Consider a binary hypothesis testing problem in which observations are modeled as
independent and identically distributed random variables under each hypothesis. Let Y_1, Y_2, \ldots, Y_n denote the observations. Let f_0 denote the
probability density function of each observation Y_i under the null hypothesis H_0 and let f_1 denote the probability density function of each observation Y_i under the alternate hypothesis H_1. In this case there are
two possible error events. Error of type I, also called
false positive, occurs when the null hypothesis is true and it is wrongly rejected. Error of type II, also called
false negative, occurs when the alternate hypothesis is true and null hypothesis is not rejected. The probability of type I error is denoted P (\mathrm{error}\mid H_0) and the probability of type II error is denoted P (\mathrm{error}\mid H_1). In some fields, the type I error is denoted by \alpha_n and the type II error is denoted by \beta_n.
Optimal error exponent for Neyman–Pearson testing In the Neyman–Pearson version of binary hypothesis testing, one is interested in minimizing the probability of type II error P (\text{error}\mid H_1) subject to the constraint that the probability of type I error P (\text{error}\mid H_0) is less than or equal to a pre-specified level \alpha. In this setting, the optimal testing procedure is a
likelihood-ratio test. Furthermore, the optimal test guarantees that the type II error probability decays exponentially in the sample size n according to \lim_{n \to \infty} \frac{- \ln P (\mathrm{error}\mid H_1)}{n} = D(f_0\parallel f_1). The error exponent D(f_0\parallel f_1) is the
Kullback–Leibler divergence between the probability distributions of the observations under the two hypotheses. This exponent is also referred to as the Chernoff–Stein lemma exponent.
Optimal error exponent for average error probability in Bayesian hypothesis testing In the
Bayesian version of binary hypothesis testing one is interested in minimizing the average error probability under both hypothesis, assuming a
prior probability of occurrence on each hypothesis. Let \pi_0 denote the prior probability of hypothesis H_0 . In this case the average error probability is given by P_\text{ave} = \pi_0 P (\text{error}\mid H_0) + (1-\pi_0)P (\text{error}\mid H_1). In this setting again a likelihood ratio test is optimal and the optimal error decays as \lim_{n \to \infty} \frac{- \ln P_\text{ave} }{n} = C(f_0,f_1) where C(f_0,f_1) represents the Chernoff-information between the two distributions defined as C(f_0,f_1) = \max_{\lambda \in [0,1]} \left[-\ln \int (f_0(x))^\lambda (f_1(x))^{(1-\lambda)} \, dx \right]. described by {{Equation|1= H_r(f_0 \parallel f_1) = \max_{0\leq s \leq 1} \frac{\Psi(s) - (1-s)r}{s} |2=1}}where \Psi(s) = \int dx\ f_0(x)^{1-s} f_1(x)^s. ==References==