The Maximum Likelihood Estimation (MLE) is the process of estimating the parameters of a model for sample data by finding the parameters that maximise the likelihood function. Consider a sample of independent and identically distributed random variables $$X = (X_1, X_2, ... , X_n)$$ with the joint density function:

$$f(X_1, X_2, ... , X_n) = f(X_1) .f(X_2) ... f(X_n).$$

The likelihood function is defined as the probability of given observations $$(x_1, x_2,...,x_n)$$ as a function of $$\theta$$

$$L(\theta; x_1, x_2, ... , x_n) = f(x_1, x_2, ... , x_n| \theta) = \prod_{i=1} ^n f(x_i | \theta).$$

The maximum likelihood estimator is then given by:

$$\hat{\theta} = argmax _{\theta} L(\theta; x_1, x_2,..., x_n ).$$

Finding the maximum of a product of terms is tedious in practice, therefore we regard the logarithm of  $$L$$ which translates the product into a summation instead, i.e.,

$$log(L(\theta; x_1, x_2, ... , x_n)) = \sum _{i=1} ^{n} log (f(x_i | \theta)).$$

Since the logarithm is an increasing function, finding the maximum of $$L$$ is equivalent to maximising $$log(L)$$, which is a much simpler problem.