Dan Morris and Mark Koning
have written an excellent introduction to the use of the statistical technique
of Bayesian analysis. They present the methods clearly in text, in formulas, in
illustrations, and in examples. They include decision trees, active links to
Internet sources, a bibliography, some history, and a glossary. The coverage is
clear and complete, though preliminary. They note that the Bayesian methods
have been in use for varieties of searches and now undergird Google’s search
algorithms, for example.
They illustrate the fundamental
Bayes formula for analyzing the degree to which a positive result on a test (+),
such as exhibiting certain symptoms, indicates one has the condition (F); for
example, whether one has the flu. Here are the equations for one of their
examples, testing positive for the flu:
P(F|+) = {P(+|F) / [P(+)]} P(F)
P(F|+) = probability of having
the flu, given the test is positive
P(+|F) = probability of having
a positive on the test, given the flu
P(+) = probability of positive
on the test, in the general population
P(F) = probability of having
the flu, in general population
The term in the curly
brackets, { }, might be considered a magnifier (not always greater than 1,
however) that adjusts our “prior” probability (likelihood) that we have the flu
by incorporating the information that we have tested positive (+), producing a “posterior”
probability.
The hypothetical numbers they
used were
P(F|+) = unknown probability
of having the flu, given symptoms,
P(+|F) = probability of having
symptoms, given the flu, 0.9
P(+) = probability of positive
on the test, in the general population, 0.2
P(F) = probability of having
the flu, in general population, 0.05
So that
P(F|+) = {P(+|F) / P(+)} P(F)
Became
P(F|+) = {0.9 / 0.2} 0.05 = 4.5
x 0.05
You are { } = 4.5 times as
likely as the general population to have the flu, given that you have the symptoms,
and there is a
4.5 x 0.05 = 0.225 or a 22.5%
chance that the symptoms indicate you have the flu.
This shows how the additional information
of having the symptoms (+) affects the likelihood you do have the flu.
More sophisticated uses of Bayes
theorem have been used successfully, including by Alan Turing and team in
cracking the German’s Enigma code in World War II.
https://www.amazon.com/Bayes-Theorem-Examples-Introduction-Beginners-ebook/dp/B01LZ1T9IX/ref=sr_1_3?s=books&ie=UTF8&qid=1542492276&sr=1-3&keywords=bayes+theorem+a+visual+introduction+for+beginners
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