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