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The laser interferometer space antenna will be the first gravitational wave observatory in space. It is scheduled to fly in the early 2030's. LISA design predicts sensitivity levels that potentially enable the detection a stochastic gravitational wave background signal. This stochastic type of signal is a superposition of signatures from sources that cannot be resolved individually and which are of various types, each one contributing with a different spectral shape. In this work we present a fast methodology to assess the detectability of a stationary, Gaussian, and isotropic stochastic signal in a set of frequency bins, combining information from the available data channels. We derive an analytic expression of the Bayes factor between the instrumental noise-only and the signal plus instrumental noise models, that allows us to compute the detectability bounds of a given signal, as a function of frequency and prior knowledge on the instrumental noise spectrum. [ABSTRACT FROM AUTHOR]