An Intuitive (and Short) Explanation of Bayes’ Theorem

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Wednesday, June 12. 2013

An Intuitive (and Short) Explanation of Bayes’ Theorem

We've been over this ground before, but somebody recently shared this with me: An Intuitive (and Short) Explanation of Bayes’ Theorem

The examples with medical tests are good:

$\displaystyle{\Pr(\mathrm{A}|\mathrm{X}) = \frac{\Pr(\mathrm{X}|\mathrm{A})\Pr(\mathrm{A})}{\Pr(\mathrm{X|A})\Pr(A)+ \Pr(\mathrm{X|\sim A})\Pr(\sim A)}
}$

And here’s the decoder key to read it:

Pr(A|X) = Chance of having cancer (A) given a positive test (X).
This is what we want to know: How likely is it to have cancer with a
positive result? In our case it was 7.8%.

Pr(X|A) = Chance of a positive test (X) given that you had cancer (A). This is the chance of a true positive, 80% in our case.

Pr(A) = Chance of having cancer (1%).

Pr(~A) = Chance of not having cancer (99%).

Pr(X|~A) = Chance of a positive test (X) given that you didn’t have cancer (~A). This is a false positive, 9.6% in our case.

Posted by Bird Dog in Our Essays, The Culture, "Culture," Pop Culture and Recreation at 16:08 | Comments (6) | Trackbacks (0)

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Overly complicated. In practice, one just normalizes the probabilities.

#1 chuck on 2013-06-12 16:48 (Reply)

Overly complicated. The denominator is a obfuscated way to write P(X) and, in practice, one just normalizes the probabilities.

#2 chuck on 2013-06-12 16:51 (Reply)

How?

#2.1 Bird Dog on 2013-06-12 16:53 (Reply)

If you add up the probabilities for all the outcomes given X, i.e., P(C|X) and P(~C|X), then they have to add to one. Thus start by omitting the denominator, compute all the outcomes, add them up, and divide everything by the sum. The article actually does that.

#2.1.1 chuck on 2013-06-12 18:35 (Reply)

Cowpies.

Envy is the rottenness of the bones.

#3 Leag on 2013-06-12 23:06 (Reply)

So there's a total chance of 89.6% of a positive result, whether or not you really had cancer, and 10.4% of a negative result? I don't think I understand this at all yet. Does the answer mean that, if you get a positive result, you still have only a 7.6% probability of really having cancer? That doesn't sound right.

#4 Texan99 on 2013-06-13 00:14 (Reply)

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Wednesday, June 12. 2013

An Intuitive (and Short) Explanation of Bayes’ Theorem