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Probability Part 2: Updating Your Beliefs with Bayes: Crash Course Statistics #14 - YouTube
Channel: CrashCourse
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Hi, Iâm Adriene Hill, and Welcome back to
Crash Course Statistics. We ended the last
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episode by talking about Conditional Probabilities
which helped us find the probability of one
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event, given that a second event had already
happened.
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But now I want to give you a better idea of
why this is true and how this formula--with
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a few small tweaks--has revolutionized the
field of statistics.
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INTRO
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In general terms, Conditional Probability
says that the probability of an event, B,
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given that event A has already happened, is
the probability of A and B happening together,
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Divided by the probability of A happening
- thatâs the general formula, but letâs
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give you a concrete example so we can visualize
it.
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Hereâs a Venn Diagram of two events, An
Email containing the words âNigerian Princeâ
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and an Email being Spam.
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So I get an email that has the words âNigerian
Princeâ in it, and I want to know what the
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probability is that this email is Spam, given
that I already know the email contains the
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words âNigerian Prince.â This is the equation.
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Alright, letâs take this part a little.
On the Venn Diagram, I can represent the fact
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that I know the words âNigerian Princeâ
already happened by only looking at the events
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where Nigerian Prince occurs, so just this
circle.
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Now inside this circle I have two areas, areas where the email is spam, and areas
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where itâs not. According to our formula,
the probability of spam given Nigerian Prince
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is the probability of spam AND Nigerian Prince
which is this region... where they overlapâŠdivided
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by Probability of Nigerian Prince which is
the whole circle that weâre looking at.
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Now...if we want to know the proportion of
times when an email is Spam given that we
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already know it has the words âNigerian
Princeâ, we need to look at how much of
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the whole Nigerian Prince circle that the
region with both Spam and Nigerian Prince
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covers.
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And actually, some email servers use a slightly
more complex version of this example to filter
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spam. These filters are called Naive Bayes
filters, and thanks to them, you donât have
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to worry about seeing the desperate pleas
of a surprisingly large number of Nigerian
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Princes.
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The Bayes in Naive Bayes comes from the Reverend
Thomas Bayes, a Presbyterian minister who
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broke up his days of prayer, with math. His
largest contribution to the field of math
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and statistics is a slightly expanded version
of our conditional probability formula.
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Bayes Theorem states that:
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The probability of B given A, is equal to
the Probability of A given B times the Probability
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of B all divided by the Probability of A
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You can see that this is just one step away
from our conditional probability formula.
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The only change is in the numerator where
P(A and B) is replaced with P(A|B)P(B). While
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the math of this equality is more than weâll
go into here, you can see with some venn-diagram-algebra
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why this is the case.
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In this form, the equation is known as Bayesâ
Theorem, and it has inspired a strong movement
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in both the statistics and science worlds.
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Just like with your emails, Bayes Theorem
allows us to figure out the probability that
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you have a piece of spam on your hands using
information that we already have, the presence
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of the words âNigerian Princeâ.
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We can also compare that probability to the
probability that you just got a perfectly
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valid email about Nigerian Princes. If you
just tried to guess your odds of an email
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being spam based on the rate of spam to non-spam
email, youâd be missing some pretty useful
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information--the actual words in the email!
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Bayesian statistics is all about UPDATING
your beliefs based on new information. When
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you receive an email, you donât necessarily
think itâs spam, but once you see the word
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Nigerian youâre suspicious. It may just
be your Aunt Judy telling you what she saw
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on the news, but as soon as you see âNigerianâ
and âPrinceâ together, youâre pretty
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convinced that this is junkmail.
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Remember our Lady Tasting Tea example... where
a woman claimed to have superior taste buds
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...that allowed her to know--with one sip--whether
tea or milk was poured into a cup first? When
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youâre watching this lady predict whether
the tea or milk was poured first, each correct
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guess makes you believe her just a little
bit more.
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A few correct guesses may not convince you,
but each correct prediction is a little more
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evidence she has some weird super-tasting
tea powers.
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Reverend Bayes described this idea of âupdatingâ
in a thought experiment.
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Say that youâre standing next to a pool
table but youâre faced away from it, so
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you canât see anything on it. You then have
your friend randomly drop a ball onto the
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table, and this is a special, very even table,
so the ball has an equal chance of landing
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anywhere on it. Your mission--is to guess
how far to the right or left this ball is.
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You have your friend drop another ball onto
the table and report whether itâs to the
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left or to the right of the original ball.
The new ball is to the right of the original,
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so, we can update our belief about where the
ball is.
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If the original is more towards the left,
than most of the new balls will fall to the
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right of our original, just because thereâs
more area there. And the further to the left
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it is, the higher the ratio of new rights
to lefts
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Since this new ball is to the right, that
means thereâs a better chance that our original
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is more toward the left side of the table
than the right, since there would be more
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âroomâ for the new ball to land.
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Each ball that lands to the right of the original
is more evidence that our original is towards
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the left of the table. But, if we get a ball
landing on the left of our original, then
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we know the original is not at the very left
edge. Again, Each new piece of information
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allows us to change our beliefs about the
location of the ball, and changing beliefs
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is what Bayesian statistics is all about.
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Outside thought experiments, Bayesian Statistics
is being used in many different ways, from
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comparing treatments in medical trials, to
helping robots learn language. Itâs being
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used by cancer researchers, ecologists, and
physicists.
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And this method of thinking about statistics...updating
existing information with whatâs come before...may
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be different from the logic of some of the
statistical tests that youâve heard of--like
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the t-test. Those Frequentist statistics can
sometimes be more like probability done in
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a vacuum. Less reliant on prior knowledge.
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When the math of probability gets hard to
wrap your head around, we can use simulations
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to help see these rules in action. Simulations
take rules and create a pretend universe that
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follows those rules.
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Letâs say youâre the boss of a company,
and you receive news that one of your employees,
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Joe, has failed a drug test. Itâs hard to
believe. You remember seeing this thing on
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YouTube that told you how to figure out the
probability that Joe really is on drugs given
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that he got a positive test.
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You canât remember exactly what the formula
is...but you could always run a simulation.
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Simulations are nice, because we can just
tell our computer some rules, and it will
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randomly generate data based on those rules.
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For example, we can tell it the base rate
of people in our state that are on drugs,
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the sensitivity (how many true positives we
get) of the drug test... and specificity (how
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many true negatives we get). Then we ask our
computer to generate 10,000 simulated people
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and tell us what percent of the time people
with positive drug tests were actually on
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drugs.
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If the drug Joe tested positive for--in this
case Glitterstim--is only used by about 5%
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of the population, and the test for Glitterstim
has a 90% sensitivity and 95% specificity,
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I can plug that in and ask the computer to
simulate 10,000 people according to these
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rules.
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And when we ran this simulation, only 49.2%
of the people who tested positive were actually
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using Glitterstim. So I should probably give
Joe another chance...or another test.
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And if I did the math, Iâd see that 49.2%
is pretty close since the theoretical answer
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is around 48.6%. Simulations can help reveal
truths about probability, even without formulas.
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Theyâre a great way to demonstrate probability
and create intuition that can stand alone
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or build on top of more mathematical approaches
to probability.
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Letâs use one to demonstrate an important
concept in probability that makes it possible
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to use samples of data to make inferences
about a population: the Law of Large Numbers.
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In fact we were secretly relying on it when
we used empirical probabilities--like how
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many times I got tails when flipping a coin
10 times--to estimate theoretical probabilities--like
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the true probability of getting tails.
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In its weak form, Law of Large Numbers tells
us that as our samples of data get bigger
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and bigger, our sample mean will be âarbitrarilyâ close to the true population mean.
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Before we go into more detail, letâs see
a simulation and if you want to follow along
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or run it on your own - instructions are in
the description below.
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In this simulation weâre picking values
from a new intelligence test--from the normal
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distribution, that has a mean of 50 and a
standard deviation of 20. When you have a
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very small sample size, say 2, your sample
means are all over the place.
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You can see that pretty much anything goes,
we see means between 5 and 95. And this makes
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sense, when we only have two data points in
our sample, itâs not that unlikely that
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we get two really small numbers, or two pretty
big numbers, which is why we see both low
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and high sample means.
Though we can tell that a lot of the means
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are around the true mean of 50 because the
histogram is the tallest at values around
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50.
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But once we increase the sample size, even
to just 100 values, you can see that the sample
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means are mostly around the real mean of 50.
In fact all of the sample means are within
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10 units of the true population mean.
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And when we go up to 1000, just about every
sample mean is very very close to the true
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mean. And when you run this simulation over
and over, youâll see pretty similar results.
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The neat thing is that the Law of Large numbers
applies to almost any distribution as long
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as the distribution doesnât have an infinite
variance.
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Take the uniform distribution which looks
like a rectangle. Imagine a 100-sided die,
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every single value is equally probable.
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Even the sample means that are selected from
a uniform distribution get closer and closer
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to the true mean of 50..
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The law of large numbers is the evidence we
need to feel confident that the mean of the
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samples we analyze is a pretty good guess
for the true population mean. And the bigger
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our samples are, the better we think the guess
is! This property allows us to make guesses
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about populations, based on samples.
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It also explains why casinos make money in
the long run over hundreds of thousands of
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payouts and losses, even if the experience
of each person varies a lot. The casino looks
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at a huge sample--every single bet and payout--whereas
your sample as an individual is smaller, and
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therefore less likely to be representative.
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Each of these concepts can help us another
way ...another way to look at the data around
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us. The Bayesian framework shows us that every
event or data point can and should âupdateâ
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your beliefs but it doesnât mean you need
to completely change your mind.
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And simulations allow us to build upon these
observations when the underlying mechanics
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arenât so clear.
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We are continuously accumulating evidence
and modifying our beliefs everyday, adding
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today's events to our conception of how the
world works. And hey, maybe one day weâll
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all start sincerely emailing each other about
Nigerian Princes.
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Then weâre gonna have to do some belief-updating. Thanks for watching. Iâll see you next time.
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