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rLlZpnT02ZU
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theta prime of A, because if--
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rLlZpnT02ZU
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let's say I just found the
bound on the total variation
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rLlZpnT02ZU
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distance, which is 0.01.
|
rLlZpnT02ZU
|
All right.
|
rLlZpnT02ZU
|
So that means that this
is going to be larger
|
rLlZpnT02ZU
|
than the max over A of P theta
minus P theta prime of A,
|
rLlZpnT02ZU
|
which means that for any A--
|
rLlZpnT02ZU
|
actually, let me write P
theta hat and P theta star,
|
rLlZpnT02ZU
|
like we said, theta
hat and theta star.
|
rLlZpnT02ZU
|
And so if I have a bound,
say, on the total variation,
|
rLlZpnT02ZU
|
which is 0.01, that
means that P theta hat--
|
rLlZpnT02ZU
|
every time I compute a
probability on P theta hat,
|
rLlZpnT02ZU
|
it's basically in the
interval P theta star of A,
|
rLlZpnT02ZU
|
the one that I really wanted
to compute, plus or minus 0.01.
|
rLlZpnT02ZU
|
This has nothing to do
with confidence interval.
|
rLlZpnT02ZU
|
This is just
telling me how far I
|
rLlZpnT02ZU
|
am from the value of
actually trying to compute.
|
rLlZpnT02ZU
|
And that's true for
all A. And that's key.
|
rLlZpnT02ZU
|
That's where this
max comes into play.
|
rLlZpnT02ZU
|
It just says, I want
this bound to hold
|
rLlZpnT02ZU
|
for all possible A's at once.
|
rLlZpnT02ZU
|
So this is actually a
very well-known distance
|
rLlZpnT02ZU
|
between probability measures.
|
rLlZpnT02ZU
|
It's the total
variation distance.
|
rLlZpnT02ZU
|
It's extremely central to
probabilistic analysis.
|
rLlZpnT02ZU
|
And it essentially tells
you that every time--
|
rLlZpnT02ZU
|
if two probability
distributions are close,
|
rLlZpnT02ZU
|
then it means that every
time I compute a probability
|
rLlZpnT02ZU
|
under P theta but
I really actually
|
rLlZpnT02ZU
|
have data from P
theta prime, then
|
rLlZpnT02ZU
|
the error is no larger
than the total variation.
|
rLlZpnT02ZU
|
OK.
|
rLlZpnT02ZU
|
So this is maybe not
the most convenient way
|
rLlZpnT02ZU
|
of finding a distance.
|
rLlZpnT02ZU
|
I mean, how are you going--
|
rLlZpnT02ZU
|
in reality, how are you
to compute this maximum
|
rLlZpnT02ZU
|
over all possible events?
|
rLlZpnT02ZU
|
I mean, it's just crazy, right?
|
rLlZpnT02ZU
|
There's an infinite
number of them.
|
rLlZpnT02ZU
|
It's much larger than the number
of intervals, for example,
|
rLlZpnT02ZU
|
so it's a bit annoying.
|
rLlZpnT02ZU
|
And so there's actually
a way to compress it
|
rLlZpnT02ZU
|
by just looking at the basically
function distance or vector
|
rLlZpnT02ZU
|
distance between probability
mass functions or probability
|
rLlZpnT02ZU
|
density functions.
|
rLlZpnT02ZU
|
So I'm going to start
with the discrete version
|
rLlZpnT02ZU
|
of the total variation.
|
rLlZpnT02ZU
|
So throughout this
chapter, I will
|
rLlZpnT02ZU
|
make the difference between
discrete random variables
|
rLlZpnT02ZU
|
and continuous random variables.
|
rLlZpnT02ZU
|
It really doesn't matter.
|
rLlZpnT02ZU
|
All it means is that when
I talk about discrete,
|
rLlZpnT02ZU
|
I will talk about
probability mass functions.
|
rLlZpnT02ZU
|
And when I talk
about continuous,
|
rLlZpnT02ZU
|
I will talk about probability
density functions.
|
rLlZpnT02ZU
|
When I talk about
probability mass functions,
|
rLlZpnT02ZU
|
I talk about sums.
|
rLlZpnT02ZU
|
When I talk about probability
density functions,
|
rLlZpnT02ZU
|
I talk about integrals.
|
rLlZpnT02ZU
|
But they're all the
same thing, really.
|
rLlZpnT02ZU
|
So let's start with the
probability mass function.
|
rLlZpnT02ZU
|
Everybody remembers what
the probability mass
|
rLlZpnT02ZU
|
function of a discrete
random variable is.
|
rLlZpnT02ZU
|
This is the function that tells
me for each possible value
|
rLlZpnT02ZU
|
that it can take,
the probability
|
rLlZpnT02ZU
|
that it takes this value.
|
rLlZpnT02ZU
|
So the Probability
Mass Function, PMF,
|
rLlZpnT02ZU
|
is just the function for
all x in the sample space
|
rLlZpnT02ZU
|
tells me the probability
that my random variable is
|
rLlZpnT02ZU
|
equal to this little value.
|
rLlZpnT02ZU
|
And I will denote it
by P sub theta of X.
|
rLlZpnT02ZU
|
So what I want is, of
course, that the sum
|
rLlZpnT02ZU
|
of the probabilities is 1.
|
rLlZpnT02ZU
|
And I want them to
be non-negative.
|
rLlZpnT02ZU
|
Actually, typically we will
assume that they are positive.
|
rLlZpnT02ZU
|
Otherwise, we can just remove
this x from the sample space.
|
rLlZpnT02ZU
|
And so then I have the total
variation distance, I mean,
|
rLlZpnT02ZU
|
it's supposed to be the
maximum overall sets of--
|
rLlZpnT02ZU
|
of subsets of E, such
that the probability
|
rLlZpnT02ZU
|
of A minus probability
of theta prime of A--
|
rLlZpnT02ZU
|
it's complicated,
but really there's
|
rLlZpnT02ZU
|
this beautiful
formula that tells me
|
rLlZpnT02ZU
|
that if I look at the total
variation between P theta
|
rLlZpnT02ZU
|
and P theta prime, it's
actually equal to just 1/2
|
rLlZpnT02ZU
|
of the sum for all X in E of the
absolute difference between P
|
rLlZpnT02ZU
|
theta X and P theta prime of X.
|
rLlZpnT02ZU
|
So that's something
you can compute.
|
rLlZpnT02ZU
|
If I give you two
probability mass functions,
|
rLlZpnT02ZU
|
you can compute
this immediately.
|
rLlZpnT02ZU
|
But if I give you
just the densities
|
rLlZpnT02ZU
|
and the original distribution,
the original definition
|
rLlZpnT02ZU
|
where you have to max
over all possible events,
|
rLlZpnT02ZU
|
it's not clear
you're going to be
|
rLlZpnT02ZU
|
able to do that very quickly.
|
rLlZpnT02ZU
|
So this is really the
one you can work with.
|
rLlZpnT02ZU
|
But the other one is
really telling you
|
rLlZpnT02ZU
|
what it is doing for you.
|
rLlZpnT02ZU
|
It's controlling the
difference of probabilities
|
rLlZpnT02ZU
|
you can compute on any event.
|
rLlZpnT02ZU
|
But here, it's just
telling you, well,
|
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