log-likelihood

Log-likelihood

The log-likelihood is the logarithm of the likelihood function:

$$\ell (\theta ;x):=\log L(\theta ;x)=\log \prod _{i=1}^{n}p(x_i\mid \theta )=\sum _{i=1}^n\log p(x_i\mid \theta ),x=(x_1,\dots ,x_n)$$

It is often easier to work with the log-likelihood than the likelihood itself, because it is more numerically stable (products of small numbers can underflow to zero, while sums of logs are more stable) and because it turns products into sums, which are easier to differentiate and optimize. Since the log is a monotone function, maximizing the log-likelihood is equivalent to maximizing the likelihood.