Neyman-Pearson Lemma

Weighing the probabilities of false alarm (a.k.a. type I error) and misdetection (a.k.a. type II error), 2 young mathematicians (Jerzy Neyman and Egon Pearson) proposed a revolutionary method of binary hypothesis testing in 1928, which minimizes the probability of misdetection for a given probability of false alarm, i.e., Neyman-Pearson lemma.

For clear description, a binary hypotheses can be denoted below, and the corresponding probabilities are represented by $p(\mathbf{x}; \mathcal{H}_0)$ and $p(\mathbf{x}; \mathcal{H}_1)$ respectively.

• Null hypothesis $\mathcal{H}_{0}$
• Alternative hypothesis $\mathcal{H}_1$

For a given probability of false alarm $\alpha$, the optimal decision minimizing the probability of misdetection can be made based on likelihood ratio $L(\mathbf{x}) \triangleq \dfrac{p(\mathbf{x}; \mathcal{H}_1)}{p(\mathbf{x}; \mathcal{H}_0)}$, i.e.,

• If $L(\mathbf{x}) > \gamma$, $\mathcal{H}_1$ is true;
• If $L(\mathbf{x}) \le \gamma$, $\mathcal{H}_0$ is true.

The threshold of likelihood ratio $\gamma$ is determined based on