The log-likelihood function 
is defined to be the natural
logarithm of the likelihood function 
. More precisely, 
, and so in particular, defining the
likelihood function for conditionally independent
observations 
, given 
, as
 |
shows, when the likelihood is positive, that
 |
For dependent observations, 
is the joint density
or mass function, rather than the product
of marginal densities. When 
, the extended log-likelihood
equals 
.
The log-likelihood function is used throughout various subfields of mathematics, both pure and applied, and has particular importance in fields such as likelihood theory.
