The log-likelihood is, as the term suggests, the natural logarithm of the likelihood.
In turn, given a sample and a parametric family of distributions (i.e., a set of distributions indexed by a parameter) that could have generated the sample, the likelihood is a function that associates to each parameter the probability (or probability density) of observing the given sample.
The following elements are needed to rigorously define the log-likelihood function:
we observe a sample
,
which is regarded as the realization of a random
vector
,
whose distribution is unknown;
the distribution of
belongs to a parametric family: there is a set
of real vectors (called the parameter
space) whose elements (called parameters) are put into correspondence with
the distributions that could have generated
;
in particular:
if
is an continuous
random vector, its
joint probability
density function belongs to a set of joint probability density functions
indexed by the parameter
;
if
is a discrete random
vector, its joint
probability mass function belongs to a set of joint probability mass
functions
indexed
by the parameter
;
when the joint probability mass (or density) function is considered as a
function of
for fixed
(i.e., for the sample
we have observed), it is called likelihood (or likelihood function) and it is
denoted by
.
So,![[eq5]](/images/log-likelihood__16.png)
if
is discrete and
if
is continuous.
Given all these elements, the log-likelihood function is the function
defined
by![[eq8]](/images/log-likelihood__21.png)
The typical example is the log-likelihood function of a sample that is made up of independent and identically distributed draws from a normal distribution.
In this case, the sample
is a
vector![[eq9]](/images/log-likelihood__23.png)
whose
entries
are draws from a normal distribution. The probability density function of a
generic draw
is![[eq11]](/images/log-likelihood__26.png)
where
and
are the parameters (mean and variance) of the normal distribution.
With the notation used in the previous section, the parameter vector
isThe
parametric family being considered is the set of all normal distributions
(that can be obtained by varying the parameters
and
).
In order to stress the fact that the probability density depends on the two
parameters, we
write![[eq13]](/images/log-likelihood__32.png)
The joint probability density of the sample
is
![[eq14]](/images/log-likelihood__34.png)
because
the joint density of a set of independent variables is equal to the product of
their marginal
densities (see the lecture on
Independent
random variables).
The likelihood function
is![[eq15]](/images/log-likelihood__35.png)
The log-likelihood function is
![[eq16]](/images/log-likelihood__36.png)
The log-likelihood function is typically used to derive the
maximum likelihood
estimator of the parameter
.
The estimator
is obtained by
solving![[eq17]](/images/log-likelihood__39.png)
that
is, by finding the parameter
that maximizes the log-likelihood of the observed sample
.
This is the same as maximizing the likelihood function
because the natural logarithm is a strictly increasing function.
One may wonder why the log of the likelihood function is taken. There are several good reasons. To understand them, suppose that the sample is made up of independent observations (as in the example above). Then, the logarithm transforms a product of densities into a sum. This is very convenient because:
the asymptotic properties of sums are easier to analyze (one can apply Laws of Large Numbers and Central Limit Theorems to these sums; see the proofs of consistency and asymptotic normality of the maximum likelihood estimator);
products are not numerically stable: they tend to converge quickly to zero or to infinity, depending on whether the densities of the single observations are on average less than or greater than 1; sums are instead more stable from a numerical standpoint; this is important because the maximum likelihood problem is often solved numerically on computers where limited machine precision does not allow to distinguish a very small number from zero and a very large number from infinity.
More example of how to derive log-likelihood functions can be found in the lectures on:
maximum likelihood (ML) estimation of the parameter of the Poisson distribution
ML estimation of the parameter of the exponential distribution
ML estimation of the parameters of a normal linear regression model
The log-likelihood and its properties are discussed in a more detailed manner in the lecture on maximum likelihood estimation.
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