ML(Maximum Likelihood) Estimator
Since the ML Estimator is a non-Bayesian estimator, it considers the vector $X$ as an unknown fixed value. Thus, it should be noted that $X$ is not arandom vector.
Since the measurement vector $z$ varies depending on the value of $x$, the probability density function of $Z$ is a function of the unkown vector $x$. This is expressed as $p_{Z}(z(x))$.
Additionally, the joint probability density function of the measurement vector union is written as $p_{Z_{k}}(z_{k}(x))$.
An ML estimator is defined as an estimate that maximizes the joint probability density function of $Z_{k}$. That is, $\hat{x}^{ML} = \text{argmax} p_{Z_{k}}(z_{k}(x))$. Since it is similar to the MAP Estimator, to maintain consistency in notation, it is common to express $p_{Z_{k}}(z_{k}(x))$ as a conditional probability density function as: