Graziella Quattrocchi & Louise Marshall Methods for Dummies 2014

Slides:



Advertisements
Similar presentations
Bayes rule, priors and maximum a posteriori
Advertisements

1 Uncertainty in rainfall-runoff simulations An introduction and review of different techniques M. Shafii, Dept. Of Hydrology, Feb
Bayesian models for fMRI data
Week 11 Review: Statistical Model A statistical model for some data is a set of distributions, one of which corresponds to the true unknown distribution.
Bayesian inference “Very much lies in the posterior distribution” Bayesian definition of sufficiency: A statistic T (x 1, …, x n ) is sufficient for 
Psychology 290 Special Topics Study Course: Advanced Meta-analysis April 7, 2014.
A Brief Introduction to Bayesian Inference Robert Van Dine 1.
Bayes for beginners Methods for dummies 27 February 2013 Claire Berna
Intro to Bayesian Learning Exercise Solutions Ata Kaban The University of Birmingham 2005.
1 Chapter 12 Probabilistic Reasoning and Bayesian Belief Networks.
Probability Notation Review Prior (unconditional) probability is before evidence is obtained, after is posterior or conditional probability P(A) – Prior.
Probability and Information Copyright, 1996 © Dale Carnegie & Associates, Inc. A brief review (Chapter 13)
I The meaning of chance Axiomatization. E Plurbus Unum.
Bayesian Models Honors 207, Intro to Cognitive Science David Allbritton An introduction to Bayes' Theorem and Bayesian models of human cognition.
Baysian Approaches Kun Guo, PhD Reader in Cognitive Neuroscience School of Psychology University of Lincoln Quantitative Methods 2011.
G. Cowan Lectures on Statistical Data Analysis Lecture 10 page 1 Statistical Data Analysis: Lecture 10 1Probability, Bayes’ theorem 2Random variables and.
Expected Value (Mean), Variance, Independence Transformations of Random Variables Last Time:
Estimating the Transfer Function from Neuronal Activity to BOLD Maria Joao Rosa SPM Homecoming 2008 Wellcome Trust Centre for Neuroimaging.
Bayes for Beginners Presenters: Shuman ji & Nick Todd.
Methods for Dummies 2009 Bayes for Beginners Georgina Torbet & Raphael Kaplan.
Dr. Gary Blau, Sean HanMonday, Aug 13, 2007 Statistical Design of Experiments SECTION I Probability Theory Review.
Renaissance Risk Changing the odds in your favour Risk forecasting & examples.
Mathematics topic handout: Conditional probability & Bayes Theorem Dr Andrew French. PAGE 1www.eclecticon.info Conditional Probability.
Bayesian vs. frequentist inference frequentist: 1) Deductive hypothesis testing of Popper--ruling out alternative explanations Falsification: can prove.
NLP. Introduction to NLP Formula for joint probability –p(A,B) = p(B|A)p(A) –p(A,B) = p(A|B)p(B) Therefore –p(B|A)=p(A|B)p(B)/p(A) Bayes’ theorem is.
Dr. Ahmed Abdelwahab Introduction for EE420. Probability Theory Probability theory is rooted in phenomena that can be modeled by an experiment with an.
Uncertainty Management in Rule-based Expert Systems
Making sense of randomness
1 Chapter 12 Probabilistic Reasoning and Bayesian Belief Networks.
Statistical Decision Theory Bayes’ theorem: For discrete events For probability density functions.
Computer Vision Group Prof. Daniel Cremers Autonomous Navigation for Flying Robots Lecture 5.3: Reasoning with Bayes Law Jürgen Sturm Technische Universität.
4 Proposed Research Projects SmartHome – Encouraging patients with mild cognitive disabilities to use digital memory notebook for activities of daily living.
12/7/20151 Math b Conditional Probability, Independency, Bayes Theorem.
Uncertainty ECE457 Applied Artificial Intelligence Spring 2007 Lecture #8.
Copyright © 2014 by McGraw-Hill Higher Education. All rights reserved. Essentials of Business Statistics: Communicating with Numbers By Sanjiv Jaggia and.
Copyright © 2006 The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Review of Statistics I: Probability and Probability Distributions.
Chapter 20 Classification and Estimation Classification – Feature selection Good feature have four characteristics: –Discrimination. Features.
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.
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.
Ch.9 Bayesian Models of Sensory Cue Integration (Mon) Summarized and Presented by J.W. Ha 1.
Probabilistic Robotics Introduction Probabilities Bayes rule Bayes filters.
Bayesian Approach Jake Blanchard Fall Introduction This is a methodology for combining observed data with expert judgment Treats all parameters.
- 1 - Outline Introduction to the Bayesian theory –Bayesian Probability –Bayes’ Rule –Bayesian Inference –Historical Note Coin trials example Bayes rule.
Statistical NLP: Lecture 4 Mathematical Foundations I: Probability Theory (Ch2)
Parameter Estimation. Statistics Probability specified inferred Steam engine pump “prediction” “estimation”
Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Chapter 4 Probability.
Bayesian Brain Probabilistic Approaches to Neural Coding 1.1 A Probability Primer Bayesian Brain Probabilistic Approaches to Neural Coding 1.1 A Probability.
Bayes for Beginners Anne-Catherine Huys M. Berk Mirza Methods for Dummies 20 th January 2016.
Probabilistic Robotics Probability Theory Basics Error Propagation Slides from Autonomous Robots (Siegwart and Nourbaksh), Chapter 5 Probabilistic Robotics.
Outline Historical note about Bayes’ rule Bayesian updating for probability density functions –Salary offer estimate Coin trials example Reading material:
Matching ® ® ® Global Map Local Map … … … obstacle Where am I on the global map?                                   
Bayesian analysis of a conceptual transpiration model with a comparison of canopy conductance sub-models Sudeep Samanta Department of Forest Ecology and.
Bayes’ Theorem Suppose we have estimated prior probabilities for events we are concerned with, and then obtain new information. We would like to a sound.
Intro to Bayesian Learning Exercise Solutions Ata Kaban The University of Birmingham.
CSE 468/568 Deadlines Lab1 grades out tomorrow (5/1) HW2 grades out by weekend Lab 3 grades out next weekend HW3 – Probability homework out Due 5/7 FINALS:
Chapter 4 Probability.
Quick Review Probability Theory
Quick Review Probability Theory
Bayes' theorem p(A|B) = p(B|A) p(A) / p(B)
Bayes for Beginners Stephanie Azzopardi & Hrvoje Stojic
Incorporating New Information to Decision Trees (posterior probabilities) MGS Chapter 6 Part 3.
Mathematical representation of Bayes theorem.
Computational models for imaging analyses
Statistical NLP: Lecture 4
Example Human males have one X-chromosome and one Y-chromosome,
Wellcome Trust Centre for Neuroimaging
Bayes for Beginners Luca Chech and Jolanda Malamud
28th September 2005 Dr Bogdan L. Vrusias
CS639: Data Management for Data Science
basic probability and bayes' rule
Presentation transcript:

Graziella Quattrocchi & Louise Marshall Methods for Dummies 2014 Bayes for Beginners Graziella Quattrocchi & Louise Marshall Methods for Dummies 2014

Question A disease occurs in 0.5% of population. A diagnostic test gives a positive result in: 99% of people with the disease 5% of people without the disease (false positive) A random person off the street is found to have a positive test result. What is the probability that this person has the disease? A: 0-30% B: 30-70% C: 70-99%

How do we figure this out? A disease occurs in 0.5% of population. 99% of people with the disease have a positive test result. 5% of people without the disease have a positive test result. A = disease B = positive test result P(A) = 0.005 (probability of having disease) P(~A) = 1 – 0.005 = 0.995 (probability of not having disease) P(B) = P(B|A) * P(A) + P(B|~A) * P(~A) = (0.99 * 0.005) + (0.05 * 0.995) = 0.055 i.e. >5% of all tests are positive

probability of disease given positive test result Conditional Probabilities P(B|A) = 0.99 probability of +ve result given disease P(~B|A) = 1 – 0.99 = 0.01 probability of -ve result given disease P(B|~A) = 0.05 probability of +ve result given no disease P(~B|~A) = 1 – 0.05 = 0.95 probability of -ve result given no disease We want: P(A|B) probability of disease given positive test result

Let’s take an example population A = disease B = positive test result P(A) = 0.005 P(B) = 0.055 Population = 1000 positive test = 55 P(A,B) disease = 5 P(A,B) Joint Probability

We already know the test result was positive…. We have to take that into account! Population = 1000 positive test = 55 P(B) disease = 5 P(A) P(A,B) Of all the people already in the purple circle, how many fall into the P(A,B) part? P(A|B) = P(A,B)/P(B)

Bayes’ Theorem P(A|B) = P(A,B)/P(B) P(B|A) = P(A,B)/P(A) The same follows for the inverse: P(B|A) = P(A,B)/P(A) Therefore, the joint probability can be expressed as: P(A,B) = P(A|B)*P(B) P(A,B) = P(B|A)*P(A) And with a bit of shuffling we get: P(A|B) = P(B|A) * P(A) P(B)

Using Bayes’ Theorem P(A) = 0.005 A = disease P(B|A) = 0.99 B = positive test P(B) = 0.055 P(A|B) = P(B|A) * P(A) P(B) P(A|B) = 0.99 * 0.005 0.055 = 0.09 A positive test result only increases your probability of having the disease to 9%, simply because the disease is very rare (relative to the false positive rate).

Some terminology P(A|B) = P(B|A) * P(A) P(B) likelihood prior posterior P(A|B) = P(B|A) * P(A) P(B) marginal probability P(A): before test result, we estimate a 0.5% chance of having the disease P(B|A): probability of a positive test result given an underlying disease P(B): probability of observing this outcome, taken over all possible values of A (disease and no disease) P(A|B): combines what you thought before obtaining the data, and the new information the data provided

Applications: Finding missing planes

Applications: Finding missing planes

Applications: Predicting election results Nate Silver Forecast the performance & career development of Major League Baseball players Correctly predicted the winner in 49/50 states during 2008 US presidential election

So, Bayes’ Rule allows us to… Represent information probabilistically Take uncertainty into account Incorporate prior knowledge and update our beliefs Invert the question (i.e. how good is our hypothesis given the data?) Used in many aspects of science…

1. Bayesian systems represent information probabilistically How wide is the pen? The pen is 8 mm wide There is a 95% chance that the pen is between 7.5 and 8.49 mm wide precision Probability density function (PDF) Represents both the average estimate of the quantity itself and the confidence in that estimate Probability O’Reilly et al, EJN 2012(35), 1169-72

The Bayesian systems integrate information using uncertainty How wide is the pen? Sensory dominance Combined estimate between the monosensory estimates precision Visual Touch Combined The Bayesian systems integrate information using uncertainty P(width|touch, vision) P(touch, vision|width) * P(width) O’Reilly et al, EJN 2012(35), 1169-72

Multisensory integration in human performance Humans do show near-Bayesian behaviour in multi-sensory integration tasks Non-optimal bias to give more weight to one sensory modality than another VISION PROPRIOCEPTION Van Beers et al, Exp Brain Res 1999;125:43-9

P(width|touch, vision) P(touch, vision|width) * P(width) 3. Bayesian system incorporates prior knowledge P(width|touch, vision) P(touch, vision|width) * P(width) The posterior estimate is biased towards the prior mean Prior permits to increase accuracy, useful considering uncertainty of observations Prior Observed Posterior 5 7 Width (mm) O’Reilly et al, EJN 2012(35), 1169-72

When stimuli are ambiguous, prior govern perception… The ideas behind Bayesian inference can be illustrated using the game of tennis. The best estimate of where the tennis ball will land can be gotten from combining about where they served before (the prior) with current information from the visual system (the likelihood). The prior is shown in blue, the likelihood distribution in red, and the posterior distribution with the white ellipse. The maximum posterior estimate is shown by the magenta ball. This estimate can be updated in light of new information from the balls trajectory The Muller-Lyer Illusion Priors could be acquired trough long experience with the environment Some others priors seem to be innate

Bayesian system incorporates prior knowledge to update our beliefs The posterior distribution as the new prior, which can be updated using new observation Bayes’ rules allow to learn from observations one after the other and shows that the more data we have, the more precise our estimate on the parameters Learning as a form of Bayesian reasoning The ideas behind Bayesian inference can be illustrated using the game of tennis. The best estimate of where the tennis ball will land can be gotten from combining about where they served before (the prior) with current information from the visual system (the likelihood). The prior is shown in blue, the likelihood distribution in red, and the posterior distribution with the white ellipse. The maximum posterior estimate is shown by the magenta ball. This estimate can be updated in light of new information from the balls trajectory Körding & Wolpert (2004) Nature

Resources Further Reading Will Penny’s slides (on his website) O’Reilly et al. (2012) How can a Bayesian approach inform neuroscience? EJN LessWrong Blog Page on Bayes History of Bayes Rule by Sharon McGrayne Previous MfD slides

Thanks to Will Penny and previous MfD presentations, and…