Table of Contents
 Bayes Theorem
 Comparative Form of Bayes Theorem
 Calculating the Probability of All Red Cards
 Positive and Negative Test Results
 Other Comparative Forms of Bayes Theorem
 Noncomparative Form of Bayes Theorem
Bayes Theorem
 Bayes Theorem formalizes the idea that the probability of a hypothesis given the evidence is, in part, a function of the probability of the evidence given the hypothesis.
 Main forms of Bayes Theorem:
 The Comparative Form formalizes the principle that a hypothesis that better predicts the evidence is more probable, other things being equal.
 The Non–comparative Form quantifies the idea that the more unexpected a prediction, the more likely the hypothesis, other things being equal.
Comparative Form of Bayes Theorem
 The Comparative Form of Bayes Theorem formalizes the principle that a hypothesis that better predicts the evidence is more probable, other things being equal.
 Specifically, it defines the numeric probability of a hypothesis in light of the evidence, based on:
 the probability of the evidence given competing hypotheses,
 the probability of competing hypotheses apart from the evidence.
 Bayes Theorem, Comparative Form for two hypothesis.

 H_{1} is the hypothesis under consideration
 H_{2} is a competing hypothesis
 Either H_{1} or H_{2} is true but not both
 E is the evidence
 P( – ) means the probability that – – –
 P( – – –  …… ) means the probability that – – – given that …..
Artificial Example of Comparative Form
 You randomly select one of two decks of cards.
 One deck is standard deck of 52 cards, half red, half black.
 The other consists of only hearts and diamonds, making all the cards red.
 The probability the selected deck is all red cards = ½
 You randomly draw a card from the selected deck. The card is red.
 Per Bayes Theorem, the probability the deck is allred given that you drew a red card = ⅔
 The ⅔ probability follows from the following, using Bayes Theorem.
 Before you selected the red card, the probability that the deck was allred = ½
 The probability of drawing a red card given that the deck is allred = 1.0
 The probability of drawing a red card given that the deck is halfred = ½
Calculating the Probability of All Red Cards
Using a Bayesian Calculator
 Abbreviations
 H_{1} = hypothesis the deck is all red cards
 H_{2} = hypothesis the deck is standard.
 E = the evidence that you selected a red card.
 Prior Probabilities, before you select a card.
 Prior probability of H_{1} = P(H_{1}) = 0.5
 Prior probability of H_{2} = P( H_{2}) = 0.5.
 Likelihoods of E given the hypotheses
 Probability of E given H_{1} = P(EH_{1}) = 1.0
 Probability of E given H_{2} = P(EH_{2}) = 0.5
 Results: Posterior Probabilities
 Probability of H_{1} given E = P(H_{1}E) = 2/3
 Probability of H_{2} given E = P(H_{2}E) = 1/3
 So,
 Probability of all red cars = P(H_{1}E) = 2/3
Using a Probability Tree
Using Bayes Formula
The probability that the deck is standard increases from 50% to 67% in light of the evidence.
Positive and Negative Test Results
View Sensitivity, Specificity, Positive and Negative Predictive Values
Bayesian Calculation of P(sick  positive)
 Issue
 You test positive for a disease. What’s the probability you’re sick?
 Given
 Sensitivity of the test = P(positive  sick) = 98.1%
 Specificity of the test = P(negative  not sick) = 99.6%.
 Prevalence of disease in the population = P(sick) = 5%
 Calculate
 P(sick  positive) = ?
 Abbreviations
 H_{1} = You’re sick
 H_{2} = You’re not sick
 E = You test positive
 Prior Probabilities, based on prevalence:
 Prior probability of H_{1} = P(H_{1}) = 0.05
 Prior probability of H_{2} = P( H_{2}) = 0.95
 Likelihoods of E given the hypotheses
 Probability of E given H_{1} = P(EH_{1}) = 0.981
 based on the sensitivity of the test
 Probability of E given H_{2} = P(EH_{2}) = 1 – 0.996 = 0.004
 based on the specificity of the test
 Probability of E given H_{1} = P(EH_{1}) = 0.981
 Results: Posterior Probabilities
 Probability of H_{1} given E = P(H_{1}E) = 0.928
 Probability of H_{2} given E = P(H_{2}E) = 0.072
 So
 P(sick  positive) = P(H_{1}E) = 92.8%
Bayesian Calculation of P(not sick  negative)
 Issue
 You test negative for a disease. What’s the probability you’re okay?
 Given
 Sensitivity of the test = P(positive  sick) = 98.1%
 Specificity of the test = P(negative  not sick) = 99.6%.
 Prevalence of the disease in the population = P(sick) = 5%
 Calculate
 P(not sick  negative) = ?
 Abbreviations
 H_{1} = You’re sick
 H_{2} = You’re not sick
 E = You test negative
 Prior Probabilities, based on prevalence:
 Prior probability of H_{1} = P(H_{1}) = 0.05
 Prior probability of H_{2} = P( H_{2}) = 0.95
 Likelihoods of E given the hypotheses
 Probability of E given H_{1} = P(EH_{1}) = 1 – 0.981 = 0.019
 based on the sensitivity of the test
 Probability of E given H_{2} = P(EH_{2}) = 0.996
 based on the specificity of the test
 Probability of E given H_{1} = P(EH_{1}) = 1 – 0.981 = 0.019
 Results: Posterior Probabilities
 Probability of H_{1} given E = P(H_{1}E) = 0.001
 Probability of H_{2} given E = P(H_{2}E) = 0.999
 So
 P(not sick  negative) = P(H_{2}E) = 99.9%
View Random Drug Test
Other Comparative Forms of Bayes Theorem
Hypothesis true or false
Multiple competing hypotheses
Noncomparative Form of Bayes Theorem
 The Noncomparative Form of Bayes Theorem quantifies the idea that the more unexpected a prediction, the more likely the hypothesis, other things being equal. That is, the lower P(E), the greater P(HE).
 H is the hypothesis under consideration
 E is the evidence
 P( – ) means the probability that – – –
 P( – – –  …… ) means the probability that – – – given that …..
 View Bayesian Estimation for the application of the Noncomparative Form to statistical estimation.