dLRdt

dLRdt is a notation encountered in statistics, signal processing, and related fields to denote the time derivative of a likelihood‑ratio quantity, most often the derivative of the log‑likelihood ratio. In many treatments, the log-likelihood ratio is defined as L(t) = log p(y(0..t)|H1) − log p(y(0..t)|H0), and dL/dt (sometimes written dLRdt) represents its instantaneous rate of change. Because notation varies, dLRdt may refer to the derivative of a different likelihood‑ratio quantity in some texts, so its precise meaning should be defined in the given context.

In discrete‑time models, practitioners typically work with the incremental log‑likelihood ratio ΔL_t = log p(y_t|H1) − log p(y_t|H0). In continuous‑time models, L(t) may evolve according to a stochastic differential equation, and dL/dt (or dLRdt) denotes the drift or instantaneous rate of change of the log‑likelihood ratio. This quantity is central to formulations of sequential detection and continuous‑time state estimation, where decisions or actions are guided by the evolution of L(t) or its rate of change.

Applications and interpretation often involve sequential probability ratio tests, adaptive detectors in communications, and neural decoding schemes. When using the dLRdt notation, it is important to specify the underlying model for L, the time variable, and any stochastic terms, to avoid ambiguity.

See also: log‑likelihood ratio, likelihood ratio, sequential probability ratio test, stochastic differential equation.