**From:** Eliezer S. Yudkowsky (*sentience@pobox.com*)

**Date:** Thu Aug 29 2002 - 15:15:38 MDT

**Next message:**Samantha Atkins: "Re: Metarationality (was: JOIN: Alden Streeter)"**Previous message:**Christian L.: "Bayesian Pop Quiz"**In reply to:**Christian L.: "Bayesian Pop Quiz"**Next in thread:**Ben Goertzel: "RE: Bayesian Pop Quiz"**Reply:**Ben Goertzel: "RE: Bayesian Pop Quiz"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ] [ attachment ]

Christian L. wrote:

* >
*

* > Reading this, I looked up Bayes' theorem in my Probability Theory book,
*

* > and under the theorem itself, it was written: "Never has any theorem
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* > been misused so much my so many".
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* >
*

* > I fail to see why anyone would hold this theorem so highly that he
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* > writes poetry about it. In fact, I do not really think that you (Eli)
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* > really understand the theorem. For instance, this example is given in
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* > my book on elementary probability theory as a direct application of
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* > Bayes Th.:
*

Heh. Well, I am not alone in holding the BPT in very high esteem. There

is a small but growing movement in science to replace the Popperian view

of proof with a Bayesian view, and you will often find "Bayesian

rationalist" used as a more precise synonym for "rationalist", so it's not

just me.

* > ** In a land there lives two kinds of people: X and Y. Among the X:s,
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* > 80% are tall. Among the Y:s, 1% are tall. The population in this
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* > country is 10% X:s and 90% Y:s.
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* >
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* > A tourist randomly meets a person, who happens to be tall. Use Bayes'
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* > theorem to calculate the probability that this person is an X. **
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* >
*

* > Can you (Eli) solve it? It ought not be a problem for someone who can
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* > "see the BPT flowing underneath the surface of all cognition, like
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* > blood beneath skin."
*

The answer is obviously 80/89. 100 Xs live in the country and 80 are

tall, 900 Ys live in the country and 9 are tall. I don't know what you

intended to prove by asking me that. There's a similar piece of math

under the definition of BPT in the glossary of GISAI:

http://intelligence.org/GISAI/meta/glossary.html#gloss_bayesian_probability_theorem

Here's also a little excerpt from a work in progress (don't know if it'll

ever be finished, so don't hold your breath):

* > Suppose you know the following: 1% of the North American population
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* > has cancer. The probability of a false negative, on a cancer test, is
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* > 2%. The probability of a false positive, on a cancer test, is 10%.
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* > You take a cancer test and it comes up positive. What is the
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* > probability that you have cancer?
*

[some discussion omitted]

* > The way the human mind works instinctively is something like this: In
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* > the beginning, you're told that around 1 in 100 people has cancer.
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* > Then, the doctor shows you a cancer test and says that if you take the
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* > test and you don't have cancer, the probability of the test coming up
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* > positive is only 10%. You take the test and it comes up positive.
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* > Inside your mind, the evidence of the test results replaces the prior
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* > probability of 1% and substitutes the new probability of 90% that you
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* > have cancer. Initially the Bayesian Probability Theorem, even if it
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* > works, seems like a very alien way of looking at the world - you take a
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* > test that has a 90% chance of working, it comes up positive, and the
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* > statistician says that your actual probability of having cancer is
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* > 49/544, roughly 9% or around one-tenth of what the intuitive
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* > probability is. Since the Bayesian Probability Theorem is often
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* > explained by teachers who don't realize how insanely powerful the BPT
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* > really is, the picture that forms in many students' minds is probably
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* > something like this: "My prior probability of having cancer is 1 in
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* > 100. I take a test which is 90% accurate and it comes up positive; in
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* > any sane world, my probability of having cancer would be 9 out of 10.
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* > But this strange thing called the Bayesian Probability Theorem says
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* > that the obvious answer of 9/10 is replaced with the bizarre answer of
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* > 49/544. I accept it, but I don't understand it."
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* >
*

* > However, there exists a way in which we can integrate the Bayesian
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* > Probability Theorem into our intuitive understanding of probabilities.
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* > Instead of imagining the prior probability of 1% being replaced with
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* > 90% after the test results arrive, imagine the test results as sliding
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* > the probability from its starting point. A test that's 90% accurate
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* > has a lot of weight, but it doesn't have quite as much weight as the 1%
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* > prior probability. A positive result on the test slides the 1%
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* > probability up to 9.007%, but not all the way to 90%. The chance of a
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* > false negative is 2%, so a negative result slides the initial 1%
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* > probability down to .0224% - getting a negative result on your test
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* > doesn't replace the initial 1% probability with a higher 2%
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* > probability! It's important, though, to remember that it isn't just
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* > the 10% chance of a false positive that matters, but also the 98%
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* > chance of a true positive if you do have cancer. If there was a 60%
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* > chance of a true positive (and hence a 40% chance of a false negative),
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* > then getting a positive result on your test would divide the groups
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* > into 8,910, 990, 40, and 60; thus, getting a positive result would make
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* > the calculation (60) / (990 + 60), 2/35 or 5.714%. The intuitive
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* > translation: The degree to which a result is evidence for X depends,
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* > not only on the strength of the statement "we'd expect to see this
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* > result if X were true", but also, vitally, on the strength of the
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* > statement "we wouldn't expect to see this result if X weren't true."
*

-- Eliezer S. Yudkowsky http://intelligence.org/ Research Fellow, Singularity Institute for Artificial Intelligence

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