stick quest chicken it's bad to the bone
stand quiz check it out it's bad to the
bone hello and welcome to stack quest
stack quest is brought to you by the
friendly folks in the genetics
department at the University of North
Carolina at Chapel Hill
today we're going to be talking about
maximum likelihood let's say we weighed
a bunch of mice the goal of maximum
likelihood is to find the optimal way to
fit a distribution to the data there are
lots of different types of distributions
for different types of data here's a
normal distribution here's what an
exponential distribution looks like and
here's what a gamma distribution looks
like and there are many more the reason
you want to fit a distribution to your
data is it can be easier to work with
and it is also more general it applies
to every experiment of the same type in
this case we think the weights might be
normally distributed
that means we think it came from this
type of distribution normally
distributed means a number of things
first we expect most of the measurements
for example Mouse weights to be close to
the mean or average and we see low and
behold in our data set most of the mice
weigh close to the average we also
expect the measurements to be relatively
symmetrical around the mean although the
measurements are not perfectly
symmetrical around the mean they are not
crazy skewed to one side either this is
pretty good normal distributions come in
all kinds of shapes and sizes they can
be skinny medium or large boned once we
settle on the shape we have to figure
out where to Center the thing is one
location better the
another before we get too technical
let's just pick any old normal
distribution and see how well it fits
the data this distribution says most of
the values you measure should be near my
average the distributions average is the
black dotted line in this case that's
different from the average of the actual
measurements unfortunately most of the
values we measured are far from the
distributions average according to a
normal distribution with a mean value
over here the probability or likelihood
of observing all these weights is low
what if we shifted the normal
distribution over so that its mean was
the same as the average weight according
to a normal distribution with a mean
value here the probability or likelihood
of observing these weights is relatively
high if we kept shifting the normal
distribution over then the probability
or likelihood of observing these
measurements would go down again we can
plot the likelihood of observing the
data over the location of the center of
the distribution we start on the left
side and we calculate the likelihood of
observing the data and then we shift the
distribution to the right and
recalculate we just do this all the way
down the data once we've tried all the
possible locations we could Center the
normal distribution on we want the
location that maximizes the likelihood
of observing the weights we measured
this location for the mean maximizes the
likelihood of observing the weights we
measured thus it is the maximum
likelihood estimate for the mean in this
case we're specifically talking about
the mean of the distribution not the
mean of the data however with a normal
distribution those two things are the
same great now we have figured out the
maximum likelihood estimate for the
me now we have to figure out the maximum
likelihood estimate for the standard
deviation again we can plot the
likelihood of observing the data over
different values for the standard
deviation now we found the standard
deviation that maximizes the likelihood
of observing the weights we measured
this is the normal distribution that has
been fit to the data by using the
maximum likelihood estimations for the
mean and the standard deviation now when
someone says that they have the maximum
likelihood estimates for the mean or the
standard deviation or for something else
you know
that they found the value for the mean
or the standard deviation or for
whatever that maximizes the likelihood
that you observed the things that you
observed terminology alert in everyday
conversation probability and likelihood
mean the same thing however in stats
land likelihood specifically refers to
this situation we've covered here where
you are trying to find the optimal value
for the mean or standard deviation for a
distribution given a bunch of observed
measurements this is how we fit a
distribution to data hooray we've made
it to the end of another exciting stat
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