my name is Ian ol Azov I'm a graduate
student at the CUNY Graduate Center and
today I want to talk to you about Bayes
theorem Bayes theorem is a fact about
probabilities a version of which was
first discovered in the 18th century by
Thomas Bayes the theorem is Bayes most
famous contribution to the mathematical
theory of probability it has a lot of
applications and some philosophers even
think it's the key to understanding what
it means to think rationally in order to
understand the theorem though we'll have
to understand a little bit about
probabilities the probability of a
proposition is the chance or likelihood
that that proposition is true suppose
you know that one student in a class of
20 has the flu but you don't know who if
you know that Sally is a student in the
class you would say the probability that
Sally has the flu is 1 in 20 or 5
percent or point zero 5 we can call this
your prior probability that Sally has
the flu because it's your probability
prior to finding out any new information
as a shorthand we'll write P of Sally
has the flu equals 0.05 suppose now that
there are five girls and 15 boys in the
class now you don't know whether the
class's flu patient is a boy or a girl
but if you were to find out that the
patient was a girl your probability that
Sally has the flu would go up to one in
five or twenty percent or 0.2 on the
other hand if you were to find out that
the patient was a boy your probability
that Sally has the flu would go down to
zero because these things are still iffy
though remember you don't yet know
whether the flu patient is a boy or girl
will call these things conditional
probabilities your probability that
Sally has the flu conditional on the flu
patient being a girl is point to your
probability that Sally has the flu
given that the flu patient is a boy is
zero as a shorthand we'll write P of
Sally has the flu given that the flu
patient is a girl equals point two and P
of Sally has the flu given that the flu
patient is a boy equals zero the little
vertical line tells you that we're
talking about conditional probabilities
now here's the thing
sometimes you don't know what your
conditional probabilities should be in
other words you know that you might
encounter some new evidence in the
future but you don't yet know how that
evidence should affect the probability
you assign to some hypothesis here's
where Bayes theorem comes in it gives
you a way of figuring out what your
conditional probabilities should be so
what does Bayes theorem actually say
remember our shorthand your probability
in some hypothesis let's call it h
conditional on some new piece of
evidence let's call it e is written P of
H given e here's what Bayes theorem
tells us P of H given e equals P of e
given H times P of H divided by P of e
in other words it tells us the three
ingredients that go into the probability
of a hypothesis conditional on some
evidence the probability of the evidence
conditional on the hypothesis the prior
probability of the hypothesis and the
prior probability of the evidence let's
look at an example imagine that one
morning you don't feel right and you go
on WebMD
to figure out what's wrong you're
browsing around until you find an
illness that catches your eye hypothesis
so the hypothesis under consideration is
that you've come down with hypothesis as
you read through the list of symptoms
you realize that you have all of them in
other words you have all of the symptoms
that you would have if you had
hypothesis so let's say P of e given H
or P of symptoms given hypothesize s
equals 0.95 you begin to freak out but
then you remember Bayes theorem it tells
you that there are two more things you
need to know in order to figure out the
probability that you have hypothesis the
prior probability that you would come
down with hypothesis and the prior
probability that you would have the
symptoms that you actually have with a
little more googling you discover that
the disease is extremely rare only one
in 100,000 people have it so P of
hypothesis is point zero zero zero zero
one now for the last ingredient what
kind of symptoms are they suppose
they're very common like a headache and
a
runny nose lots of people have those
google tells you one in a hundred so P
of symptoms your prior probability that
you would come down with the symptoms
you have is point zero one at last you
know everything that you need to know in
order to figure out the probability that
you have hypothesize s given your
symptoms Bayes theorem tells you that P
of hypothesis given symptoms equals P of
symptoms given hypothesize s times P of
hypothesis divided by P of symptoms in
other words P of hypothesis given
symptoms equals 0.001 five or a little
less than one in a thousand Bayes
theorem is very helpful because in
figuring out what to make of some new
piece of evidence people often ignore
the prior probability of the hypothesis
or treat P of H given E as P of e given
H this mistake is sometimes known as the
base rate fallacy in the case we just
looked at P of H given E is very
different from P of e given H 1 is less
than one tenth of a percent and the
other is 95 percent without Bayes
theorem you might have gotten a lot more
worked up about hypothesis than you need
it to be wrapping up then Bayes theorem
is a formula that tells you how to
calculate conditional probabilities or
the probability you should assign to
some hypothesis given a piece of
evidence even if you forget the formula
though try to remember that the
conditional probability of H given E is
determined by three things the
conditional probability of e given H the
prior probability of H and the prior
probability of e if you leave one of
those three things out you don't have a
complete picture