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  • Hi, I'm Adriene Hill, and Welcome back to Crash Course Statistics. We ended the last

  • episode by talking about Conditional Probabilities which helped us find the probability of one

  • event, given that a second event had already happened.

  • But now I want to give you a better idea of why this is true and how this formula--with

  • a few small tweaks--has revolutionized the field of statistics.

  • INTRO

  • In general terms, Conditional Probability says that the probability of an event, B,

  • given that event A has already happened, is the probability of A and B happening together,

  • Divided by the probability of A happening - that's the general formula, but let's

  • give you a concrete example so we can visualize it.

  • Here's a Venn Diagram of two events, An Email containing the wordsNigerian Prince

  • and an Email being Spam.

  • So I get an email that has the wordsNigerian Princein it, and I want to know what the

  • probability is that this email is Spam, given that I already know the email contains the

  • wordsNigerian Prince.” This is the equation.

  • Alright, let's take this part a little. On the Venn Diagram, I can represent the fact

  • that I know the wordsNigerian Princealready happened by only looking at the events

  • where Nigerian Prince occurs, so just this circle.

  • Now inside this circle I have two areas, areas where the email is spam, and areas

  • where it's not. According to our formula, the probability of spam given Nigerian Prince

  • is the probability of spam AND Nigerian Prince which is this region... where they overlapdivided

  • by Probability of Nigerian Prince which is the whole circle that we're looking at.

  • Now...if we want to know the proportion of times when an email is Spam given that we

  • already know it has the wordsNigerian Prince”, we need to look at how much of

  • the whole Nigerian Prince circle that the region with both Spam and Nigerian Prince

  • covers.

  • And actually, some email servers use a slightly more complex version of this example to filter

  • spam. These filters are called Naive Bayes filters, and thanks to them, you don't have

  • to worry about seeing the desperate pleas of a surprisingly large number of Nigerian

  • Princes.

  • The Bayes in Naive Bayes comes from the Reverend Thomas Bayes, a Presbyterian minister who

  • broke up his days of prayer, with math. His largest contribution to the field of math

  • and statistics is a slightly expanded version of our conditional probability formula.

  • Bayes Theorem states that:

  • The probability of B given A, is equal to the Probability of A given B times the Probability

  • of B all divided by the Probability of A

  • You can see that this is just one step away from our conditional probability formula.

  • The only change is in the numerator where P(A and B) is replaced with P(A|B)P(B). While

  • the math of this equality is more than we'll go into here, you can see with some venn-diagram-algebra

  • why this is the case.

  • In this form, the equation is known as Bayes' Theorem, and it has inspired a strong movement

  • in both the statistics and science worlds.

  • Just like with your emails, Bayes Theorem allows us to figure out the probability that

  • you have a piece of spam on your hands using information that we already have, the presence

  • of the wordsNigerian Prince”.

  • We can also compare that probability to the probability that you just got a perfectly

  • valid email about Nigerian Princes. If you just tried to guess your odds of an email

  • being spam based on the rate of spam to non-spam email, you'd be missing some pretty useful

  • information--the actual words in the email!

  • Bayesian statistics is all about UPDATING your beliefs based on new information. When

  • you receive an email, you don't necessarily think it's spam, but once you see the word

  • Nigerian you're suspicious. It may just be your Aunt Judy telling you what she saw

  • on the news, but as soon as you seeNigerianandPrincetogether, you're pretty

  • convinced that this is junkmail.

  • Remember our Lady Tasting Tea example... where a woman claimed to have superior taste buds

  • ...that allowed her to know--with one sip--whether tea or milk was poured into a cup first? When

  • you're watching this lady predict whether the tea or milk was poured first, each correct

  • guess makes you believe her just a little bit more.

  • A few correct guesses may not convince you, but each correct prediction is a little more

  • evidence she has some weird super-tasting tea powers.

  • Reverend Bayes described this idea ofupdatingin a thought experiment.

  • Say that you're standing next to a pool table but you're faced away from it, so

  • you can't see anything on it. You then have your friend randomly drop a ball onto the

  • table, and this is a special, very even table, so the ball has an equal chance of landing

  • anywhere on it. Your mission--is to guess how far to the right or left this ball is.

  • You have your friend drop another ball onto the table and report whether it's to the

  • left or to the right of the original ball. The new ball is to the right of the original,

  • so, we can update our belief about where the ball is.

  • If the original is more towards the left, than most of the new balls will fall to the

  • right of our original, just because there's more area there. And the further to the left

  • it is, the higher the ratio of new rights to lefts

  • Since this new ball is to the right, that means there's a better chance that our original

  • is more toward the left side of the table than the right, since there would be more

  • roomfor the new ball to land.

  • Each ball that lands to the right of the original is more evidence that our original is towards

  • the left of the table. But, if we get a ball landing on the left of our original, then

  • we know the original is not at the very left edge. Again, Each new piece of information

  • allows us to change our beliefs about the location of the ball, and changing beliefs

  • is what Bayesian statistics is all about.

  • Outside thought experiments, Bayesian Statistics is being used in many different ways, from

  • comparing treatments in medical trials, to helping robots learn language. It's being

  • used by cancer researchers, ecologists, and physicists.

  • And this method of thinking about statistics...updating existing information with what's come before...may

  • be different from the logic of some of the statistical tests that you've heard of--like

  • the t-test. Those Frequentist statistics can sometimes be more like probability done in

  • a vacuum. Less reliant on prior knowledge.

  • When the math of probability gets hard to wrap your head around, we can use simulations

  • to help see these rules in action. Simulations take rules and create a pretend universe that

  • follows those rules.

  • Let's say you're the boss of a company, and you receive news that one of your employees,

  • Joe, has failed a drug test. It's hard to believe. You remember seeing this thing on

  • YouTube that told you how to figure out the probability that Joe really is on drugs given

  • that he got a positive test.

  • You can't remember exactly what the formula is...but you could always run a simulation.

  • Simulations are nice, because we can just tell our computer some rules, and it will

  • randomly generate data based on those rules.

  • For example, we can tell it the base rate of people in our state that are on drugs,

  • the sensitivity (how many true positives we get) of the drug test... and specificity (how

  • many true negatives we get). Then we ask our computer to generate 10,000 simulated people

  • and tell us what percent of the time people with positive drug tests were actually on

  • drugs.

  • If the drug Joe tested positive for--in this case Glitterstim--is only used by about 5%

  • of the population, and the test for Glitterstim has a 90% sensitivity and 95% specificity,

  • I can plug that in and ask the computer to simulate 10,000 people according to these

  • rules.

  • And when we ran this simulation, only 49.2% of the people who tested positive were actually

  • using Glitterstim. So I should probably give Joe another chance...or another test.

  • And if I did the math, I'd see that 49.2% is pretty close since the theoretical answer

  • is around 48.6%. Simulations can help reveal truths about probability, even without formulas.

  • They're a great way to demonstrate probability and create intuition that can stand alone

  • or build on top of more mathematical approaches to probability.

  • Let's use one to demonstrate an important concept in probability that makes it possible

  • to use samples of data to make inferences about a population: the Law of Large Numbers.

  • In fact we were secretly relying on it when we used empirical probabilities--like how

  • many times I got tails when flipping a coin 10 times--to estimate theoretical probabilities--like

  • the true probability of getting tails.

  • In its weak form, Law of Large Numbers tells us that as our samples of data get bigger

  • and bigger, our sample mean will be 'arbitrarily' close to the true population mean.

  • Before we go into more detail, let's see a simulation and if you want to follow along

  • or run it on your own - instructions are in the description below.

  • In this simulation we're picking values from a new intelligence test--from the normal

  • distribution, that has a mean of 50 and a standard deviation of 20. When you have a

  • very small sample size, say 2, your sample means are all over the place.

  • You can see that pretty much anything goes, we see means between 5 and 95. And this makes

  • sense, when we only have two data points in our sample, it's not that unlikely that

  • we get two really small numbers, or two pretty big numbers, which is why we see both low

  • and high sample means. Though we can tell that a lot of the means

  • are around the true mean of 50 because the histogram is the tallest at values around

  • 50.

  • But once we increase the sample size, even to just 100 values, you can see that the sample

  • means are mostly around the real mean of 50. In fact all of the sample means are within

  • 10 units of the true population mean.

  • And when we go up to 1000, just about every sample mean is very very close to the true

  • mean. And when you run this simulation over and over, you'll see pretty similar results.

  • The neat thing is that the Law of Large numbers applies to almost any distribution as long

  • as the distribution doesn't have an infinite variance.

  • Take the uniform distribution which looks like a rectangle. Imagine a 100-sided die,

  • every single value is equally probable.

  • Even the sample means that are selected from a uniform distribution get closer and closer

  • to the true mean of 50..

  • The law of large numbers is the evidence we need to feel confident that the mean of the

  • samples we analyze is a pretty good guess for the true population mean. And the bigger

  • our samples are, the better we think the guess is! This property allows us to make guesses

  • about populations, based on samples.

  • It also explains why casinos make money in the long run over hundreds of thousands of

  • payouts and losses, even if the experience of each person varies a lot. The casino looks

  • at a huge sample--every single bet and payout--whereas your sample as an individual is smaller, and

  • therefore less likely to be representative.

  • Each of these concepts can help us another way ...another way to look at the data around

  • us. The Bayesian framework shows us that every event or data point can and shouldupdate

  • your beliefs but it doesn't mean you need to completely change your mind.

  • And simulations allow us to build upon these observations when the underlying mechanics

  • aren't so clear.

  • We are continuously accumulating evidence and modifying our beliefs everyday, adding

  • today's events to our conception of how the world works. And hey, maybe one day we'll

  • all start sincerely emailing each other about Nigerian Princes.

  • Then we're gonna have to do some belief-updating. Thanks for watching. I'll see you next time.

Hi, I'm Adriene Hill, and Welcome back to Crash Course Statistics. We ended the last

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確率パート2: ベイズを使ってあなたの信念を更新する: クラッシュコース統計学 #14 (Probability Part 2: Updating Your Beliefs with Bayes: Crash Course Statistics #14)

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    林宜悉 に公開 2021 年 01 月 14 日
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