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,if is discrete and if is continuous.
Given all these elements, the log-likelihood function is the function defined by
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 vectorwhose entries are draws from a normal distribution. The probability density function of a generic draw iswhere 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
The joint probability density of the sample is 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
The log-likelihood function is
The log-likelihood function is typically used to derive the maximum likelihood estimator of the parameter . The estimator is obtained by solvingthat 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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