Maximum Likelihood Estimation (MLE)

In statistics, maximul likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data.

MLE Definition

$\hat{\theta }$ is the MLE if $\hat{\theta }$ maximizes $L(\theta ;y_1,y_2,...y_n)$ where $L$ is the Likelihood Function.

General template for deriving MLE

• We always use the log likelihood since it makes it much easier to derive. Think of $\ln (abc)=\ln a+\ln b+\ln c$
• You then take the derivative, and set it to 0, since you want to maximize it.

We define

• $L(\theta$) as the likelihood
• $l(\theta )$ as the log likelihood