5.3 Likelihood function
Definition 5.3 (Likelihood function) For the observed data \(x\), the function \(L(\theta|x)=f(x|\theta)\), considered as a function of \(\theta\), is called likelihood function. Mathematically \(\\\)
\[\begin{equation} L(\theta | x) = \prod_{i=1}^{n} f(x_i | \theta) \tag{5.1} \end{equation}\]
Example 5.4 Let \(X_1,\ldots,X_n\) be a sequence of (conditionally) random variables iid \(\mathcal{Ber}(\theta) \equiv \mathcal{B}(1,\theta)\). The likelihood function is given by \[\begin{equation} L(\theta|x) = \prod_{i=1}^{n} {1 \choose x_i} \theta^{x_i} (1-\theta)^{1-x_i} = \theta^{\sum_{i=1}^n x_i} (1-\theta)^{n - \sum_{i=1}^n x_i} \tag{5.2} \end{equation}\]
According to (Berger 1985, 27), the intuition behind the name ‘likelihood function’ is that the more likely the \(\theta\) is, the larger \(f(x|\theta)\).
Exercise 5.4 Whach the video Probability is not Likelihood. Find out why!!! from the channel StatQuest with Josh Starmer.