---- > [!definition] Definition. ([[likelihood]]) > A **likelihood function** $L$ is a [[joint probability distribution|joint probability]] (or [[Density|probability density]]) of [[training data|observed data]], viewed as a function of the parameters $\b \theta$ of a statistical model. > [!basicexample] > Suppose we draw $n$ [[independent random variables|independent]] [[random variable|random]] [[real numbers]] in the range $[0,\infty)$ from the (properly normalized) [[exponential random variable|exponential probability density]] $P(x)=\mu e ^{-\mu x}$, i.e., we have a [[random variable|random vector]] $\b X$ for which the $X_{i} \sim \mu e^{-\mu x}$, $i \in [n]$ are [[independent random variables|independent]]. \ The [[probability distribution|probability]] to draw values $X_{1}=x_{1},\dots,X_{n}=x_{n}$ when $\mu$ is viewed as a parameter is thus $\begin{align} p(x_{1},\dots,x_{n};\mu)= & p(x_{1} ; \mu) \cdots p(x_{n} ; \mu) \\ = & \mu e ^{-\mu x_{1}} \cdots \mu e ^{-\mu x_{n}} \\ = & \mu^{n}e^{-\mu \sum_{i=1}^{n} x_{n}}. \end{align}$ Hence $L(\mu)=\mu^{n}e^{-\mu \sum_{i=1}^{n}x_{n}}$. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```