Kind: captions
Language: en
picture this you wake up one morning and
you feel a little bit sick no particular
symptoms just not 100% And so you go to
the doctor and she also doesn't know
what's going on with you so she suggests
they run a battery of tests and after a
week goes by the results come back turns
out you tested positive for a very rare
disease that affects about .1% of the
population and it's a nasty disease
horrible consequences you don't want it
so you asked the doctor you know how
certain is it that I have this disease
and she says well the test will
correctly identify 99% of people that
have the disease and only incorrectly
identify 1% of people who don't have the
disease so that sounds pretty bad I mean
what are the chances that you actually
have this disease I think most people
would say 99% because that's the
accuracy of the test but that is not
actually correct you need B theorem to
get some perspective
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base theorem can give you the
probability that some hypothesis say
that you actually have the disease is
true given an event that you tested
positive for the disease to calculate
this you need to take the prior
probability the hypothesis was true that
is How likely you thought it was that
you had this disease before you got the
test results and multiply it by the
probability of the event given the hyp
hypothesis is true that is the
probability that you would test positive
if you had the disease and then divide
that by the total probability of the
event occurring that is testing positive
this term is a combination of your
probability of having the disease and
correctly testing positive plus your
probability of not having the disease
and being falsely identified the prior
probability that a hypothesis is true is
often the hardest part of this equation
to figure out and sometimes it's no
better than a guess but in this case a
reasonable starting point is the
frequency of the disease in the
population so. 1% and if you plug in the
rest of the numbers you find that you
have a 9% chance of actually having the
disease after testing positive which is
incredibly low if you think about it now
this isn't some sort of crazy magic it's
actually Common Sense applied to
mathematics just think about a sample
size of a thousand people now one person
out of that thousand is likely to
actually have the disease and the test
would likely identify them correctly as
having the disease but out of the 999
other people 1% or 10 people would
falsely be identified as having the
disease so if you're one of those people
who has a positive test result and
everyone's just selected at random well
you're actually part of a group of 11
where only one person has the disease so
your chances of actually having it are 1
in 11 9% it just makes sense when baze
first came up with this theorem he
didn't actually think it was revolu
tionary he didn't even think it was
worthy of publication he didn't submit
it to the Royal Society of which he was
a member and in fact it was discovered
in his papers after he died and he had
abandoned it for more than a decade his
relatives asked his friend Richard Price
to dig through his papers and see if
there was anything worth publishing in
there and that's where price discovered
what we now know as the origins of baz
theorem baz originally considered a
thought experiment where he was sitting
with his back to a perfectly flat ly
Square table and then he would ask an
assistant to throw a ball onto the table
now this ball could obviously land and
end up anywhere on the table and he
wanted to figure out where it was so
what he'd ask his assistant to do was
throw on another ball and then tell him
if it landed to the left or to the right
or in front behind of the first ball and
he would note that down and then ask for
more and more balls to be thrown on the
table what he realized was that through
this method he could keep updating his
idea of where the first ball was now of
course he would never be completely
certain but with each new piece of
evidence he would get more and more
accurate and that's how baze saw the
world it wasn't that he thought the
world was not determined that reality
didn't quite exist but it was that we
couldn't know it perfectly and all we
could hope to do was update our
understanding as more and more evidence
became available when Richard Price
introduced Bay theorem he made an
analogy to a man coming out of a cave
maybe he'd lived his whole life in there
and he saw the sunrise for the first
time and kind of thought to himself is
is this a one-off is this a quirk or
does the sun always do this and then
every day after that as the sun rose
again he could get a little bit more
confident that well that was the way the
world works so Bas theem wasn't really a
formula intended to be used just once it
was intended to be used multiple times
each time gaining new evidence and
updating your probability that something
is true so if we go back to the first
example when you tested positive for a
disease what would happen if you went to
another doctor get a second opinion and
get that test run again but let's say by
a different lab just to be sure that
those tests are independent and let's
say that test also comes back as
positive now what is the probability
that you actually have the disease well
you can use base formula again except
this time for your prior probability
that you have the disease you have to
put in the posterior probability the the
probability that we worked out before
which is 9% because you've already had
one positive test and if you crunch
those numbers the new probability based
on two positive tests is 91% there's a
91% chance that you actually have the
disease which kind of makes sense two
positive results by different labs are
unlikely to just be chance but you'll
notice that probability is still not as
high as the accuracy the reported
accuracy of the test baz theorem has
found a number of practical applications
including notably filtering your spam
you know traditional spam filters
actually do a kind of bad job there's
too many false positives too much of
your email ends up in spam but using a
ban filter you can look at the various
words that appear in emails and use Baye
theorem to give a probability that the
email is Spam given that those words
appear
now Bas theorem tells us how to update
our beliefs in light of new evidence but
it can't tell us how to set our prior
beliefs and so it's possible for some
people to hold that certain things are
true with 100% certainty and other
people to hold those same things are
true with 0% certainty what Bas theorem
shows us is that in those cases there is
absolutely no evidence nothing anyone
could do to change their minds and so as
Nate silver points out in his book The
Signal and the noise
we should probably not have debates
between people with 100% prior certainty
and 0% prior certainty because well
really they'll never convince each other
of
anything most of the time when people
talk about base theorem they discuss how
counterintuitive it is and how we don't
really have an inbuilt sense of it but
recently my concern has been the
opposite that maybe we're too good at
internalizing the thinking behind Bay
theorem and the reason I'm worried about
that is because because I think in life
we can get used to particular
circumstances we can get used to results
maybe getting rejected or failing at
something or uh getting paid a low wage
and we can internalize that as though we
are that man emerging from the cave and
we see the sun rise every day and every
day and we keep updating our beliefs to
uh a point of near certainty that we
think that that is basically the way
that nature is it's the way the world is
and there's nothing that we can do to
change it you know there's Nelson
Mandela's quote that everything is
impossible until it's done and I think
that is kind of a very beian viewpoint
on the world if you have uh no instances
of something happening then what is your
prior for that event it will seem
completely impossible your prior may be
zero until it actually happens you know
the thing we forget in Bas theorem is
that our actions play a role in
determining outcomes and determining how
true things actually are but if we
internalize that something is true and
maybe we're 100% sure that it's true and
there's nothing we can do to change it
well then we're going to keep on doing
the same thing and we're going to keep
on getting the same result it's a
self-fulfilling prophecy so I think a
really good understanding of Bas theorem
implies that experimentation is
essential if you've been doing the same
thing for a long time and getting the
same result that you're not necessarily
happy with maybe it's time to
change so is there something like that
that you've been thinking about if so
let me know in the
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comments hey this episode of veritasium
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You on patreon and by audible audible is
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information including an unmatched
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you thinking about trying something new
and you haven't tried a audible yet you
should give them a shot and for viewers
of this channel they offer a free 30-day
trial just by going to
audible.com/veritasium you know the book
I've been listening to on Audible
recently is called the theory that would
not die by Cheryl Burch mcgrain and it
is an incredible indepth look at baze
theorem and I've learned a lot just
listening to this book including the
crazy fact that b never came up with the
mathematical formulation of his rule
that was done independently by the
mathematician Pierre Simone llas so
really I think he deserves a lot of the
credit for this Theory uh but baze gets
naming rights because he was first and
if you want you can download this book
and uh listen to it as I have when I've
just been uh driving in the car or going
to the gym which I'm doing again um and
so if there's a a part of your day that
you feel is kind of boring then I can
highly recommend trying out audiobooks
from audible just go to
audible.com/veritasium so as always I
want to thank Audible for supporting me
and I want to thank you for watching