logexpE

LogexpE is a term used in probability and statistics to denote the quantity log E[e^{X}], where X is a real-valued random variable and E denotes expectation. In several texts, logexpE is presented as the one-sided instance of the cumulant generating function, which is defined as K(t) = log E[e^{tX}] and evaluated at t = 1.

Definition: For a random variable X with E[e^{X}] finite, logexpE(X) is defined as log E[e^{X}]. If X is vector-valued, a related quantity is log E[e^{⟨θ, X⟩}] for a given vector θ, but the standard logexpE focuses on the scalar case with t = 1.

Properties: logexpE is finite precisely when the exponential moment E[e^{X}] exists. It is always nonnegative when X has a nonnegative mean, and it equals zero if X is almost surely zero. As a function of the distribution of X, logexpE is connected to the cumulant generating function K(t) by logexpE(X) = K(1). The quantity is convex in the distribution of X and plays a role in tail bounds and moment analysis.

Computation: If the distribution is known analytically, logexpE can be computed directly. In empirical settings, with samples X1, …, Xn, log E[e^{X}] ≈ log( (1/n) ∑ e^{Xi} ), often evaluated with numerical stability tricks such as the log-sum-exp technique.

Applications: logexpE appears in large deviations theory, risk assessment, and the analysis of exponential families. It also underpins log-partition function computations and can be used to derive Chernoff bounds and moment-based approximations.

See also: cumulant generating function, moment generating function, log-sum-exp, exponential family, large deviations.