Most Likelihood Estimation
on Math
1. Likelihood
a function of the parameters of a statistical model given data
- Discrete probability distribution
${\displaystyle {\mathcal {L}}(\theta \mid x)=p_{\theta }(x)=P_{\theta }(X=x)}$
- Continuous probability distribution
${\displaystyle {\mathcal {L}}(\theta \mid x)=f_{\theta }(x)}$
2. MLE
a method of estimating the parameters of a statistical model, given observations
The method defines a maximum likelihood estimate:
$\hat\theta_{\text{MLE}} = \arg \max_{\theta} L(\theta; {x_i})$
natural logarithm of the likelihood function, called the log-likelihood:
or the average log-likelihood:
Reference