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  • 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.

  • Hey, this episode of Veritasium was

  • supported in part by viewers like you on

  • Patreon and by Audible. Audible is a

  • leading provider of spoken audio

  • information including an unmatched

  • selection of audiobooks: original,

  • programming, news, comedy and more.

  • So if you're thinking about trying something new and

  • you haven't tried 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

  • 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.

Picture this: You wake up one morning and

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A2 初級

貝葉斯陷阱 (The Bayesian Trap)

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    林宜悉 發佈於 2021 年 01 月 14 日
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