# of observing an [[sigma-algebra|event]] The **surprise** upon observing an event $A$ is its **negative log-likelihood**: $ \begin{align*} \mathrm{surprise}(A) &:= - \log_{2} \Pr(A) \text{ bits} \\ &= -\ln \Pr(A) \text{ nats} \end{align*} $ ie gaining $1$ bit of information from an event means we have cut the sample space in half ![[negative log likelihood graph.png|200]] # of observing the value of a [[random variable]] i.e. if $x \sim p$ then we say $- \log_{2} p(x)$ measures the surprise expected surprise is called the [[entropy]]