In this video I want to provide an introduction to maximum likelihood estimation
by means of an example. So the example
I'm gonna be talking about here is the example
of the UK population, so
the idea is that they're something like seventy-million
individuals in the population and,
of course, we don't have the entirety of the data on them, we only have a
sample from that particular population, and from the sample we have
maybe N individuals. Ok, so
let's say we were interested in evaluating
what the probability is that an individual
was male, given that we only have
a sample from the population. How
might we go about thinking about this? Well the idea is that we could
suppose that there is some sort some of
probability distribution function which
basically determines the probability that
given individual was either going to be male, or female.
Ok, well the idea is that we can actually define
function like this. Our probability density function which we gonna call
F, and we are going to call it f of x(i),
given the probability p, and I should stress that
p is a population quantity on
in general is the thing which we trying to estimate I'm just an issue that we
know you
he to stop so X I
is gains be a I'm either which is gonna take on
the Palio one it an individual is male
I'm it's gonna take on values 0 if the individual with FEMA
I'm FK is actually gonna be given point he
with he is be probability that an individual is male
power exxon times 1mon sp
to one minus excited okay say
this is quite complicated less talks three how actually works
say the idea is that if an individual
in all sample is mile then
book value just act a comment so this is a functional one now
given that we have the probability but we just substitute
in one for exxon a week and a half heats the Powell on
times one minus p to them one minus
XI website just once it's gonna be 0
head and he's our 0 is just one say we just gonna be late
with the so our function
here at tells us if
that individual was male what is the probability that
we would have called observation ans
that's just me maquina frequency an individual is my
similarly really thinking about an individual being female say this is a
function of 08
getting p I'm this would just be given by
heats the house 0 times one minus p
to the one minus serious it can be 11 is P the powwow
well peace 0 is just once a
is just gonna be one times 1mon sp so
our function act a value eights to you
11 is key which is actually the probability that an individual is female
say in by circumstances you can see that
our function at is telling us the probability that we would have got
up observation if indeed
it was the case that's we can think about all observations
as being almost like possible point okay so that's the second thoughts for
one observation how about if we have a
supplemental summations say in a sense of what week why not it's fun as we must
define
a from militants the function and full
be an individual said this to be actual x1
x2 3 to
X and given that we know it busy P
all an individual be my well
that's a hearty calculate in the second songs where
we're considering observations be random sample in other words
they all independent of one another it's just gonna be
individual density functions multiplied together so that just gonna be
the probability all x1
OP the power x1 times one minus heats the power
1161 times he
to power x2 times one minus he to the
one minus XD and then be continue on
all the way up tp the axe n times
11 is P to the one minus accent
so we've done is we've multiplying together the individual
density functions at affects IQ the
and we can actually write this whole thing a little bit more neatly blind
writing
as a products and so we can use product notation
this talk to point him means that we're gonna
calculate the productive all the terms and it's gonna be product from mike was
12 an
all he all X on times
11 sp the one minus exxon
ans this product I'm here what it means is it means we can multiply together
all of the terms which is specified on excites
is gone it might be a problem of the summation notation
up when talking about products ads
when we saw multiplied together each of these individual
density functions were actually can obtain is
in a discrete case we can be talked about what's the probability
that the run a parable X one actually took home
are you affects one's what was the probability that random variable
X which determines sex if an individual
oxy walls that's eight mile first it visually
0 actually pulled out was and we're also talking about what was the probability
around america XT all say was
let's say its about particular value St which we obtain
in all sample and we talking about
joint probability all these things occurring up into
we're talking about what's the probability run relax n
actually to call a particular Ali 0 accent
so it's kinda but the joint probability we would have got
this sample observations from out population
and this joint probability is what we define
as the likelihood so the likelihood discrete case
is the probability that we would have got love these observations
given that there was a problem in CP of an individual
be my in general we don't know the prompt
p and indeed it's probably the thing which one estimate
say how does one do this from
a likelihood perspective well the idea is that what we want to do is we won't
see
maximize the probability that we would have got
this opera observations I have a choice it P so what we're asking to do is begin
to differentiate
its likelihood function with respect see p
unsaid equal to 0 say the idea is that we try to maximize the likelihood
or rather which using the probability to maximize
the likelihood that we would have obtained not set observations
given that the individual XI
has a probability distribution which specified all this about
so that's the idea with like which one to maximize
sensei some sort of probability joint probability of getting
ap salt levels the nation's the problem with the French chasing
this function down here as it sounds is that
because its products that's actually quite a complicated thing to do we have
to
sense to use the chain rule all the product well
ons because we go and things in our product is actually going to be a really
complicated
to differentiate say as it sounds it looks like it's
Minnesota or might be lost but in fact is not
I'm we can talk about potential solutions to this problem
in the next few videos and how we can actually derived
the maximum likelihood estimator 0 he in this case the probability that an
individual is my