This document discusses Bayesian inference and Bayes' theorem. It provides examples of how Bayesian inference can be used to calculate conditional probabilities based on new evidence or data. Specifically, it discusses using Bayesian inference to determine the probability a die is "good" based on rolling results, and the probability a woman has breast cancer based on positive or negative mammogram results.

1. +
Bayesian’s Theory
and Inference
Slides edited by Valerio Di Fonzo for www.globalpolis.org
Based on the work of Mine Çetinkaya-Rundel of OpenIntro
The slides may be copied, edited, and/or shared via the CC BY-SA license
Some images may be included under fair use guidelines (educational purposes)

2. Bayes' Theorem
The conditional probability formula we have seen so far is a
special case of the Bayes' Theorem, which is applicable even
when events have more than just two outcomes.
Bayes’ Theorem

3. Bayesian Inference

5. In which hand is the “good die”?
Before you make a final decision, you will be able to collect data by asking to roll the die in one hand and
know whether the outcome of the roll is greater than or equal to 4. Think about what it means to roll a
number greater than or equal to 4 with the two types of dice we have. We're going to ask two questions.
What is the probability of rolling a value greater than or equal to 4 with a six-sided die? And what is that
probability with a 12-sided die? With a six-sided die, the sample space is made up of numbers between 1
and 6. We're interested in an outcome greater than or equal to 4, the probability of getting such an
outcome is then 3 out of 6, or 1 out of 2 or, 50%. With a 12-sided die, the sample space is bigger, number
is between 1 and 12. And once again, we're interested in outcomes 4 or greater. The probability of getting
such an outcome is 9 out of 12, or 3 4ths, or 75%.

8. After rolling the die in the right hand you know that the result is greater or
equal to 4. Therefore, now, how do the probabilities you assign to the same set
of hypothesis change? In other words, what are now the probabilities to get
the good die in the right hand after this first roll?

14. Example of Bayesian Inference
Breast cancer screening
● American Cancer Society estimates that about 1.7% of women
have breast
cancer.http://www.cancer.org/cancer/cancerbasics/cancer-prevalence
● Susan G. Komen For The Cure Foundation states that
mammography correctly identifies about 78% of women who truly
have breast
cancer.http://ww5.komen.org/BreastCancer/AccuracyofMammogra
ms.html
● An article published in 2003 suggests that up to 10% of all
mammograms result in false positives for patients who do not have
cancer.http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1360940
Note: These percentages are approximate, and very difficult to estimate.

15. When a patient goes through breast cancer screening there are two
competing claims: patient had cancer and patient doesn't have
cancer. If a mammogram yields a positive result, what is the
probability that patient actually has cancer?
Note: Tree diagrams are useful for inverting probabilities:
we are given P(+|C) and asked for P(C|+).

16. Suppose a woman who gets tested once and obtains a positive result wants to
get tested again. In the second test, what should we assume to be the probability
of this specific woman having cancer?
What is the probability that this woman has cancer if this second
mammogram also yielded a positive result?
(a) 0.0936
(b) 0.088
(c) 0.48
(d) 0.52
posterior
(a) 0.017;
(b) 0.12 (posterior);
(c) 0.0133;
(d) 0.88;