The analysis of high spectral resolution spectroscopic and spectropolarimetric observations constitutes a very powerful way of inferring the dynamical, thermodynamical, and magnetic properties of distant objects. However, these techniques starve photons, making it difficult to use them for all purposes. A common problem is not being able to detect a signal because it is buried on the noise at the wavelength of some interesting spectral feature. This problem is especially relevant for spectropolarimetric observations, because only a small fraction of the received light is typically polarized. We present in this paper a Bayesian technique for detecting spectropolarimetric signals. The technique is based on applying the nonparametric relevance vector machine to the observations, which allows us to compute the evidence for the presence of the signal and compute the more probable signal. The method is suited for analyzing data from experimental instruments onboard space missions and rockets aiming at detecting spectropolarimetric signals in unexplored regions of the spectrum, such as the Chromospheric Lyman-Alpha Spectro-Polarimeter (CLASP) sounding rocket experiment.