Hypothesis testing and error exponents

Let H-zero assert law P and H-one assert law Q on a finite alphabet, with every probability strictly positive. Observe n iid samples under whichever hypothesis is true. We define alpha as P of rejecting H-zero and beta as Q of accepting H-zero. A randomized test is allowed at a likelihood-ratio threshold to meet an error constraint exactly. Different references sometimes use alpha for acceptance probability, so their symbols must be translated through the underlying event.

The Neyman–Pearson principle compares likelihood ratios: among rules with a fixed allowed alpha, reject P on observations with the largest Q-to-P ratio. These are the outcomes most favorable to Q relative to P. Randomization at a boundary can fill the remaining error budget. This finite-sample rule is separate from an asymptotic theorem; its threshold depends on the constraint and the actual product likelihood distribution.

Now assign fixed positive prior probabilities to both hypotheses and minimize the weighted average of alpha and beta. The optimal decision compares the prior-weighted likelihoods. Its error exponent is Chernoff information, the negative logarithm of a minimized mixed power sum. With the displayed base-two convention, error has scale two to the minus n times the exponent, up to factors whose logarithms are smaller than n. The priors are fixed as n grows; exponentially changing priors define another regime.

Stein's asymmetric constraint allows the null rejection error to remain below a fixed positive tolerance while making the other error exponentially small. Bayesian optimization instead balances the two weighted errors, so its exponent generally differs from either directed divergence. Using natural logarithms multiplies both exponents by ln two and writes the exponential scale with e. Mixing a bits-valued divergence with exp of minus n times that number introduces a systematic conversion error.

For symmetric binary models, the likelihood rule can reduce to majority voting, whose finite error is an exact binomial tail. A Chernoff or Bhattacharyya bound can upper-bound that error at finite n, but need not equal it. An exponent statement discards subexponential factors and therefore cannot specify a decimal error probability at a chosen short length. When that probability matters, calculate the tail or use a validated finite-sample bound.

Support and sampling assumptions also matter. If one law assigns zero probability where the other is positive, a single observation can sometimes settle a hypothesis and divergence can become infinite. With dependent observations, likelihoods need not factor and a one-sample exponent need not simply multiply by n. The strictly positive finite iid setting keeps these issues controlled, while making clear which changes require a new theorem rather than a symbolic substitution.