video_id stringclasses 7
values | text stringlengths 2 29.3k |
|---|---|
rLlZpnT02ZU | if you do it for each
simple event, it's little x. |
rLlZpnT02ZU | It's actually simple. |
rLlZpnT02ZU | Now, if we have continuous
random variables-- so |
rLlZpnT02ZU | by the way, I didn't mention,
but discrete means Bernoulli. |
rLlZpnT02ZU | Binomial, but not only those
that have finite support, |
rLlZpnT02ZU | like Bernoulli has
support of size 2, |
rLlZpnT02ZU | binomial NP has
support of size n-- |
rLlZpnT02ZU | there's n possible values it
can take-- but also Poisson. |
rLlZpnT02ZU | Poisson distribution can
take an infinite number |
rLlZpnT02ZU | of values, all the
positive integers, |
rLlZpnT02ZU | non-negative integers. |
rLlZpnT02ZU | And so now we have also
the continuous ones, |
rLlZpnT02ZU | such as Gaussian, exponential. |
rLlZpnT02ZU | And what characterizes
those guys is that they |
rLlZpnT02ZU | have a probability density. |
rLlZpnT02ZU | So the density,
remember the way I |
rLlZpnT02ZU | use my density is
when I want to compute |
rLlZpnT02ZU | the probability of
belonging to some event A. |
rLlZpnT02ZU | The probability of X falling to
some subset of the real line A |
rLlZpnT02ZU | is simply the integral of
the density on this set. |
rLlZpnT02ZU | That's the famous area
under the curve thing. |
rLlZpnT02ZU | So since for each possible
value, the probability at X-- |
rLlZpnT02ZU | so I hope you
remember that stuff. |
rLlZpnT02ZU | That's just probably
something that you |
rLlZpnT02ZU | must remember from probability. |
rLlZpnT02ZU | But essentially, we know that
the probability that X is equal |
rLlZpnT02ZU | to little x is 0 for a
continuous random variable, |
rLlZpnT02ZU | for all possible X's. |
rLlZpnT02ZU | There's just none of them
that actually gets weight. |
rLlZpnT02ZU | So what we have to do is to
describe the fact that it's |
rLlZpnT02ZU | in some little region. |
rLlZpnT02ZU | So the probability that it's in
some interval, say, a, b, this |
rLlZpnT02ZU | is the integral between A
and B of f theta of X, dx. |
rLlZpnT02ZU | So I have this density,
such as the Gaussian one. |
rLlZpnT02ZU | And the probability that I
belong to the interval a, |
rLlZpnT02ZU | b is just the area under
the curve between A and B. |
rLlZpnT02ZU | If you don't remember that,
please take immediate remedy. |
rLlZpnT02ZU | So this function f, just
like P, is non-negative. |
rLlZpnT02ZU | And rather than summing
to 1, it integrates to 1 |
rLlZpnT02ZU | when I integrate it over
the entire sample space E. |
rLlZpnT02ZU | And now the total
variation, well, it |
rLlZpnT02ZU | takes basically the same form. |
rLlZpnT02ZU | I said that you
essentially replace sums |
rLlZpnT02ZU | by integrals when you're
dealing with densities. |
rLlZpnT02ZU | And here, it's just
saying, rather than having |
rLlZpnT02ZU | 1/2 of the sum of
the absolute values, |
rLlZpnT02ZU | you have 1/2 of the integral
of the absolute value |
rLlZpnT02ZU | of the difference. |
rLlZpnT02ZU | Again, if I give
you two densities |
rLlZpnT02ZU | and if you're not too bad at
calculus, which you will often |
rLlZpnT02ZU | be, because there's lots of them
you can actually not compute. |
rLlZpnT02ZU | But if I gave you, for example,
two Gaussian densities, |
rLlZpnT02ZU | exponential minus x squared,
blah, blah, blah, and I say, |
rLlZpnT02ZU | just compute the total
variation distance, |
rLlZpnT02ZU | you could actually
write it as an integral. |
rLlZpnT02ZU | Now, whether you can
actually reduce this integral |
rLlZpnT02ZU | to some particular
number is another story. |
rLlZpnT02ZU | But you could technically do it. |
rLlZpnT02ZU | So now, you have actually
a handle on this thing |
rLlZpnT02ZU | and you could technically
ask Mathematica, |
rLlZpnT02ZU | whereas asking
Mathematica to take |
rLlZpnT02ZU | the max over all possible
events is going to be difficult. |
rLlZpnT02ZU | All right. |
rLlZpnT02ZU | So the total variation
has some properties. |
rLlZpnT02ZU | So let's keep on the
board the definition that |
rLlZpnT02ZU | involves, say, the densities. |
rLlZpnT02ZU | So think Gaussian in your mind. |
rLlZpnT02ZU | And you have two Gaussians,
one with mean theta |
rLlZpnT02ZU | and one with mean theta prime. |
rLlZpnT02ZU | And I'm looking at the total
variation between those two |
rLlZpnT02ZU | guys. |
rLlZpnT02ZU | So if I look at P theta minus-- |
rLlZpnT02ZU | sorry. |
rLlZpnT02ZU | TV between P theta and
P theta prime, this |
rLlZpnT02ZU | is equal to 1/2 of the integral
between f theta, f theta prime. |
rLlZpnT02ZU | And when I don't write it-- |
rLlZpnT02ZU | so I don't write the
X, dx but it's there. |
rLlZpnT02ZU | And then I integrate over E. |
rLlZpnT02ZU | So what is this
thing doing for me? |
rLlZpnT02ZU | It's just saying,
well, if I have-- so |
rLlZpnT02ZU | think of two Gaussians. |
rLlZpnT02ZU | For example, I have one that's
here and one that's here. |
rLlZpnT02ZU | So this is let's say f
theta, f theta prime. |
rLlZpnT02ZU | This guy is doing what? |
rLlZpnT02ZU | It's computing the absolute
value of the difference |
rLlZpnT02ZU | between f and f theta prime. |
rLlZpnT02ZU | You can check for yourself
that graphically, this I |
rLlZpnT02ZU | can represent as an area
not under the curve, |
rLlZpnT02ZU | but between the curves. |
rLlZpnT02ZU | So this is this guy. |
rLlZpnT02ZU | Now, this guy is really the
integral of the absolute value. |
rLlZpnT02ZU | So this thing here,
this area, this |
rLlZpnT02ZU | is 2 times the total variation. |
rLlZpnT02ZU | The scaling 1/2
really doesn't matter. |
rLlZpnT02ZU | It's just if I want to have
an actual correspondence |
rLlZpnT02ZU | between the maximum and the
other guy, I have to do this. |
rLlZpnT02ZU | So this is what it looks like. |
rLlZpnT02ZU | So we have this definition. |
rLlZpnT02ZU | And so we have a couple of
properties that come into this. |
rLlZpnT02ZU | The first one is
that it's symmetric. |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.