Likelihood ratio

For two parameter values ${\theta }_1$ and ${\theta }_0$, the likelihood ratio is $f(x\mid {\theta }_1)/f(x\mid {\theta }_0)$.

A log-likelihood ratio compares two likelihoods on an additive scale. For independent observations its increments add. Under distributions $P_0,P_1$ with suitable integrability and common support, $E_0[\log (d{P}_1/d{P}_0)]=-D(P_0\Vert P_1)\le 0$ and $E_1[\log (d{P}_1/d{P}_0)]=D(P_1\Vert P_0)\ge 0$, where $D$ is Kullback-Leibler divergence. This sign change motivates likelihood-based sequential monitoring.