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rLlZpnT02ZU | just because-- well, that's
just we can say a set, |
rLlZpnT02ZU | a confidence set. |
rLlZpnT02ZU | But people like to
say confidence region. |
rLlZpnT02ZU | So an interval is just a
one-dimensional conference |
rLlZpnT02ZU | region. |
rLlZpnT02ZU | And it has to be an
interval as well. |
rLlZpnT02ZU | So a confidence interval
of level 1 minus alpha-- |
rLlZpnT02ZU | so we refer to the quality
of a confidence interval |
rLlZpnT02ZU | is actually called it's level. |
rLlZpnT02ZU | It takes value 1 minus alpha
for some positive alpha. |
rLlZpnT02ZU | And so the confidence level-- |
rLlZpnT02ZU | the level of the confidence
interval is between 0 and 1. |
rLlZpnT02ZU | The closer to 1 it is, the
better the confidence interval. |
rLlZpnT02ZU | The closer to 0,
the worse it is. |
rLlZpnT02ZU | And so for any
random interval-- so |
rLlZpnT02ZU | a confidence interval
is a random interval. |
rLlZpnT02ZU | The bounds of this interval
depends on random data. |
rLlZpnT02ZU | Just like we had
X bar plus/minus |
rLlZpnT02ZU | 1 over square root of
n, for example, or 2 |
rLlZpnT02ZU | over square root
of n, this X bar |
rLlZpnT02ZU | was the random thing that would
make fluctuate those guys. |
rLlZpnT02ZU | And so now I have an interval. |
rLlZpnT02ZU | And now I have its boundaries,
but now the boundaries |
rLlZpnT02ZU | are not allowed to depend
on my unknown parameter. |
rLlZpnT02ZU | Otherwise, it's not a
confidence interval, |
rLlZpnT02ZU | just like an
estimator that depends |
rLlZpnT02ZU | on the unknown parameter
is not an estimator. |
rLlZpnT02ZU | The confidence interval
has to be something |
rLlZpnT02ZU | that I can compute
once I collect data. |
rLlZpnT02ZU | And so what I want is that--
so there's this weird notation. |
rLlZpnT02ZU | The fact that I write theta-- |
rLlZpnT02ZU | that's the probability
that I contains theta. |
rLlZpnT02ZU | You're used to seeing
theta belongs to I. |
rLlZpnT02ZU | But here, I really
want to emphasize |
rLlZpnT02ZU | that the randomness is
in I. And so the way |
rLlZpnT02ZU | you actually say
it when you read |
rLlZpnT02ZU | this formula is the probability
that I contains theta |
rLlZpnT02ZU | is at least 1 minus alpha. |
rLlZpnT02ZU | So it better be close to 1. |
rLlZpnT02ZU | You want 1 minus alpha
to be very close to 1, |
rLlZpnT02ZU | because it's really
telling you that whatever |
rLlZpnT02ZU | random variable I'm giving
you, my error bars are actually |
rLlZpnT02ZU | covering the right theta. |
rLlZpnT02ZU | And I want this to be true. |
rLlZpnT02ZU | But I want this--
since I don't know |
rLlZpnT02ZU | what my confidence--
my parameter of theta |
rLlZpnT02ZU | is, I want this to hold
true for all possible values |
rLlZpnT02ZU | of the parameters that nature
may have come up with from. |
rLlZpnT02ZU | So I want this-- so there's
theta that changes here, |
rLlZpnT02ZU | so the distribution
of the interval |
rLlZpnT02ZU | is actually changing
with theta hopefully. |
rLlZpnT02ZU | And theta is changing
with this guy. |
rLlZpnT02ZU | So regardless of the value
of theta that I'm getting, |
rLlZpnT02ZU | I want that the probability
that it contains the theta |
rLlZpnT02ZU | is actually larger
than 1 minus alpha. |
rLlZpnT02ZU | So I'll come back
to it in a second. |
rLlZpnT02ZU | I just want to say
that here, we can |
rLlZpnT02ZU | talk about asymptotic level. |
rLlZpnT02ZU | And that's typically when
you use central limit |
rLlZpnT02ZU | theorem to compute this guy. |
rLlZpnT02ZU | Then you're not guaranteed
that the value is |
rLlZpnT02ZU | at least 1 minus
alpha for every n, |
rLlZpnT02ZU | but it's actually in the limit
larger than 1 minus alpha. |
rLlZpnT02ZU | So maybe for each fixed n
it's going to be not true. |
rLlZpnT02ZU | But for as no goes
to infinity, it's |
rLlZpnT02ZU | actually going to become true. |
rLlZpnT02ZU | If you want this to
hold for every n, |
rLlZpnT02ZU | you actually need to use things
such as Hoeffding's inequality |
rLlZpnT02ZU | that we described at some
point, that hold for every n. |
rLlZpnT02ZU | So as a rule of thumb, if you
use the central limit theorem, |
rLlZpnT02ZU | you're dealing with
a confidence interval |
rLlZpnT02ZU | with asymptotic
level 1 minus alpha. |
rLlZpnT02ZU | And the reason is
because you actually |
rLlZpnT02ZU | want to get the quintiles
of the normal-- the Gaussian |
rLlZpnT02ZU | distribution that comes from
the central limit theorem. |
rLlZpnT02ZU | And if you want to use
Hoeffding's, for example, |
rLlZpnT02ZU | you might actually get away with
a confidence interval that's |
rLlZpnT02ZU | actually true even
non-asymptotically. |
rLlZpnT02ZU | It's just the regular
confidence interval. |
rLlZpnT02ZU | So this is the
formal definition. |
rLlZpnT02ZU | It's a bit of a mouthful. |
rLlZpnT02ZU | But we actually-- the best
way to understand them |
rLlZpnT02ZU | is to build them. |
rLlZpnT02ZU | Now, at some point I said-- |
rLlZpnT02ZU | and I think it was
part of the homework-- |
rLlZpnT02ZU | so here, I really
say the probability |
rLlZpnT02ZU | the true parameter belongs
to the confidence interval |
rLlZpnT02ZU | is actually 1 minus alpha. |
rLlZpnT02ZU | And so that's because here,
this confidence interval |
rLlZpnT02ZU | is still a random variable. |
rLlZpnT02ZU | Now, if I start plugging
in numbers instead |
rLlZpnT02ZU | of the random
variables X1 to Xn, |
rLlZpnT02ZU | I start putting 1,
0, 0, 1, 0, 0, 1, |
rLlZpnT02ZU | like I did for the kiss
example, then in this case, |
rLlZpnT02ZU | the random interval is actually
going to be 0.42, 0.65. |
rLlZpnT02ZU | And this guy, the probability
that theta belongs to it |
rLlZpnT02ZU | is not 1 minus alpha. |
rLlZpnT02ZU | It's either 0 if
it's not in there |
rLlZpnT02ZU | or it's 1 if it's in there. |
rLlZpnT02ZU | So here is the
example that we had. |
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