Saturday, November 17, 2018

Review: BAYES THEOREM: A Visual Introduction for Beginners

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)

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.

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