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Bayesian’s Theory

and Inference

Slides edited by Valerio Di Fonzo for www.globalpolis.orgBased 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)

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

Bayesian Inference

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%.

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?

● 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

Breast cancer screening

Note: These percentages are approximate, and very difficult to estimate.

Example of Bayesian Inference

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|+).

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;