We use recent Hubble Space Telescope colour-magnitude diagrams of the resolved stellar populations of a sample of local dSph galaxies (Carina, Leo I, Leo II and Ursa Minor) to infer the star formation histories of these systems, SFR(t). Applying a new variational calculus maximum likelihood method, which includes a full Bayesian analysis and allows a non-parametric estimate of the function one is solving for, we infer the star formation histories of the systems studied. This method has the advantage of yielding an objective answer, as one need not assume a priori the form of the function one is trying to recover. The results are checked independently using Saha's W statistic. The total luminosities of the systems are used to normalize the results into physical units and derive SN type II rates. We derive the luminosity-weighted mean star formation history of this sample of galaxies.