Question

Explain the reason why, in many MLE problems, we need to take the natural log of the likelihood function Ccalled the log likelihood function.)

Answer 1

There are many reasons, some of them are as follows:

1.Likelihood is a big product of many functions; taking log of it simplifies its computation. It also helps us in avoiding the exponential function present in the likelihood ( especially in normal family distributions).

2. Log is a monotonically increasing function hence, maximising the log likelihood is equivalent to maximising the likelihood function.

3. Likelihood is the product of likelihood for each point. By taking its log we change the products into sum.

4. Likelihood being product of a large no. Of small probabilities, would become very small and thus making the computer run out of its floating point precision quickly resulting in an underflow. Hence it is convenient to take log of likelihood because then we have the sum of log probabilities.