufficiency Data reduction in terms of a particular statistic can be thought of as a partition of the sample space X. Definition T is sufficient for θ if the conditional
More informationLecture 10: Point Estimation
Lecture 10: Point Estimation MSU-STT-351-Sum-17B (P. Vellaisamy: MSU-STT-351-Sum-17B) Probability & Statistics for Engineers 1 / 31 Basic Concepts of Point Estimation A point estimate of a parameter θ,
More informationContinuous Distributions
Quantitative Methods 2013 Continuous Distributions 1 The most important probability distribution in statistics is the normal distribution. Carl Friedrich Gauss (1777 1855) Normal curve A normal distribution
More informationMetropolis-Hastings algorithm
Metropolis-Hastings algorithm Dr. Jarad Niemi STAT 544 - Iowa State University March 27, 2018 Jarad Niemi (STAT544@ISU) Metropolis-Hastings March 27, 2018 1 / 32 Outline Metropolis-Hastings algorithm Independence
More informationStatistics 6 th Edition
Statistics 6 th Edition Chapter 5 Discrete Probability Distributions Chap 5-1 Definitions Random Variables Random Variables Discrete Random Variable Continuous Random Variable Ch. 5 Ch. 6 Chap 5-2 Discrete
More informationProbability Theory and Simulation Methods. April 9th, Lecture 20: Special distributions
April 9th, 2018 Lecture 20: Special distributions Week 1 Chapter 1: Axioms of probability Week 2 Chapter 3: Conditional probability and independence Week 4 Chapters 4, 6: Random variables Week 9 Chapter
More informationINSTITUTE AND FACULTY OF ACTUARIES. Curriculum 2019 SPECIMEN EXAMINATION
INSTITUTE AND FACULTY OF ACTUARIES Curriculum 2019 SPECIMEN EXAMINATION Subject CS1A Actuarial Statistics Time allowed: Three hours and fifteen minutes INSTRUCTIONS TO THE CANDIDATE 1. Enter all the candidate
More informationSOCIETY OF ACTUARIES EXAM STAM SHORT-TERM ACTUARIAL MATHEMATICS EXAM STAM SAMPLE QUESTIONS
SOCIETY OF ACTUARIES EXAM STAM SHORT-TERM ACTUARIAL MATHEMATICS EXAM STAM SAMPLE QUESTIONS Questions 1-307 have been taken from the previous set of Exam C sample questions. Questions no longer relevant
More informationChapter 4: Asymptotic Properties of MLE (Part 3)
Chapter 4: Asymptotic Properties of MLE (Part 3) Daniel O. Scharfstein 09/30/13 1 / 1 Breakdown of Assumptions Non-Existence of the MLE Multiple Solutions to Maximization Problem Multiple Solutions to
More information2 of PU_2015_375 Which of the following measures is more flexible when compared to other measures?
PU M Sc Statistics 1 of 100 194 PU_2015_375 The population census period in India is for every:- quarterly Quinqennial year biannual Decennial year 2 of 100 105 PU_2015_375 Which of the following measures
More informationEstimation after Model Selection
Estimation after Model Selection Vanja M. Dukić Department of Health Studies University of Chicago E-Mail: vanja@uchicago.edu Edsel A. Peña* Department of Statistics University of South Carolina E-Mail:
More informationExam STAM Practice Exam #1
!!!! Exam STAM Practice Exam #1 These practice exams should be used during the month prior to your exam. This practice exam contains 20 questions, of equal value, corresponding to about a 2 hour exam.
More information**BEGINNING OF EXAMINATION** A random sample of five observations from a population is:
**BEGINNING OF EXAMINATION** 1. You are given: (i) A random sample of five observations from a population is: 0.2 0.7 0.9 1.1 1.3 (ii) You use the Kolmogorov-Smirnov test for testing the null hypothesis,
More informationGOV 2001/ 1002/ E-200 Section 3 Inference and Likelihood
GOV 2001/ 1002/ E-200 Section 3 Inference and Likelihood Anton Strezhnev Harvard University February 10, 2016 1 / 44 LOGISTICS Reading Assignment- Unifying Political Methodology ch 4 and Eschewing Obfuscation
More informationBayesian Linear Model: Gory Details
Bayesian Linear Model: Gory Details Pubh7440 Notes By Sudipto Banerjee Let y y i ] n i be an n vector of independent observations on a dependent variable (or response) from n experimental units. Associated
More informationLecture 2. Probability Distributions Theophanis Tsandilas
Lecture 2 Probability Distributions Theophanis Tsandilas Comment on measures of dispersion Why do common measures of dispersion (variance and standard deviation) use sums of squares: nx (x i ˆµ) 2 i=1
More information1 Bayesian Bias Correction Model
1 Bayesian Bias Correction Model Assuming that n iid samples {X 1,...,X n }, were collected from a normal population with mean µ and variance σ 2. The model likelihood has the form, P( X µ, σ 2, T n >
More informationStochastic Volatility (SV) Models
1 Motivations Stochastic Volatility (SV) Models Jun Yu Some stylised facts about financial asset return distributions: 1. Distribution is leptokurtic 2. Volatility clustering 3. Volatility responds to
More informationECE 295: Lecture 03 Estimation and Confidence Interval
ECE 295: Lecture 03 Estimation and Confidence Interval Spring 2018 Prof Stanley Chan School of Electrical and Computer Engineering Purdue University 1 / 23 Theme of this Lecture What is Estimation? You
More informationThe normal distribution is a theoretical model derived mathematically and not empirically.
Sociology 541 The Normal Distribution Probability and An Introduction to Inferential Statistics Normal Approximation The normal distribution is a theoretical model derived mathematically and not empirically.
More informationBinomial Random Variables. Binomial Random Variables
Bernoulli Trials Definition A Bernoulli trial is a random experiment in which there are only two possible outcomes - success and failure. 1 Tossing a coin and considering heads as success and tails as
More informationConfidence Intervals Introduction
Confidence Intervals Introduction A point estimate provides no information about the precision and reliability of estimation. For example, the sample mean X is a point estimate of the population mean μ
More informationProbability. An intro for calculus students P= Figure 1: A normal integral
Probability An intro for calculus students.8.6.4.2 P=.87 2 3 4 Figure : A normal integral Suppose we flip a coin 2 times; what is the probability that we get more than 2 heads? Suppose we roll a six-sided
More informationSubject CS1 Actuarial Statistics 1 Core Principles. Syllabus. for the 2019 exams. 1 June 2018
` Subject CS1 Actuarial Statistics 1 Core Principles Syllabus for the 2019 exams 1 June 2018 Copyright in this Core Reading is the property of the Institute and Faculty of Actuaries who are the sole distributors.
More informationIntroduction to Probability and Inference HSSP Summer 2017, Instructor: Alexandra Ding July 19, 2017
Introduction to Probability and Inference HSSP Summer 2017, Instructor: Alexandra Ding July 19, 2017 Please fill out the attendance sheet! Suggestions Box: Feedback and suggestions are important to the
More informationExtended Model: Posterior Distributions
APPENDIX A Extended Model: Posterior Distributions A. Homoskedastic errors Consider the basic contingent claim model b extended by the vector of observables x : log C i = β log b σ, x i + β x i + i, i
More informationLearning From Data: MLE. Maximum Likelihood Estimators
Learning From Data: MLE Maximum Likelihood Estimators 1 Parameter Estimation Assuming sample x1, x2,..., xn is from a parametric distribution f(x θ), estimate θ. E.g.: Given sample HHTTTTTHTHTTTHH of (possibly
More informationVersion A. Problem 1. Let X be the continuous random variable defined by the following pdf: 1 x/2 when 0 x 2, f(x) = 0 otherwise.
Math 224 Q Exam 3A Fall 217 Tues Dec 12 Version A Problem 1. Let X be the continuous random variable defined by the following pdf: { 1 x/2 when x 2, f(x) otherwise. (a) Compute the mean µ E[X]. E[X] x
More informationPractice Exercises for Midterm Exam ST Statistical Theory - II The ACTUAL exam will consists of less number of problems.
Practice Exercises for Midterm Exam ST 522 - Statistical Theory - II The ACTUAL exam will consists of less number of problems. 1. Suppose X i F ( ) for i = 1,..., n, where F ( ) is a strictly increasing
More informationA random variable (r. v.) is a variable whose value is a numerical outcome of a random phenomenon.
Chapter 14: random variables p394 A random variable (r. v.) is a variable whose value is a numerical outcome of a random phenomenon. Consider the experiment of tossing a coin. Define a random variable
More informationWhat was in the last lecture?
What was in the last lecture? Normal distribution A continuous rv with bell-shaped density curve The pdf is given by f(x) = 1 2πσ e (x µ)2 2σ 2, < x < If X N(µ, σ 2 ), E(X) = µ and V (X) = σ 2 Standard
More informationThe binomial distribution p314
The binomial distribution p314 Example: A biased coin (P(H) = p = 0.6) ) is tossed 5 times. Let X be the number of H s. Fine P(X = 2). This X is a binomial r. v. The binomial setting p314 1. There are
More informationChapter 8. Introduction to Statistical Inference
Chapter 8. Introduction to Statistical Inference Point Estimation Statistical inference is to draw some type of conclusion about one or more parameters(population characteristics). Now you know that a
More informationBusiness Statistics 41000: Probability 4
Business Statistics 41000: Probability 4 Drew D. Creal University of Chicago, Booth School of Business February 14 and 15, 2014 1 Class information Drew D. Creal Email: dcreal@chicagobooth.edu Office:
More information1/2 2. Mean & variance. Mean & standard deviation
Question # 1 of 10 ( Start time: 09:46:03 PM ) Total Marks: 1 The probability distribution of X is given below. x: 0 1 2 3 4 p(x): 0.73? 0.06 0.04 0.01 What is the value of missing probability? 0.54 0.16
More informationModeling Co-movements and Tail Dependency in the International Stock Market via Copulae
Modeling Co-movements and Tail Dependency in the International Stock Market via Copulae Katja Ignatieva, Eckhard Platen Bachelier Finance Society World Congress 22-26 June 2010, Toronto K. Ignatieva, E.
More informationStatistics and Probability
Statistics and Probability Continuous RVs (Normal); Confidence Intervals Outline Continuous random variables Normal distribution CLT Point estimation Confidence intervals http://www.isrec.isb-sib.ch/~darlene/geneve/
More informationIEOR E4602: Quantitative Risk Management
IEOR E4602: Quantitative Risk Management Basic Concepts and Techniques of Risk Management Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com
More informationChapter 4: Estimation
Slide 4.1 Chapter 4: Estimation Estimation is the process of using sample data to draw inferences about the population Sample information x, s Inferences Population parameters µ,σ Slide 4. Point and interval
More informationStatistical Tables Compiled by Alan J. Terry
Statistical Tables Compiled by Alan J. Terry School of Science and Sport University of the West of Scotland Paisley, Scotland Contents Table 1: Cumulative binomial probabilities Page 1 Table 2: Cumulative
More informationMaximum Likelihood Estimation
Maximum Likelihood Estimation The likelihood and log-likelihood functions are the basis for deriving estimators for parameters, given data. While the shapes of these two functions are different, they have
More informationPROBABILITY AND STATISTICS
Monday, January 12, 2015 1 PROBABILITY AND STATISTICS Zhenyu Ye January 12, 2015 Monday, January 12, 2015 2 References Ch10 of Experiments in Modern Physics by Melissinos. Particle Physics Data Group Review
More informationBivariate Birnbaum-Saunders Distribution
Department of Mathematics & Statistics Indian Institute of Technology Kanpur January 2nd. 2013 Outline 1 Collaborators 2 3 Birnbaum-Saunders Distribution: Introduction & Properties 4 5 Outline 1 Collaborators
More informationNormal distribution Approximating binomial distribution by normal 2.10 Central Limit Theorem
1.1.2 Normal distribution 1.1.3 Approimating binomial distribution by normal 2.1 Central Limit Theorem Prof. Tesler Math 283 Fall 216 Prof. Tesler 1.1.2-3, 2.1 Normal distribution Math 283 / Fall 216 1
More informationLikelihood Methods of Inference. Toss coin 6 times and get Heads twice.
Methods of Inference Toss coin 6 times and get Heads twice. p is probability of getting H. Probability of getting exactly 2 heads is 15p 2 (1 p) 4 This function of p, is likelihood function. Definition:
More information2. The sum of all the probabilities in the sample space must add up to 1
Continuous Random Variables and Continuous Probability Distributions Continuous Random Variable: A variable X that can take values on an interval; key feature remember is that the values of the variable
More informationAsymmetric Type II Compound Laplace Distributions and its Properties
CHAPTER 4 Asymmetric Type II Compound Laplace Distributions and its Properties 4. Introduction Recently there is a growing trend in the literature on parametric families of asymmetric distributions which
More informationMATH 446/546 Homework 1:
MATH 446/546 Homework 1: Due September 28th, 216 Please answer the following questions. Students should type there work. 1. At time t, a company has I units of inventory in stock. Customers demand the
More informationNormal Distribution. Definition A continuous rv X is said to have a normal distribution with. the pdf of X is
Normal Distribution Normal Distribution Definition A continuous rv X is said to have a normal distribution with parameter µ and σ (µ and σ 2 ), where < µ < and σ > 0, if the pdf of X is f (x; µ, σ) = 1
More informationvalue BE.104 Spring Biostatistics: Distribution and the Mean J. L. Sherley
BE.104 Spring Biostatistics: Distribution and the Mean J. L. Sherley Outline: 1) Review of Variation & Error 2) Binomial Distributions 3) The Normal Distribution 4) Defining the Mean of a population Goals:
More informationSemiparametric Modeling, Penalized Splines, and Mixed Models
Semi 1 Semiparametric Modeling, Penalized Splines, and Mixed Models David Ruppert Cornell University http://wwworiecornelledu/~davidr January 24 Joint work with Babette Brumback, Ray Carroll, Brent Coull,
More informationEstimation Appendix to Dynamics of Fiscal Financing in the United States
Estimation Appendix to Dynamics of Fiscal Financing in the United States Eric M. Leeper, Michael Plante, and Nora Traum July 9, 9. Indiana University. This appendix includes tables and graphs of additional
More informationStochastic Volatility and Jumps: Exponentially Affine Yes or No? An Empirical Analysis of S&P500 Dynamics
Stochastic Volatility and Jumps: Exponentially Affine Yes or No? An Empirical Analysis of S&P5 Dynamics Katja Ignatieva Paulo J. M. Rodrigues Norman Seeger This version: April 3, 29 Abstract This paper
More informationMAS1403. Quantitative Methods for Business Management. Semester 1, Module leader: Dr. David Walshaw
MAS1403 Quantitative Methods for Business Management Semester 1, 2018 2019 Module leader: Dr. David Walshaw Additional lecturers: Dr. James Waldren and Dr. Stuart Hall Announcements: Written assignment
More informationChapter 7 - Lecture 1 General concepts and criteria
Chapter 7 - Lecture 1 General concepts and criteria January 29th, 2010 Best estimator Mean Square error Unbiased estimators Example Unbiased estimators not unique Special case MVUE Bootstrap General Question
More information