A novel method for the inversion of band-limited seismic traces to full bandwidth reflectivity traces, is based on a probabilistic spiky model of the reflectivity trace, in which position indicators and amplitudes of the spikes occur as random variables, and relies on relative entropy inference from information theory. First, an a priori model for general reflectivity traces in the prospect is derived from nearby wells. Second, the a priori distribution is updated into an a posteriori distribution for the specific trace being studied by the addition of the Fourier data of the seismic trace within a passband. Uncertainty about the Fourier coefficients can be accounted for by specification of a noise variance, which implicitly is infinite outside the passband. The update with relative entropy inference is justified because of its relationship with Bayesian inference. Application of maximum a posteriori (MAP) estimation to the a posteriori distribution results in the most likely spiky reflectivity trace of full bandwidth. A numerical algorithm for obtaining the MAP estimates of spike positions and spike amplitudes is derived from the concept of continuation and is described in detail. The algorithm avoids searching among all possible patterns of spike positions.