Maximum likelihood estimation, or MLE for short, is a tool for determining the best among a set of hypotheses, often called a statistical model.  Let \(M = \{h_{\theta} :\: \theta \in \Theta \}\) be the model, labeled by the parameter \(\theta\), let \(S\) be the sample space, and \(P_{\theta}\) the distribution associated with \(h_{\theta}\).  Then define the maximum likelihood estimator \(\hat{\theta}\) as a function over the sample space:  \[ \hat{\theta}(s) = \left\{ \theta :\: \forall h_{\theta'} \bigl(P_{\theta'}(s)ds \leq P_{\theta}(s)ds \bigr) \right\}. \]  So the estimator is a set, typically a singleton, of values of \(\theta\) for which the likelihood of \(h_{\theta}\) on the data \(s\) is maximal. The associated best hypothesis we denote with \(h_{\hat{\theta}}\). This can again be illustrated for the tea tasting lady.