Sanov's Theorem

Sanov's Theorem provides a way to estimate how unlikely it is for a set of observed data, sampled from a known distribution, to significantly differ from what we expect. It quantifies the probability that the empirical distribution (what we observe) falls outside a specific set of outcomes, showing that such deviations decrease exponentially fast as the sample size grows. In essence, it formalizes how rare large deviations are, allowing us to evaluate the likelihood of observing unusual patterns in data, which is useful for statistical testing and assessing the reliability of observed phenomena.