## 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 $$\\$$

$$$L(\theta | x) = \prod_{i=1}^{n} f(x_i | \theta) \tag{5.1}$$$

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 $$$L(\theta|x) = \prod_{i=1}^{n} {1 \choose x_i} \theta^{x_i} (1-\theta)^{1-x_i} = \theta^{\sum_{i=1}^n x_i} (1-\theta)^{n - \sum_{i=1}^n x_i} \tag{5.2}$$$

According to , 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.

### References

Berger, James O. 1985. Statistical Decision Theory and Bayesian Analysis. 2nd ed. Springer Science & Business Media. https://www.springer.com/gp/book/9780387960982.