[통계] Estimation - MLE (Maximum Likelihood Estimation)

Maximum Likelihood Estimation

Since log is a monotonically increasing function, the value $\hat{\theta}$ maximizes the function $\ln{L(\theta ; y)}$. This means that if the value on the x-axis increases, the value on the y-axis also increases (see figure below). This is important because it ensures that the maximum value of the log of the probability occurs at the same point as the original probability function. Therefore we can work with the simpler log-likelihood instead of the original likelihood.

Monotonic transformation is a way of transforming a set of numbers into another set that preserves the order of the original set, it is a function mapping real numbers into real numbers, which satisfies the property, that if $x>y$, then $f(x)>f(y)$, simply it is a strictly increasing function.