Picture this: You wake up one morning and
you feel a little bit sick. No particular
symptoms, just not 100%.
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 0.1% of the
population and it's a nasty disease,
horrible consequences, you don't want it.
So you ask 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 Bayes' Theorem
to get some perspective.
Bayes' 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 of the hypothesis was true - that
is, how likely you thought it was that
you have this disease before you got the
test results - and multiply it by the
probability of the event given the
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 0.1%. 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 1000
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 Bayes
first came up with this theorem he
didn't actually think it was
revolutionary. 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 is anything worth
publishing in there. And that's where
Price discovered what we now know as
the origins of Bayes' Theorem. Bayes
originally considered a thought
experiment where he was sitting with his
back to a perfectly flat, perfectly
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 asked his assistant to do was
to 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 Bayes 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 Bayes' 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 Sun rise 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 Bayes' Theorem 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 Bayes formula
again, except this time for your prior
probability that you have the disease,
you have to put in the posterior
probability, the probability that we
worked out before which is 9%,
because you've already had one positive
test. 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. 2 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. Bayes' 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 Bayesian filter, you can look at
the various words that appear in e-mails,
and use Bayes' Theorem to give a
probability that the email is spam, given
that those words appear. Now Bayes' 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 a
100% certainty, and other
people to hold those same things are
true with 0% certainty. What
Bayes' 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 a 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
Bayes' Theorem, they discussed 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 Bayes' Theorem,
and the reason I'm worried about
that is 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
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 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 Bayesian viewpoint on the
world, if you have no instances of
something happening, then what is your
prior for that event? It will seem
completely impossible your prior may be
0 until it actually happens. You know, the
thing we forget in Bayes' 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 a
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 Bayes'
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 comments.
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You know, the book I've
been listening to on Audible recently is
called: 'The Theory That Would Not Die' by
Sharon Bertsch McGrayne, and it is an
incredible in-depth look at Bayes'
Theorem, and I've learned a lot just
listening to this book, including the crazy
fact that Bayes never came up with the
mathematical formulation of his rule
that was done independently by the
mathematician Pierre-Simon Laplace so,
really I think he deserves a lot of a
credit for this theory, but Bayes gets
naming rights because he was first, and
if you want, you can download this book
and listen to it, as I have, when I've
just been driving in the car or going to
the gym, which I'm doing again, and so if
there's 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.