  Detecting stochastic gravitational wave backgrounds (SGWBs) with The Laser
Interferometer Space Antenna (LISA) is among the mission science objectives.
Disentangling SGWBs of astrophysical and cosmological origin is a challenging
task, further complicated by the noise level uncertainties. In this study, we
introduce a Bayesian methodology to infer upon SGWBs, taking inspiration from
Gaussian stochastic processes. We investigate the suitability of the approach
for signal of unknown spectral shape. We do by discretely exploring the model
hyperparameters, a first step towards a more efficient transdimensional
exploration. We apply the proposed method to a representative astrophysical
scenario: the inference on the astrophysical foreground of Extreme Mass Ratio
Inspirals, recently estimated in~\cite{Pozzoli2023}. We find the algorithm to
be capable of recovering the injected signal even with large priors, while
simultaneously providing estimate of the noise level.
