logconjugate

Logconjugate is a term found in some mathematical and statistical writings to describe a duality or transform that emerges when problems are formulated in logarithmic space. It is not a standardized concept, and its precise meaning can vary across sources.

A commonly described idea defines the log-conjugate of a nonnegative function f as the convex conjugate of its logarithm: f_log^*(y) = sup_x { y^T x - log f(x) }. In this view, multiplicative relationships in f translate into additive terms in log f, which can simplify certain optimization or variational calculations.

This construction requires that log f be well defined and that the supremum be finite for the dual variable y. If f is log-convex, then log f is convex, which supports the existence of the conjugate. The specifics—such as domains and regularity—depend on the application.

Applications of logconjugate ideas appear in exponential-family modeling, where log-linear forms are natural, and in variational inference or convex optimization problems that use log-space formulations. It may also arise in the study of multiplicative noise or multiplicative models where the log transformation converts products into sums.

Related notions include convex conjugation (the Legendre-Fenchel transform), log-convexity, and conjugate priors in Bayesian statistics. Because the term is used variably, practitioners typically cite the exact definition from their source when employing logconjugate.