Bayes Theorem, a.k.a. Bayes Rule

Slides:



Advertisements
Similar presentations
Lecture XXIII.  In general there are two kinds of hypotheses: one concerns the form of the probability distribution (i.e. is the random variable normally.
Advertisements

Psychology 290 Special Topics Study Course: Advanced Meta-analysis April 7, 2014.
Probability & Statistical Inference Lecture 7 MSc in Computing (Data Analytics)
Probability & Statistical Inference Lecture 6 MSc in Computing (Data Analytics)
Chapter Seventeen HYPOTHESIS TESTING
Course overview Tuesday lecture –Those not presenting turn in short review of a paper using the method being discussed Thursday computer lab –Turn in short.
Introduction to Bayesian statistics Three approaches to Probability  Axiomatic Probability by definition and properties  Relative Frequency Repeated.
Psych 548, Miyamoto, Win '15 1 Set Up for Students Your computer should already be turned on and logged in. Open a browser to the Psych 548 website ( you.
Statistics: Unlocking the Power of Data Lock 5 STAT 101 Dr. Kari Lock Morgan Bayesian Inference SECTION 11.1, 11.2 Bayes rule (11.2) Bayesian inference.
Lecture 9: p-value functions and intro to Bayesian thinking Matthew Fox Advanced Epidemiology.
INFERENTIAL STATISTICS – Samples are only estimates of the population – Sample statistics will be slightly off from the true values of its population’s.
Chapter 9 Title and Outline 1 9 Tests of Hypotheses for a Single Sample 9-1 Hypothesis Testing Statistical Hypotheses Tests of Statistical.
Statistical inference: confidence intervals and hypothesis testing.
June 18, 2008Stat Lecture 11 - Confidence Intervals 1 Introduction to Inference Sampling Distributions, Confidence Intervals and Hypothesis Testing.
Bayesian Inference, Basics Professor Wei Zhu 1. Bayes Theorem Bayesian statistics named after Thomas Bayes ( ) -- an English statistician, philosopher.
Bayes for Beginners Presenters: Shuman ji & Nick Todd.
METHODSDUMMIES BAYES FOR BEGINNERS. Any given Monday at pm “I’m sure this makes sense, but you lost me about here…”
Statistics: Unlocking the Power of Data Lock 5 STAT 101 Dr. Kari Lock Morgan 12/4/12 Bayesian Inference SECTION 11.1, 11.2 More probability rules (11.1)
Copyright © Cengage Learning. All rights reserved. 10 Inferences Involving Two Populations.
CHAPTER 17: Tests of Significance: The Basics
Week 71 Hypothesis Testing Suppose that we want to assess the evidence in the observed data, concerning the hypothesis. There are two approaches to assessing.
Bayesian vs. frequentist inference frequentist: 1) Deductive hypothesis testing of Popper--ruling out alternative explanations Falsification: can prove.
DNA Identification: Bayesian Belief Update Cybergenetics © TrueAllele ® Lectures Fall, 2010 Mark W Perlin, PhD, MD, PhD Cybergenetics, Pittsburgh,
1 9 Tests of Hypotheses for a Single Sample. © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. 9-1.
Bayesian Inference, Review 4/25/12 Frequentist inference Bayesian inference Review The Bayesian Heresy (pdf)pdf Professor Kari Lock Morgan Duke University.
Populations III: evidence, uncertainty, and decisions Bio 415/615.
Artificial Intelligence CIS 342 The College of Saint Rose David Goldschmidt, Ph.D.
"Classical" Inference. Two simple inference scenarios Question 1: Are we in world A or world B?
Bayesian Approach For Clinical Trials Mark Chang, Ph.D. Executive Director Biostatistics and Data management AMAG Pharmaceuticals Inc.
Ch15: Decision Theory & Bayesian Inference 15.1: INTRO: We are back to some theoretical statistics: 1.Decision Theory –Make decisions in the presence of.
Representativeness, Similarity, & Base Rate Neglect Psychology 466: Judgment & Decision Making Instructor: John Miyamoto 10/27/2015: Lecture 05-1 Note:
Bayes Theorem. Prior Probabilities On way to party, you ask “Has Karl already had too many beers?” Your prior probabilities are 20% yes, 80% no.
When “Less Is More” – A Critique of the Heuristics & Biases Approach to Judgment and Decision Making Psychology 466: Judgment & Decision Making Instructor:
MPS/MSc in StatisticsAdaptive & Bayesian - Lect 71 Lecture 7 Bayesian methods: a refresher 7.1 Principles of the Bayesian approach 7.2 The beta distribution.
Statistical Inference Statistical inference is concerned with the use of sample data to make inferences about unknown population parameters. For example,
Bayesian Statistics, Modeling & Reasoning What is this course about? P548: Intro Bayesian Stats with Psych Applications Instructor: John Miyamoto 01/04/2016:
Markov-Chain-Monte-Carlo (MCMC) & The Metropolis-Hastings Algorithm P548: Intro Bayesian Stats with Psych Applications Instructor: John Miyamoto 01/19/2016:
Outline Historical note about Bayes’ rule Bayesian updating for probability density functions –Salary offer estimate Coin trials example Reading material:
The Psychology of Inductive Inference Psychology 355: Cognitive Psychology Instructor: John Miyamoto 5/26/2016: Lecture 09-4 Note: This Powerpoint presentation.
Presented by Mo Geraghty and Danny Tran June 10, 2016.
FIXETH LIKELIHOODS this is correct. Bayesian methods I: theory.
Naive Bayes Classifier. REVIEW: Bayesian Methods Our focus this lecture: – Learning and classification methods based on probability theory. Bayes theorem.
Statistics for Decision Making Hypothesis Testing QM Fall 2003 Instructor: John Seydel, Ph.D.
The Representativeness Heuristic then: Risk Attitude and Framing Effects Psychology 355: Cognitive Psychology Instructor: John Miyamoto 6/1/2016: Lecture.
Computing with R & Bayesian Statistical Inference P548: Intro Bayesian Stats with Psych Applications Instructor: John Miyamoto 01/11/2016: Lecture 02-1.
Bayesian Statistics, Modeling & Reasoning What is this course about?
Review of Probability.
MCMC Output & Metropolis-Hastings Algorithm Part I
Use of Pseudo-Priors in Bayesian Model Comparison
Statistics 350 Lecture 4.
Set Up for Instructor MGH Display: Try setting your resolution to 1024 by 768 Run Powerpoint. For most reliable start up: Start laptop & projector before.
Statistical inference: distribution, hypothesis testing
Set Up for Instructor MGH Display: Try setting your resolution to 1024 by 768 Run Powerpoint. For most reliable start up: Start laptop & projector before.
Brief History of Cognitive Psychology
9 Tests of Hypotheses for a Single Sample CHAPTER OUTLINE
More about Posterior Distributions
(Very Brief) Introduction to Bayesian Statistics
Bayesian Reasoning & Base-Rate Neglect
My Office Hours I will stay after class on both Monday and Wednesday, i.e., 1:30 Mon/Wed in MGH 030. Can everyone stay if they need to? Psych 548, Miyamoto,
Introduction to Bayesian Model Comparison
One-Sample Models (Continuous DV) Then: Simple Linear Regression
Set Up for Instructor Classroom Support Services (CSS), 35 Kane Hall,
Intro to Bayesian Hierarchical Modeling
Set Up for Instructor Classroom Support Services (CSS), 35 Kane Hall,
Hierarchical Models of Memory Retention
Wellcome Trust Centre for Neuroimaging
Set Up for Instructor Classroom Support Services (CSS), 35 Kane Hall,
Set Up for Instructor Classroom Support Services (CSS), 35 Kane Hall,
Psychology 355: Cognitive Psychology Instructor: John Miyamoto /2011
CS639: Data Management for Data Science
Presentation transcript:

Bayes Theorem, a.k.a. Bayes Rule P548: Intro Bayesian Stats with Psych Applications Instructor: John Miyamoto 01/06/2016: Lecture 01-2 Note: This Powerpoint presentation may contain macros that I wrote to help me create the slides. The macros aren’t needed to view the slides. You can disable or delete the macros without any change to the presentation.

Outline Bayes Rule Odds Form of Bayes Rule Application of Bayes Rule to the Interpretation of a Medical Test Overview: Bayesian Statistical Inference versus Classical Statistical Inference ------------------------------------------------------------------------------------------------ Learn R & RStudio – write a Bayesian inference function Bayes Rule Psych 548: Miyamoto, Win ‘16

Bayes Rule Reverend Thomas Bayes, 1702 – 1761 British Protestant minister & mathematician Bayes Rule is fundamentally important to: Bayesian statistics Bayesian decision theory Bayesian models in psychology Next: Explanation of Bayes Rule Psych 548, Miyamoto, Win '16

Bayes Rule – Explanation Likelihood of the Data Prior Probability of the Hypothesis Posterior Probability of the Hypothesis Normalizing Constant Formula for Computing P(Data) Psych 548, Miyamoto, Win '16

Bayes Rule – Explanation Information Needed to Compute P( Data ) P(Data | Hypothesis) P(Data | Not-Hypothesis) P(Hypothesis) P(Not-Hypothesis) = 1 - P(Hypothesis) Same As Previous Slide w-o Emphasis Rectangles Psych 548, Miyamoto, Win '16

Bayes Rule – Explanation Prior Probability of the Hypothesis Posterior Probability of the Hypothesis Likelihood of the Data Normalizing Constant Odds Form of Bayes Rule Psych 548, Miyamoto, Win '16

Bayes Rule – Odds Form Bayes Rule for H given D Bayes Rule for not-H given D Odds Form of Bayes Rule Explanation of Odds form of Bayes Rule Psych 548, Miyamoto, Win '16

Likelihood Ratio (diagnosticity) Bayes Rule (Odds Form) Posterior Odds Prior Odds (base rate) Likelihood Ratio (diagnosticity) Talk a bit about the concept of the odds. E.g., if you think that Obama has 2 to 1 odds to beat McCain, you think Obama is twice as likely as McCain to win. E.g., JM’s personal odds are more like 4 to 1 for Obama to beat McCain. E.g., if the odds are even (1 to 1) that Obama will beat McCain, then their chances are equal (50%). H = a hypothesis, e.g.., hypothesis that the patient has cancer = the negation of the hypothesis, e.g.., the hypothesis that the patient does not have cancer D = the data, e.g., a + result for a cancer test Interpretation of a Medical Test Result Psych 548, Miyamoto, Win '16

Bayesian Analysis of a Medical Test Result (Look at Handout) QUESTION: A physician knows from past experience in his practice that 1% of his patients have cancer (of a specific type) and 99% of his patients do not have the cancer. He also knows the probabilities of a positive test result (+ result) given cancer and given no cancer. These probabilities are: P(+ test | Cancer) = .792 and P(+ test | no cancer) = .096 Suppose Mr. X has a positive test result. What is the probability that Mr. X has cancer? Write down your intuitive answer. (Note to JM: Write estimates on board) Solution to this problem Psych 548, Miyamoto, Win '16

Given Information in the Diagnostic Inference from a Medical Test Result P(+ test | Cancer) = .792 (true positive rate a.k.a. hit rate) P(+ test | no cancer) = .096 (false positive rate a.k.a. false alarm rate) P(Cancer) = Prior probability of cancer = .01 P(No Cancer) = Prior probability of no cancer = 1 - P(Cancer) = .99 Mr. X has a + test result. What is the probability that Mr. X has cancer? Solution to this problem Psych 548, Miyamoto, Win '16

Bayesian Analysis of a Medical Test Result P(+ test | Cancer) = 0.792 and P(+ test | no cancer) = 0.096 P(Cancer) = Prior probability of cancer = 0.01 P(No Cancer) = Prior probability of no cancer = 0.99 P(Cancer | + test) = 1 / (12 + 1) = 0.077 Digression concerning What Are Odds? Psych 548, Miyamoto, Win '16

Digression: Converting Odds to Probabilities If X / (1 – X) = Y = the odds of X versus not-X Then X = Y(1 – X) = Y – XY So X + XY = Y So X(1 + Y) = Y So X = Y / (1 + Y) Conclusion: If Y are the odds for an event, then, Y / (1 + Y) is the probability of the event Return to Slide re Medical Test Inference Psych 548, Miyamoto, Win '16

Bayesian Analysis of a Medical Test Result P(+ test | Cancer) = 0.792 and P(+ test | no cancer) = 0.096 P(Cancer) = Prior probability of cancer = 0.01 P(No Cancer) = Prior probability of no cancer = 0.99 P(Cancer | + test) = (1/12) / (1 + 1/12) = 1 / (12 + 1) = 0.077 Compare the Normative Result to Physician’s Judgments Psych 548, Miyamoto, Win '16

Continue with the Medical Test Problem P(Cancer | + Result) = (.792)(.01)/(.103) = .077 Posterior odds against cancer are (.077)/(1 - .077) or about 1 chance in 12. Notice: The test is very diagnostic but still P(cancer | + result) is low because the base rate is low. David Eddy found that about 95 out of 100 physicians stated that P(cancer | +result) is about 75% in this case (very close to the 79% likelihood of a + result given cancer). General Characteristics of Bayesian Inference Psych 548, Miyamoto, Win '16

General Characteristics of Bayesian Inference The decision maker (DM) is willing to specify the prior probability of the hypotheses of interest. DM can specify the likelihood of the data given each hypothesis. Using Bayes Rule, we infer the probability of the hypotheses given the data Comparison Between Bayesian & Classical Stats - END Psych 548, Miyamoto, Win '16

How Does Bayesian Stats Differ from Classical Stats? Bayesian: Common Aspects Statistical Models Credible Intervals – sets of parameters that have high posterior probability Bayesian: Divergent Aspects Given data, compute the full posterior probability distribution over all parameters Generally null hypothesis testing is nonsensical. Posterior probabilities are meaningful; p-values are half-assed. MCMC approximations to posterior distributions. Classical: Common Aspects Statistical Models Confidence Intervals – which parameter values are tenable after viewing the data. Classical: Divergent Aspects No prior distributions in general, so this idea is meaningless or self- deluding. Null hypothesis te%sting P-values MCMC approximations are sometimes useful but not for computing posterior distributions. Sequential Presentation of the Common & Divergent Aspects Psych 548, Miyamoto, Win '16

How Does Bayesian Stats Differ from Classical Stats? Bayesian: Common Aspects Statistical Models Credible Intervals – sets of parameters that have high posterior probability Bayesian: Divergent Aspects Given data, compute the full posterior probability distribution over all parameters Generally null hypothesis testing is nonsensical. Posterior probabilities are meaningful; p-values are half-assed. MCMC approximations to posterior distributions. Classical: Common Aspects Statistical Models Confidence Intervals – which parameter values are tenable after viewing the data. Classical: Divergent Aspects No prior distributions in general, so this idea is meaningless or self- deluding. Null hypothesis te%sting P-values MCMC approximations are sometimes useful but not for computing posterior distributions. Repeat This Slide With Emphasis on Divergent Aspects Psych 548, Miyamoto, Win '16

How Does Bayesian Stats Differ from Classical Stats? Bayesian: Common Aspects Statistical Models Credible Intervals – sets of parameters that have high posterior probability Bayesian: Divergent Aspects Given data, compute the full posterior probability distribution over all parameters Generally null hypothesis testing is nonsensical. Posterior probabilities are meaningful; p-values are half-assed. MCMC approximations to posterior distributions. Classical: Common Aspects Statistical Models Confidence Intervals – which parameter values are tenable after viewing the data. Classical: Divergent Aspects No prior distributions in general, so this idea is meaningless or self-deluding. Null hypothesis testing P-values MCMC approximations are sometimes useful but not for computing posterior distributions. Repeat This Slide Without Emphasis Rectangles Psych 548, Miyamoto, Win '16

How Does Bayesian Stats Differ from Classical Stats? Bayesian: Common Aspects Statistical Models Credible Intervals – sets of parameters that have high posterior probability Bayesian: Divergent Aspects Given data, compute the full posterior probability distribution over all parameters Generally null hypothesis testing is nonsensical. Posterior probabilities are meaningful; p-values are half-assed. MCMC approximations to posterior distributions. Classical: Common Aspects Statistical Models Confidence Intervals – which parameter values are tenable after viewing the data. Classical: Divergent Aspects No prior distributions in general, so this idea is meaningless or self- deluding. Null hypothesis testing P-values MCMC approximations are sometimes useful but not for computing posterior distributions. Next Topic: Some Lessons in R - END Psych 548, Miyamoto, Win '16

Next Go to: Demo01-2: wUsing R to Think About Bayesian Inference END Psych 548: Miyamoto, Win ‘16