This link has been bookmarked by 103 people . It was first bookmarked on 02 Mar 2006, by Hugh Bristic.
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These two extreme examples help demonstrate that the mammography result
doesn't replace your old
information about the patient's chance of having cancer; the
mammography slides the
estimated probability in
the direction of the result. A positive result slides the
original
probability upward; a negative result slides the probability
downward. -
Most people encountering problems of this type for the first time carry
out the mental operation of replacing
the original 1% probability with the 80% probability that a woman with
cancer gets a positive mammography. It may seem like a good idea,
but it just doesn't work. - 28 more annotations...
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The chance that a patient with a "positive" result has breast cancer is
then the proportion of group A within the combined group A + C, or P*M
/
[P*M + (1 - P)*M], which, cancelling the common factor M from the
numerator and denominator, is P / [P + (1 - P)] or P / 1 or just
P. -
Which is common sense. Take, for example, the "test" of flipping
a coin; if the coin comes up heads, does it tell you anything about
whether a patient has breast cancer? No; the coin has a 50%
chance
of coming up heads if the patient has breast cancer, and also a 50%
chance of coming up heads if the patient does not have breast
cancer. Therefore there is no reason to call either heads or
tails
a "positive" result. It's not the probability being "50/50" that
makes the coin a bad test; it's that the two probabilities, for "cancer
patient turns up heads" and "healthy patient turns up heads", are the
same. -
prior probability
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conditional
probabilities -
the priors
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revised probability or the posterior probability
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if the two
conditional probabilities are equal, the posterior probability equals
the prior probability -
One thing that's confusing
about this notation is that the order of implication is read
right-to-left, as in Hebrew or Arabic -
Reading from left to right, "|" means "given"; reading from
right to left, "|" means
"implies" or "leads to". -
Looking at this applet, it's easier to see why the final answer depends
on all three probabilities; it's the differential
pressure between the two conditional probabilities, p(blue|pearl) and p(blue|~pearl), that slides the prior probability p(pearl) to the posterior
probability p(pearl|blue). -
Even when the prior probability changes, the
differential pressure of the two conditional probabilities always
slides
the probability in the same direction.
If you learn the egg is painted blue, the probability the egg contains
a pearl always goes up - but
it
goes up from the prior
probability, so you need to know the prior probability in order to
calculate the final answer. -
A study by Gigerenzer and Hoffrage in 1995 showed that some ways of
phrasing story problems are much more evocative of correct Bayesian
reasoning. The least
evocative phrasing used probabilities. A slightly more evocative
phrasing used frequencies instead of probabilities -
The most effective presentation found so far is what's known as natural frequencies
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the information about the prior
probability
is included in presenting the conditional probabilities -
In this case, you might as well just say that 30% of eggs are painted blue, since
the probability of an egg being painted blue is independent of whether
the egg contains a pearl. -
If the bottom bar were renormalized
to the same length as the top bar, it would look like the left sector
had expanded. This is why the proportion of "women with breast
cancer" in the group "women with positive mammographies" is higher than
the proportion of "women with breast cancer" in the general population
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although the proportion is still not very high. -
The evidence of
the positive mammography slides the prior probability of 1% to the
posterior probability of 7.8%. -
You might intuit that since the test could have returned positive for
health, but didn't, then the failure of the test to return positive
must
mean that the woman has a higher chance of having breast cancer -
Law of
Conservation of Probability - not a standard term, but the conservation
rule is exact. If you take the revised probability of breast
cancer after a positive result, times the probability of a positive result,
and add that to the revised probability of breast cancer after a
negative result, times the probability
of a negative result, then you must always arrive at the prior
probability. -
p(A&B) is the same as p(B&A), but p(A|B) is not the same thing as
p(B|A) -
For
example, the two quantities p(cancer)
and p(~cancer) have 1
degree of freedom between them, because of the general law p(A) + p(~A) = 1. -
p(positive|cancer) and p(~positive|cancer) also have
only one degree of freedom between them; either a woman with breast
cancer gets a positive mammography or she doesn't. On the other
hand, p(positive|cancer)
and p(positive|~cancer)
have two degrees of
freedom. You can have a mammography test that returns positive
for
80% of cancerous patients and 9.6% of healthy patients, or that returns
positive for 70% of cancerous patients and 2% of healthy patients, or
even a health test that returns "positive" for 30% of cancerous
patients
and 92% of healthy patients. -
p(positive&cancer) =
p(positive|cancer) * p(cancer) -
You should recognize this operation
from the graph; it's the projection of the top bar into the bottom
bar. p(cancer) is
the
left sector of the top bar, and p(positive|cancer)
determines how much of that sector projects into the bottom bar, and
the
left sector of the bottom bar is p(positive&cancer). -
Similarly, if we know the number of patients with breast cancer and
positive mammographies, and also the number of patients with breast
cancer, we can estimate the chance that a woman with breast cancer gets
a positive mammography by dividing: p(positive|cancer) =
p(positive&cancer) / p(cancer). -
What about p(positive&cancer), p(positive&~cancer), p(~positive&cancer), and p(~positive&~cancer)?
You might at first be tempted to think that there are only two degrees
of freedom for these four quantities - that you can, for example, get p(positive&~cancer) by
multiplying p(positive) *
p(~cancer), and thus that all four quantities can be found given
only the two quantities p(positive)
and p(cancer). This
is not the case! p(positive&~cancer)
= p(positive) * p(~cancer) only if the two probabilities are statistically independent - if the
chance that a woman has breast cancer has no bearing on whether she has
a positive mammography. -
groups A, B, C, and D
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it follows that the entire set of 16 probabilities
contains only three degrees of freedom. Remember that in our
problems we always needed three
pieces of information - the prior probability and the two conditional
probabilities
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16 Jun 08
François ParmentierHere you will find an attempt to offer an intuitive explanation of Bayesian reasoning
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People do not employ Bayesian reasoning intuitively, find it very difficult to
learn Bayesian reasoning when tutored, and rapidly forget Bayesian methods once
the tutoring is over -
Bayesian problems in your dreams.
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the vast majority of doctors in these studies seem to have thought that if
around 80% of women with breast cancer have positive mammographies, then the
probability of a women with a positive mammography having breast cancer must be
around 80%. -
a positive result on the mammography does increase the estimated probability,
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Even if mammography
in this world detects
breast cancer in 8 out of 10 cases, while returning a false positive on
a woman without breast cancer in only 1 out of 10 cases, there will
still be a hundred thousand false positives for every real case of
cancer detected.
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10 Mar 07
Olifante *"science itself is a special case of Bayes' Theorem; experimental evidence is Bayesian evidence. [...] Karl Popper's falsificationism [...] is the old philosophy that the Bayesian revolution is currently dethroning."
bayesian statistics falsification evidence theorem conditional_probability
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Bayes' Theorem
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16 Oct 05
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02 Jan 05
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Your friends and colleagues are talking about something called "Bayes' Theorem" or "Bayes' Rule", or something called Bayesian reasoning. They sound really enthusiastic about it, too, so you google and find a webpage about Bayes' Theorem and...
It's this equation. That's all. Just one equation. The page you found gives a definition of it, but it doesn't say what it is, or why it's useful, or why your friends would be interested in it. It looks like this random statistics thing.
So you came here. Maybe you don't understand what the equation says. Maybe you understand it in theory, but every time you try to apply it in practice you get mixed up trying to remember the difference between p(a|x) and p(x|a), and whether p(a)*p(x|a) belongs in the numerator or the denominator. Maybe you see the theorem, and you understand the theorem, and you can use the theorem, but you can't understand why your friends and/or research colleagues seem to think it's the secret of the universe. Maybe your friends are all wearing Bayes' Theorem T-shirts, and you're feeling left out. Maybe you're a girl looking for a boyfriend, but the boy you're interested in refuses to date anyone who "isn't Bayesian". What matters is that Bayes is cool, and if you don't know Bayes, you aren't cool.
Why does a mathematical concept generate this strange enthusiasm in its students? What is the so-called Bayesian Revolution now sweeping through the sciences, which claims to subsume even the experimental method itself as a special case? What is the secret that the adherents of Bayes know? What is the light that they have seen?
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01 Jan 05
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23 Nov 04
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Bayes' Theorem for the curious and bewildered; an excruciatingly gentle introduction.
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11 Oct 04
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05 Oct 04
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02 Sep 04
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Your friends and colleagues are talking about something called "Bayes' Theorem" or "Bayes' Rule", or something called Bayesian reasoning. They sound really enthusiastic about it, too, so you google and find a webpage about Bayes' Theorem and...
It's this equation. That's all. Just one equation. The page you found gives a definition of it, but it doesn't say what it is, or why it's useful, or why your friends would be interested in it. It looks like this random statistics thing. -
Your friends and colleagues are talking about something called "Bayes' Theorem" or "Bayes' Rule", or something called Bayesian reasoning. They sound really enthusiastic about it, too, so you google and find a webpage about Bayes' Theorem and...
It's this equation. That's all. Just one equation. The page you found gives a definition of it, but it doesn't say what it is, or why it's useful, or why your friends would be interested in it. It looks like this random statistics thing. - 1 more annotations...
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Your friends and colleagues are talking about something called "Bayes' Theorem" or "Bayes' Rule", or something called Bayesian reasoning. They sound really enthusiastic about it, too, so you google and find a webpage about Bayes' Theorem and...
It's this equation. That's all. Just one equation. The page you found gives a definition of it, but it doesn't say what it is, or why it's useful, or why your friends would be interested in it. It looks like this random statistics thing.
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31 Aug 04
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An Intuitive Explanation of Bayesian Reasoning
Bayes' Theorem
for the curious and bewildered;
an excruciatingly gentle introduction.
By Eliezer Yudkowsky
Your friends and colleagues are talking about something called "Bayes' Theorem" or "Bayes' Rule", or something called Bayesian reasoning. They sound really enthusiastic about it, too, so you google and find a webpage about Bayes' Theorem and... -
An Intuitive Explanation of Bayesian Reasoning
Bayes' Theorem
for the curious and bewildered;
an excruciatingly gentle introduction.
By Eliezer Yudkowsky
Your friends and colleagues are talking about something called "Bayes' Theorem" or "Bayes' Rule", or something called Bayesian reasoning. They sound really enthusiastic about it, too, so you google and find a webpage about Bayes' Theorem and... - 1 more annotations...
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An Intuitive Explanation of Bayesian Reasoning
Bayes' Theorem
for the curious and bewildered;
an excruciatingly gentle introduction.
By Eliezer Yudkowsky
Your friends and colleagues are talking about something called "Bayes' Theorem" or "Bayes' Rule", or something called Bayesian reasoning. They sound really enthusiastic about it, too, so you google and find a webpage about Bayes' Theorem and...
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19 Aug 04
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28 Jul 04
Ben ButtigiegFacinating article about Bayesian Reasoning
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other." In one sense, this is correct - any correlation, no matter how weak, is fair prey for Bayes' Theorem; but Bayes' Theorem distinguishes between weak and strong evidence. That is, Bayes' Theorem not only tells us what is and isn't evidence, it also describes the strength of evidence. Bayes' Theorem not only tells us when to revise our probabilities, but how much to revise our probabilities. A correlation between hope and biological warfare may exist, but it's a lot weaker than the speaker wants it to be; he is revising his probabilities much too far.
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26 Jul 04
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Or so they claim. Here you will find an attempt to offer an intuitive explanation of Bayesian reasoning - an excruciatingly gentle introduction that invokes all the human ways of grasping numbers, from natural frequencies to spatial visualization. The intent is to convey, not abstract rules for manipulating numbers, but what the numbers mean, and why the rules are what they are (and cannot possibly be anything else). When you are finished reading this page, you will see Bayesian problems in your dreams.
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25 Jul 04
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24 Jul 04
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Your friends and colleagues are talking about something called "Bayes' Theorem" or "Bayes' Rule", or something called Bayesian reasoning. They sound really enthusiastic about it, too, so you google and find a webpage about Bayes' Theorem and...
It's this equation. That's all. Just one equation. The page you found gives a definition of it, but it doesn't say what it is, or why it's useful, or why your friends would be interested in it. It looks like this random statistics thing.
So you came here. Maybe you don't understand what the equation says. Maybe you understand it in theory, but every time you try to apply it in practice you get mixed up trying to remember the difference between p(a|x) and p(x|a), and whether p(a)*p(x|a) belongs in the numerator or the denominator. Maybe you see the theorem, and you understand the theorem, and you can use the theorem, but you can't understand why your friends and/or research colleagues seem to think it's the secret of the universe. Maybe your friends are all wearing Bayes' Theorem T-shirts, and you're feeling left out. Maybe you're a girl looking for a boyfriend, but the boy you're interested in refuses to date anyone who "isn't Bayesian". What matters is that Bayes is cool, and if you don't know Bayes, you aren't cool. -
Your friends and colleagues are talking about something called "Bayes' Theorem" or "Bayes' Rule", or something called Bayesian reasoning. They sound really enthusiastic about it, too, so you google and find a webpage about Bayes' Theorem and...
It's this equation. That's all. Just one equation. The page you found gives a definition of it, but it doesn't say what it is, or why it's useful, or why your friends would be interested in it. It looks like this random statistics thing.
So you came here. Maybe you don't understand what the equation says. Maybe you understand it in theory, but every time you try to apply it in practice you get mixed up trying to remember the difference between p(a|x) and p(x|a), and whether p(a)*p(x|a) belongs in the numerator or the denominator. Maybe you see the theorem, and you understand the theorem, and you can use the theorem, but you can't understand why your friends and/or research colleagues seem to think it's the secret of the universe. Maybe your friends are all wearing Bayes' Theorem T-shirts, and you're feeling left out. Maybe you're a girl looking for a boyfriend, but the boy you're interested in refuses to date anyone who "isn't Bayesian". What matters is that Bayes is cool, and if you don't know Bayes, you aren't cool.
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23 Jul 04
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22 Jul 04
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p(A|X) = p(X|A)*p(A)
p(X|A)*p(A) p(X|~A)*p(~A)
Why wait so long to introduce Bayes' Theorem, instead of just showing it at the beginning? Well... because I've tried that before; and what happens, in my experience, is that people get all tangled up in trying to apply Bayes' Theorem as a set of poorly grounded mental rules; instead of the Theorem helping, it becomes one more thing to juggle mentally, so that in addition to trying to remember how many women with breast cancer have positive mammographies, the reader is also trying to remember whether it's p(X|A) in the numerator or p(A|X), and whether a positive mammography result corresponds to A or X, and which side of p(X|A) is the implication, and what the terms are in the denominator, and so on. In this excruciatingly gentle introduction, I tried to show all the workings of Bayesian reasoning without ever introducing the explicit Theorem as something extra to memorize, hopefully reducing the number of factors the reader needed to mentally juggle.
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Here you will find an attempt to offer an intuitive explanation of Bayesian reasoning - an excruciatingly gentle introduction that invokes all the human ways of grasping numbers, from natural frequencies to spatial visualization. The intent is to convey, not abstract rules for manipulating numbers, but what the numbers mean, and why the rules are what they are (and cannot possibly be anything else). When you are finished reading this page, you will see Bayesian problems in your dreams.
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