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Bayes Theorem & A Quick Intro to Bayesian Statistics. Psychology 548 Bayesian Statistics, Modeling & Reasoning Instructor: John Miyamoto 1/8/2014: Lecture 01-2.

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bayes theorem a quick intro to bayesian statistics

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.

outline
Outline
  • Bayes Rule
  • Example: Bayesian Statistical Computation

Psych 548, Miyamoto, Win '14

bayes rule
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

bayes rule explanation
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

bayes rule explanation1
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

bayes rule odds form
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

bayes rule odds form1
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

bayesian analysis of a medical test result
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

given information in the diagnostic inference from a medical test result
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

bayesian analysis of a medical test result1
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

digression converting odds to probabilities
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

bayesian analysis of a medical test result2
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

continue with the medical test problem
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

general characteristics of bayesian inference
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

how does bayesian stats differ from classical stats
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

slide19
Bayesian Assumption: Model Parameters Are Random VariablesThey Are Sampled from Probability Distributions of Possibilities

Psych 548, Miyamoto, Win '14