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6. Monte Carlo Simulation

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6. Monte Carlo Simulation
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  • JOHN GUTTAG: Welcome to Lecture 6.
  • As usual, I want to start by posting some relevant reading.
  • For those who don't know, this lovely picture
  • is of the Casino at Monte Carlo, and shortly you'll
  • see why we're talking about casinos and gambling today.
  • Not because I want to encourage you to gamble your life
  • savings away.
  • A little history about Monte Carlo simulation,
  • which is the topic of today's lecture.
  • The concept was invented by the Polish American mathematician,
  • Stanislaw Ulam.
  • Probably more well known for his work on thermonuclear weapons
  • than on mathematics, but he did do
  • a lot of very important mathematics
  • earlier in his life.
  • The story here starts that he was ill,
  • recovering from some serious illness,
  • and was home and was bored and was
  • playing a lot of games of solitaire, a game I
  • suspect you've all played.
  • Being a mathematician, he naturally wondered,
  • what's the probability of my winning this stupid game which
  • I keep losing?
  • And so he actually spent quite a lot of time
  • trying to work out the combinatorics,
  • so that he could actually compute the probability.
  • And despite being a really amazing mathematician,
  • he failed.
  • The combinatorics were just too complicated.
  • So he thought, well suppose I just play lots of hands
  • and count the number I win, divide by the number
  • of hands I played.
  • Well then he thought about it and said,
  • well, I've already played a lot of hands and I haven't won yet.
  • So it probably will take me years
  • to play enough hands to actually get a good estimate,
  • and I don't want to do that.
  • So he said, well, suppose instead of playing the game,
  • I just simulate the game on a computer.
  • He had no idea how to use a computer,
  • but he had friends in high places.
  • And actually talked to John von Neumann,
  • who is often viewed as the inventor of the stored program
  • computer.
  • And said, John, could you do this on your fancy new ENIAC
  • machine?
  • And on the lower right here, you'll
  • see a picture of the ENIAC.
  • It was a very large machine.
  • It filled a room.
  • And von Neumann said, sure, we could probably
  • do it in only a few hours of computation.
  • Today we would think of a few microseconds,
  • but those machines were slow.
  • Hence was born Monte Carlo simulation,
  • and then they actually used it in the design of the hydrogen
  • bomb.
  • So it turned out to be not just useful for cards.
  • So what is Monte Carlo simulation?
  • It's a method of estimating the values
  • of an unknown quantity using what is
  • called inferential statistics.
  • And we've been using inferential statistics
  • for the last several lectures.
  • The key concepts-- and I want to be careful about these things
  • will be coming back to them--
  • are the population.
  • So think of the population as the universe
  • of possible examples.
  • So in the case of solitaire, it's
  • a universe of all possible games of solitaire
  • that you could possibly play.
  • I have no idea how big that is, but it's really big,
  • Then we take that universe, that population,
  • and we sample it by drawing a proper subset.
  • Proper means not the whole thing.
  • Usually more than one sample to be useful.
  • Certainly more than 0.
  • And then we make an inference about the population
  • based upon some set of statistics we do on the sample.
  • So the population is typically a very large set of examples,
  • and the sample is a smaller set of examples.
  • And the key fact that makes them work
  • is that if we choose the sample at random,
  • the sample will tend to exhibit the same properties
  • as the population from which it is drawn.
  • And that's exactly what we did with the random walk, right?
  • There were a very large number of different random walks
  • you could take of say, 10,000 steps.
  • We didn't look at all possible random walks of 10,000 steps.
  • We drew a small sample of, say 100 such walks,
  • computed the mean of those 100, and said,
  • we think that's probably a good expectation
  • of what the mean would be of all the possible walks of 10,000
  • steps.
  • So we were depending upon this principle.
  • And of course the key fact here is that the sample
  • has to be random.
  • If you start drawing the sample and it's not random,
  • then there's no reason to expect it
  • to have the same properties as that of the population.
  • And we'll go on throughout the term,
  • and talk about the various ways you can get fooled and think
  • of a random sample when exactly you don't.
  • All right, let's look at a very simple example.
  • People like to use flipping coins because coins are easy.
  • So let's assume we have some coin.
  • All right, so I bought two coins slightly larger
  • than the usual coin.
  • And I can flip it.
  • Flip it once, and let's consider one flip,
  • and let's assume it came out heads.
  • I have to say the coin I flipped is not actually a $20 gold
  • piece, in case any of you were thinking of stealing it.
  • All right, so we've got one flip, and it came up heads.
  • And now I can ask you the question--
  • if I were to flip the same coin an infinite number of times,
  • how confident would you be about answering
  • that all infinite flips would be heads?
  • Or even if I were to flip it once more,
  • how confident would you be that the next flip would be heads?
  • And the answer is not very.
  • Well, suppose I flip the coin twice,
  • and both times it came up heads.
  • And I'll ask you the same question--
  • do you think that the next flip is likely to be heads?
  • Well, maybe you would be more inclined to say yes
  • and having only seen one flip, but you wouldn't really
  • jump to say, sure.
  • On the other hand, if I flipped it 100 times and all 100 flips
  • came up heads, well, you might be suspicious
  • that my coin only has a head on both sides, for example.
  • Or is weighted in some funny way that it mostly comes up heads.
  • And so a lot of people, maybe even me, if you said,
  • I flipped it 100 times and it came up heads.
  • What do you think the next one will be?
  • My best guess would be probably heads.
  • How about this one?
  • So here I've simulated 100 flips,
  • and we have 50 heads here, two heads here, And 48 tails.
  • And now if I said, do you think that the probability
  • of the next flip coming up heads--
  • is it 52 out of 100?
  • Well, if you had to guess, that should be the guess you make.
  • Based upon the available evidence,
  • that's the best guess you should probably make.
  • You have no reason to believe it's a fair coin.
  • It could well be weighted.
  • We don't see it with coins, but we see weighted dice
  • all the time.
  • We shouldn't, but they exist.
  • You can buy them on the internet.
  • So typically our best guess is what we've seen,
  • but we really shouldn't have very much confidence
  • in that guess.
  • Because well, could've just been an accident.
  • Highly unlikely even if the coin is fair
  • that you'd get 50-50, right?
  • So why when we see 100 samples and they all come up heads
  • do we feel better about guessing heads for the 101st
  • than we did when we saw two samples?
  • And why don't we feel so good about guessing 52 out of 100
  • when we've seen a hundred flips that came out 52 and 48?
  • And the answer is something called variance.
  • When I had all heads, there was no variability in my answer.
  • I got the same answer all the time.
  • And so there was no variability, and that intuitively--
  • and in fact, mathematically-- should make us feel confident
  • that, OK, maybe that's really the way the world is.
  • On the other hand, when almost half are heads and almost half
  • are tails, there's a lot of variance.
  • Right, it's hard to predict what the next one will be.
  • And so we should have very little confidence
  • that it isn't an accident that it happened
  • to be 52-48 in one direction.
  • So as the variance grows, we need larger samples
  • to have the same amount of confidence.
  • All right, let's look at that with a detailed example.
  • We'll look at roulette in keeping with the theme of Monte
  • Carlo simulation.
  • This is a roulette wheel that could well be at Monte Carlo.
  • There's no need to simulate roulette, by the way.
  • It's a very simple game, but as we've
  • seen with our earlier examples, it's
  • nice when we're learning about simulations to simulate things
  • where we actually can know what the actual answer is
  • so that we can then understand our simulation better.
  • For those of you who don't know how roulette is played--
  • is there anyone here who doesn't know how roulette is played?
  • Good for you.
  • You grew up virtuous.
  • All right, so-- well all right.
  • Maybe I won't go there.
  • So you have a wheel that spins around,
  • and in the middle are a bunch of pockets.
  • Each pocket has a number and a color.
  • You bet in advance on what number
  • you think is going to come up, or what color you
  • think is going to come up.
  • Then somebody drops a ball in that wheel, gives it a spin.
  • And through centrifugal force, the ball
  • stays on the outside for a while.
  • But as the wheel slows down and heads towards the middle,
  • and eventually settles in one of those pockets.
  • And you win or you lose.
  • Now you can bet on it, and so let's look
  • at an example of that.
  • So here is a roulette game.
  • I've called it fair roulette, because it's
  • set up in such a way that in principle, if you bet,
  • your expected value should be 0.
  • You'll win some, you'll lose some,
  • but it's fair in the sense that it's not either
  • a negative or positive sum game.
  • So as always, we have an underbar underbar in it.
  • Well we're setting up the wheel with 36 pockets on it,
  • so you can bet on the numbers 1 through 36.
  • That's way range work, you'll recall.
  • Initially, we don't know where the ball is,
  • so we'll say it's none.
  • And here's the key thing is, if you make a bet,
  • this tells you what your odds are.
  • That if you bet on a pocket and you win,
  • you get [? len ?] of pockets minus 1.
  • So This is why it's a fair game, right?
  • You bet $1.
  • If you win, you get $36, your dollar plus $35 back.
  • If you lose, you lose.
  • All right, self dot spin will be random dot
  • choice among the pockets.
  • And then there is simply bet, where you just
  • can choose an amount to bet and the pocket you want to bet on.
  • I've simplified it.
  • I'm not allowing you to bet here on colors.
  • All right, so then we can play it.
  • So here is play roulette.
  • I've made game the class a parameter,
  • because later we'll look at other kinds of roulette games.
  • You tell it how many spins.
  • What pocket you want to bet on.
  • For simplicity, I'm going to bet on this same pocket
  • all the time.
  • Pick your favorite lucky number and how much you want to bet,
  • and then we'll have a simulation just like the ones
  • we've already looked at.
  • So the number you get right starts at 0.
  • For I and range number of spins, we'll do a spin.
  • And then tote pocket plus equal game dot that pocket.
  • And it will come back either 0 if you've lost,
  • or 35 if you've won.
  • And then we'll just print the results.
  • So we can do it.
  • In fact, let's run it.
  • So here it is.
  • I guess I'm doing a million games here, so quite a few.
  • Actually I'm going to do two.
  • What happens when you spin it 100 times?
  • What happens when you spin it a million times?
  • And we'll see what we get.
  • So what we see here is that we do 100 spins.
  • The first time I did it my expected return was minus 100%.
  • I lost everything I bet.
  • Not so unlikely, given that the odds
  • are pretty long that you could do 100 times without winning.
  • Next time I did a 100, my return was a positive 44%, and then
  • a positive 28%.
  • So you can see, for 100 spins it's highly variable what
  • the expected return is.
  • That's one of the things that makes
  • gambling attractive to people.
  • If you go to a casino, 100 spins would be a pretty long night
  • at the table.
  • And maybe you'd won 44%, and you'd
  • feel pretty good about it.
  • What about a million spins?
  • Well people aren't interested in that, but the casino is, right?
  • They don't really care what happens with 100 spins.
  • They care what happens with a million spins.
  • What happens when everybody comes every night to play.
  • And there what we see is--
  • you'll notice much less variance.
  • Happens to be minus 0.04 plus 0.6 plus 0.79.
  • So it's still not 0, but it's certainly,
  • these are all closer to 0 than any of these are.
  • We know it should be 0, but it doesn't
  • happen to be in these examples.
  • But not only are they closer to 0, they're closer together.
  • There is much less variance in the results, right?
  • So here I show you these three numbers,
  • and ask what do you expect to happen?
  • You have no clue, right?
  • So I don't know, maybe I'll win a lot.
  • Maybe I'll lose everything.
  • I show you these three numbers, you're going to look at it
  • and say, well you know, I'm going
  • to be somewhere between around 0 and maybe 1%.
  • But you're never going to guess it's
  • going to be radically different from that.
  • And if I were to change this number to be even higher,
  • it would go even closer to 0.
  • But we won't bother.
  • OK, so these are the numbers we just
  • looked at, because I said the seed to be the same.
  • So what's going on here is something
  • called the law of large numbers, or sometimes Bernoulli's law.
  • This is a picture of Bernoulli on the stamp.
  • It's one of the two most important theorems in all
  • of statistics, and we'll come to the second most important
  • theorem in the next lecture.
  • Here it says, "in repeated independent tests
  • with the same actual probability, the chance
  • that the fraction of times the outcome differs
  • from p converges to 0 as the number of trials
  • goes to infinity."
  • So this says if I were to spin this fair roulette
  • wheel an infinite number of times,
  • the expected-- the return would be 0.
  • The real true probability from the mathematics.
  • Well, infinite is a lot, but a million
  • is getting closer to infinite.
  • And what this says is the closer I get to infinite,
  • the closer it will be to the true probability.
  • So that's why we did better with a million than with a hundred.
  • And if I did a 100 million, we'd do way better
  • than I did with a million.
  • I want to take a minute to talk about a way this law is
  • often misunderstood.
  • This is something called the gambler's fallacy.
  • And all you have to do is say, let's
  • go watch a sporting event.
  • And you'll watch a batter strike out
  • for the sixth consecutive time.
  • The next time they come to the plate,
  • the idiot announcer says, well he struck out six times
  • in a row.
  • He's due for a hit this time, because he's usually
  • a pretty good hitter.
  • Well that's nonsense.
  • It says, people somehow believe that if deviations
  • from expected occur, they'll be evened out in the future.
  • And we'll see something similar to this that is true,
  • but this is not true.
  • And there is a great story about it.
  • This is told in a book by [INAUDIBLE] and [INAUDIBLE].
  • And this truly happened in Monte Carlo, with Roulette.
  • And you could either bet on black or red.
  • Black came up 26 times in a row.
  • Highly unlikely, right?
  • 2 to the 26th is a giant number.
  • And what happened is, word got out on the casino floor
  • that black had kept coming up way too often.
  • And people more or less panicked to rush to the table
  • to bet on red, saying, well it can't keep coming up black.
  • Surely the next one will be red.
  • And as it happened when the casino totaled up its winnings,
  • it was a record night for the casino.
  • Millions of francs got bet, because people were
  • sure it would have to even out.
  • Well if we think about it, probability
  • of 26 consecutive reds is that.
  • A pretty small number.
  • But the probability of 26 consecutive reds
  • when the previous 25 rolls were red is what?
  • No, that.
  • AUDIENCE: Oh, I thought you meant [INAUDIBLE].
  • JOHN GUTTAG: No, if you had 25 reds and then
  • you spun the wheel once more, the probability
  • of it having 26 reds is now 0.5, because these
  • are independent events.
  • Unless of course the wheel is rigged, and we're assuming
  • it's not.
  • People have a hard time accepting this,
  • and I know it seems funny.
  • But I guarantee there will be some point in the next month
  • or so when you will find yourself thinking this way,
  • that something has to even out.
  • I did so badly on the midterm, I will
  • have to do better on the final.
  • That was mean, I'm sorry.
  • All right, speaking of means--
  • see?
  • Professor [? Grimm's ?] not the only one
  • who can make bad jokes.
  • There is something-- it's not the gambler's fallacy--
  • that's often confused with it, and that's
  • called regression to the mean.
  • This term was coined in 1885 by Francis Galton
  • in a paper, of which I've shown you a page from it here.
  • And the basic conclusion here was--
  • what this table says is if somebody's parents are
  • both taller than average, it's likely
  • that the child will be smaller than the parents.
  • Conversely, if the parents are shorter than average,
  • it's likely that the child will be taller than average.
  • Now you can think about this in terms of genetics and stuff.
  • That's not what he did.
  • He just looked at a bunch of data,
  • and the data actually supported this.
  • And this led him to this notion of regression to the mean.
  • And here's what it is, and here's
  • the way in which it is subtly different from the gambler's
  • fallacy.
  • What he said here is, following an extreme event--
  • parents being unusually tall--
  • the next random event is likely to be less extreme.
  • He didn't know much about genetics,
  • and he kind of assumed the height of people were random.
  • But we'll ignore that.
  • OK, but the idea is here that it will be less extreme.
  • So let's look at it in roulette.
  • If I spin a fair roulette wheel 10 times and get 10 reds,
  • that's an extreme event.
  • Right, here's a probability of basically 1.1024.
  • Now the gambler's fallacy says, if I
  • were to spin it another 10 times,
  • it would need to even out.
  • As in I should get more blacks than you would usually
  • get to make up for these excess reds.
  • What regression to the mean says is different.
  • It says, it's likely that in the next 10 spins,
  • you will get fewer than 10 reds.
  • You will get a less extreme event.
  • Now it doesn't have to be 10.
  • If I'd gotten 7 reds instead of 5, you'd consider that extreme,
  • and you would bet that the next 10 would have fewer than 7.
  • But you wouldn't bet that it would have fewer than 5.
  • Because of this, if you now look at the average of the 20 spins,
  • it will be closer to the mean of 50% reds
  • than you got from the extreme first spins.
  • So that's why it's called regression to the mean.
  • The more samples you take, the more likely
  • you'll get to the mean.
  • Yes?
  • AUDIENCE: So, roulette wheel spins
  • are supposed to be independent.
  • JOHN GUTTAG: Yes.
  • AUDIENCE: So it seems like the second 10--
  • JOHN GUTTAG: Pardon?
  • AUDIENCE: It seems like the second 10 times
  • that you spin it.
  • that shouldn't have to [INAUDIBLE].
  • JOHN GUTTAG: Has nothing to do with the first one.
  • AUDIENCE: But you said it's likely [INAUDIBLE].
  • JOHN GUTTAG: Right, because you have an extreme event, which
  • was unlikely.
  • And now if you have another event,
  • it's likely to be closer to the average
  • than the extreme was to the average.
  • Precisely because it is independent.
  • That makes sense to everybody?
  • Yeah?
  • AUDIENCE: Isn't that the same as the gambler's fallacy, then?
  • By saying that, because this was super unlikely,
  • the next one [INAUDIBLE].
  • JOHN GUTTAG: No, the gambler's fallacy here--
  • and it's a good question, and indeed people often
  • do get these things confused.
  • The gambler's fallacy would say that the second 10
  • spins would--
  • we would expect to have fewer than 5 reds,
  • because you're trying to even out the unusual number of reds
  • in the first Spin
  • Whereas here we're not saying we would have fewer than 5.
  • We're saying we'd probably have fewer than 10.
  • That it'll be closer to the mean,
  • not that it would be below the mean.
  • Whereas the gambler's fallacy would say
  • it should be below that mean to quote, even out, the first 10.
  • Does that makes sense?
  • OK, great questions.
  • Thank you.
  • All right, now you may not know this,
  • but casinos are not in the business of being fair.
  • And the way they don't do that is in Europe,
  • they're not all red and black.
  • They sneak in one green.
  • And so now if you bet red, well sometimes
  • it isn't always red or black.
  • And furthermore, there is this 0.
  • They index from 0 rather than from one, and so
  • you don't get a full payoff.
  • In American roulette, they manage to sneak in two greens.
  • They have a 0 in a double 0.
  • Tilting the odds even more in favor of the casino.
  • So we can do that in our simulation.
  • We'll look at European roulette as a subclass of fair roulette.
  • I've just added this extra pocket, 0.
  • And notice I have not changed the odds.
  • So what you get if you get your number is no higher,
  • but you're a little bit less likely to get it
  • because we snuck in that 0.
  • Than American roulette is a subclass of European roulette
  • in which I add yet another pocket.
  • All right, we can simulate those.
  • Again, nice thing about simulations,
  • we can play these games.
  • So I've simulated 20 trials of 1,000 spins, 10,000 spins,
  • 100,000, and a million.
  • And what do we see as we look at this?
  • Well, right away we can see that fair roulette is usually
  • a much better bet than either of the other two.
  • That even with only 1,000 spins the return is negative.
  • And as we get more and more as I got to a million,
  • it starts to look much more like closer to 0.
  • And these, we have reason to believe at least,
  • are much closer to true expectation
  • saying that, while you break even in fair roulette,
  • you'll lose 2.7% in Europe and over 5% in Las Vegas,
  • or soon in Massachusetts.
  • All right, we're sampling, right?
  • That's why the results will change,
  • and if I ran a different simulation
  • with a different seed I'd get different numbers.
  • Whenever you're sampling, you can't be guaranteed
  • to get perfect accuracy.
  • It's always possible you get a weird sample.
  • That's not to say that you won't get exactly the right answer.
  • I might have spun the wheel twice
  • and happened to get the exact right answer of the return.
  • Actually not twice, because the math
  • doesn't work out, but 35 times and gotten
  • exactly the right answer.
  • But that's not the point.
  • We need to be able to differentiate
  • between what happens to be true and what we actually know,
  • in a rigorous sense, is true.
  • Or maybe don't know it, but have real good reason
  • to believe it's true.
  • So it's not just a question of faith.
  • And that gets us to what's in some sense
  • the fundamental question of all computational statistics,
  • is how many samples do we need to look
  • at before we can have real, justifiable confidence
  • in our answer?
  • As we've just seen--
  • not just, a few minutes ago-- with the coins,
  • our intuition tells us that it depends
  • upon the variability in the underlying possibilities.
  • So let's look at that more carefully.
  • We have to look at the variation in the data.
  • So let's look at first something called variance.
  • So this is variance of x.
  • Think of x as just a list of data examples, data items.
  • And the variance is we first compute the average
  • of value, that's mu.
  • So mu is for the mean.
  • For each little x and big X, we compare the difference
  • of that and the mean.
  • How far is it from the mean?
  • And square of the difference, and then we just sum them.
  • So this takes, how far is everything from the mean?
  • We just add them all up.
  • And then we end up dividing by the size of the set,
  • the number of examples.
  • Why do we have to do this division?
  • Well, because we don't want to say something has high variance
  • just because it has many members, right?
  • So this sort of normalizes is by the number of members,
  • and this just sums how different the members are from the mean.
  • So if everything is the same value,
  • what's the variance going to be?
  • If I have a set of 1,000 6's, what's the variance?
  • Yes?
  • AUDIENCE: 0.
  • JOHN GUTTAG: 0.
  • You think this is going to be hard, but I came prepared.
  • I was hoping this would happen.
  • Look out, I don't know where this is going to go.
  • [FIRES SLINGSHOT]
  • AUDIENCE: [LAUGHTER]
  • JOHN GUTTAG: All right, maybe it isn't the best technology.
  • I'll go home and practice.
  • And then the thing you're more familiar
  • with is the standard deviation.
  • And if you look at the standard deviation is,
  • it's simply the square root of the variance.
  • Now, let's understand this a little bit
  • and first ask, why am I squaring this here,
  • especially because later on I'm just going
  • to take a square root anyway?
  • Well squaring it has one virtue, which
  • is that it means I don't care whether the difference is
  • positive or negative.
  • And I shouldn't, right?
  • I don't care which side of the mean it's on,
  • I just care it's not near the mean.
  • But if that's all I wanted to do I
  • could take the absolute value.
  • The other thing we see with squaring
  • is it gives the outliers extra emphasis, because I'm
  • squaring that distance.
  • Now you can think that's good or bad,
  • but it's worth knowing it's a fact.
  • The more important thing to think about
  • is standard deviation all by itself is a meaningless number.
  • You always have to think about it in the context of the mean.
  • If I tell you the standard deviation is 100,
  • you then say, well-- and I ask you whether it's big or small,
  • you have no idea.
  • If the mean is 100 and the standard deviation is 100,
  • it's pretty big.
  • If the mean is a billion and the standard deviation is 100,
  • it's pretty small.
  • So you should never want to look at just the standard deviation.
  • All right, here is just some code
  • to compute those, easy enough.
  • Why am I doing this?
  • Because we're now getting to the punch line.
  • We often try and estimate values just by giving the mean.
  • So we might report on an exam that the mean grade was 80.
  • It's better instead of trying to describe
  • an unknown value by it--
  • an unknown parameter by a single value,
  • say the expected return on betting a roulette wheel,
  • to provide a confidence interval.
  • So what a confidence interval is is
  • a range that's likely to contain the unknown value,
  • and a confidence that the unknown value is
  • within that range.
  • So I might say on a fair roulette
  • wheel I expect that your return will be between minus 1%
  • and plus 1%, and I expect that to be true 95% of the time
  • you play the game if you play 100 rolls, spins.
  • If you take 100 spins of the roulette wheel,
  • I expect that 95% of the time your return
  • will be between this and that.
  • So here, we're saying the return on betting a pocket 10 times,
  • 10,000 times in European roulette is minus 3.3%.
  • I think that was the number we just saw.
  • And now I'm going to add to that this margin of error,
  • which is plus or minus 3.5% with a 95% level of confidence.
  • What does this mean?
  • If I were to conduct an infinite number of trials
  • of 10,000 bets each, my expected average return
  • would indeed be minus 3.3%, and it
  • would be between these values 95% of the time.
  • I've just subtracted and added this 3.5,
  • saying nothing about what would happen
  • in the other 5% of the time.
  • How far away I might be from this,
  • this is totally silent on that subject.
  • Yes?
  • AUDIENCE: I think you want 0.2 not 9.2.
  • JOHN GUTTAG: Oh, let's see.
  • Yep, I do.
  • Thank you.
  • We'll fix it on the spot.
  • This is why you have to come to lecture
  • rather than just reading the slides,
  • because I make mistakes.
  • Thank you, Eric.
  • All right, so it's telling me that, and that's all it means.
  • And it's amazing how often people don't quite
  • know what this means.
  • For example, when they look at a political pole
  • and they see how many votes somebody is expected to get.
  • And they see this confidence interval and say,
  • what does that really mean?
  • Most people don't know.
  • But it does have a very precise meaning, and this is it.
  • How do we compute confidence intervals?
  • Most of the time we compute them using something
  • called the empirical rule.
  • Under some assumptions, which I'll get to a little bit later,
  • the empirical rule says that if I take the data, find the mean,
  • compute the standard deviation as we've just seen,
  • 68% of the data will be within one standard deviation in front
  • of or behind the mean.
  • Within one standard deviation of the mean.
  • 95% will be within 1.96 standard deviations.
  • And that's what people usually use.
  • Usually when people talk about confidence intervals,
  • they're talking about the 95% confidence interval.
  • And they use this 1.6 number.
  • And 99.7% of the data will be within three
  • standard deviations.
  • So you can see if you are outside the third standard
  • deviation, you are a pretty rare bird,
  • for better or worse depending upon which side.
  • All right, so let's apply the empirical rule
  • to our roulette game.
  • So I've got my three roulette games as before.
  • I'm going to run a simple simulation.
  • And the key thing to notice is really
  • this print statement here.
  • Right, that I'll print the mean, which I'm rounding.
  • And then I'm going to give the confidence intervals,
  • plus or minus, and I'll just take the standard deviation
  • times 1.6 times 100, y times 100,
  • because I'm showing you percentages.
  • All right so again, very straightforward code.
  • Just simulation, just like the ones we've been looking at.
  • And well, I'm just going--
  • I don't think I'll bother running it for you
  • in the interest of time.
  • You can run it yourself.
  • But here's what I got when I ran it.
  • So when I simulated betting a pocket for 20 trials,
  • we see that the--
  • of 1,000 spins each, for 1,000 spins
  • the expected return for fair roulette happened to be 3.68%.
  • A bit high.
  • But you'll notice the confidence interval plus or minus
  • 27 includes the actual answer, which is 0.
  • And we have very large confidence intervals
  • for the other two games.
  • If you go way down to the bottom where I've spun, spun the wheel
  • many more times, what we'll see is
  • that my expected return for fair roulette is much closer to 0
  • than it was here.
  • But more importantly, my confidence interval
  • is much smaller, 0.8.
  • So now I really have constrained it pretty well.
  • Similarly, for the other two games you will see--
  • maybe it's more accurate, maybe it's less accurate,
  • but importantly the confidence interval is smaller.
  • So I have good reason to believe that the mean I'm computing
  • is close to the true mean, because my confidence
  • interval has shrunk.
  • So that's the really important concept here,
  • is that we don't just guess--
  • compute the value in the simulation.
  • We use, in this case, the empirical rule
  • to tell us how much faith we should have in that value.
  • All right, the empirical rule doesn't always work.
  • There are a couple of assumptions.
  • One is that the mean estimation error is 0.
  • What is that saying?
  • That I'm just as likely to guess high as gas low.
  • In most experiments of this sort, most simulations,
  • that's a very fair assumption.
  • There's no reason to guess I'd be systematically off
  • in one direction or another.
  • It's different when you use this in a laboratory experiment,
  • where in fact, depending upon your laboratory technique,
  • there may be a bias in your results in one direction.
  • So we have to assume that there's no bias in our errors.
  • And we have to assume that the distribution of errors
  • is normal.
  • And we'll come back to this in just a second.
  • But this is a normal distribution,
  • called the Gaussian.
  • Under those two assumptions the empirical rule
  • will always hold.
  • All right, let's talk about distributions,
  • since I just introduced one.
  • We've been using a probability distribution.
  • And this captures the notion of the relative frequency
  • with which some random variable takes on different values.
  • There are two kinds. , Discrete and these when the values are
  • drawn from a finite set of values.
  • So when I flip these coins, there
  • are only two possible values, head or tails.
  • And so if we look at the distribution of heads
  • and tails, it's pretty simple.
  • We just list the probability of heads.
  • We list the probability of tails.
  • We know that those two probabilities must add up to 1,
  • and that fully describes our distribution.
  • Continuous random variables are a bit trickier.
  • They're drawn from a set of reals between two numbers.
  • For the sake of argument, let's say
  • those two numbers are 0 and 1.
  • Well, we can't just enumerate the probability
  • for each number.
  • How many real numbers are there between 0 and 1?
  • An infinite number, right?
  • And so I can't say, for each of these infinite numbers, what's
  • the probability of it occurring?
  • Actually the probability is close to 0 for each of them.
  • Is 0, if they're truly infinite.
  • So I need to do something else, and what
  • I do that is what's called the probability density function.
  • This is a different kind of PDF than the one Adobe sells.
  • So there, we don't give the probability
  • of the random variable taking on a specific value.
  • We give the probability of it lying
  • somewhere between two values.
  • And then we define a curve, which shows how it works.
  • So let's look at an example.
  • So we'll go back to normal distributions.
  • This is-- for the continuous normal distribution,
  • it's described by this function.
  • And for those of you who don't know about the magic number e,
  • this is one of many ways to define it.
  • But I really don't care whether you remember this.
  • I don't care whether you know what e is.
  • I don't care if you know what this is.
  • What we really want to say is, it looks like this.
  • In this case, the mean is 0.
  • It doesn't have to be 0.
  • I've [INAUDIBLE] a mean of 0 and a standard deviation of 1.
  • This is called the so-called standard normal distribution.
  • But it's symmetric around the mean.
  • And that gets back to, it's equally likely
  • that our errors are in either direction, right?
  • So it peaks at the mean.
  • The peak is always at the mean.
  • That's the most probable value, and it's
  • symmetric about the mean.
  • So if we look at it, for example, and I say,
  • what's the probability of the number being between 0 and 1?
  • I can look at it here and say, all right,
  • let's draw a line here, and a line here.
  • And then I can integrate the curve under here.
  • And that tells me the probability
  • of this random variable being between 0 and 1.
  • If I want to know between minus 1 and 1.
  • I just do this and then I integrate over that area.
  • All right, so the area under the curve in this case
  • defines the likelihood.
  • Now I have to divide and normalize to actually get
  • the answer between 0 and 1.
  • So the question is, what fraction
  • of the area under the curve is between minus 1 and 1?
  • And that will tell me the probability.
  • So what does the empirical rule tell us?
  • What fraction is between minus 1 and 1, roughly?
  • Yeah?
  • 68%, right?
  • So that tells me 68% of the area under this curve
  • is between minus 1 and 1, because my standard deviation
  • is 1, roughly 68%.
  • And maybe your eyes will convince you
  • that's a reasonable guess.
  • OK, we'll come back and look at this in a bit more detail
  • on Monday of next week.
  • And also look at the question of,
  • why does this work in so many cases
  • where we don't actually have a normal distribution
  • to start with?

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MIT 6.0002 Introduction to Computational Thinking and Data Science, Fall 2016
View the complete course: http://ocw.mit.edu/6-0002F16
Instructor: John Guttag

Prof. Guttag discusses the Monte Carlo simulation, Roulette

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