In
statistics
, a
likelihood function
(often
simply the
likelihood
) is a function of
the
parameters
of a
statistical
model
,
defined as follows: the
likelihood of a set of
parameter values given some observed outcomes
is
equal
to the probability
of those observed outcomes given those
parameter
values.
Likelihood functions
play a key role in
statistical
inference
,
especially methods
of estimating a parameter from
a set
of
statistics
.
In
non-technical parlance, "likelihood" is usually
a synonym for "
probability
" but in statistical usage, a
clear
technical
distinction is made. One may ask "If I were to
flip a fair coin 100
times,
what is the probability of it landing
heads-up every time?" or
"Given
that I have flipped a coin 100 times and it has
landed heads-up 100
times,
what is the likelihood that the coin is
fair?" but it would be
improper
to switch "likelihood" and "probability" in the
two sentences.
If
a probability distribution depends on a
parameter, one may on one hand
consider—for
a given value of the parameter—the probability
(density) of the
different outcomes, and on the
other
hand consider—for a given outcome—the
probability (density) this
outcome has occurred for different values of
the
parameter. The first approach
interprets the probability distribution as
a
function of the outcome, given
a fixed
parameter value, while the second
interprets it as a function of
the
parameter, given a fixed outcome. In the
latter case the function is
called the "likelihood function" of the
parameter,
and indicates how likely a
parameter value is in light of the
observed
outcome.