Bayes Theorem & A Quick Intro to Bayesian Statistics

1. Bayes Theorem & A Quick Intro to Bayesian Statistics Psychology 548Bayesian Statistics, Modeling & Reasoning Instructor: John Miyamoto 1/8/2014: 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.

2. Outline • Bayes Rule • Example: Bayesian Statistical Computation Psych 548, Miyamoto, Win '14

3. Bayes Rule • Reverend Thomas Bayes, 1702 – 1761British 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 '14

4. Bayes Rule – Explanation Prior Probability of the Hypothesis Posterior Probability of the Hypothesis Likelihood of the Data NormalizingConstant Odds Form of Bayes Rule Psych 548, Miyamoto, Win '14

5. Bayes Rule – Explanation Prior Probability of the Hypothesis Posterior Probability of the Hypothesis Likelihood of the Data NormalizingConstant Odds Form of Bayes Rule Psych 548, Miyamoto, Win '14

6. 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 '14

7. Bayes Rule (Odds Form) Prior Odds(base rate) Likelihood Ratio(diagnosticity) Posterior Odds 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 '14

8. Bayesian Analysis of a Medical Test Result 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 '14

9. 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 '14

10. 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 '14

11. Digression: Converting Odds to Probabilities • If X / (1 – X) = Y • 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 '14

12. 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 '14

13. 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 '14

14. 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 '14

15. 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. END Psych 548, Miyamoto, Win '14

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19. Bayesian Assumption: Model Parameters Are Random VariablesThey Are Sampled from Probability Distributions of Possibilities Psych 548, Miyamoto, Win '14