  We introduce a new method for solving maximum likelihood problems through variational calculus, and apply it to the case of recovering an unknown star formation history, $SFR(t)$, from a resulting HR diagram. This approach allows a totally non-parametric solution which has the advantage of requiring no initial assumptions on the $SFR(t)$. As a full maximum likelihood statistical model is used, we take advantage of all the information available in the HR diagram, rather than concentrating on particular features such as turn off points or luminosity functions. We test the method using a series of synthetic HR diagrams produced from known $SFR(t)$, and find it to be quite successful under noise conditions comparable to those present in current observations. At this point we restrict the analysis to situations where the metallicity of the system is known, as is the case with the resolved populations of the dwarf spheroidal companions to the Milky Way or the solar neighbourhood Hipparcos data. We also include tests to quantify the way uncertainties in the assumed metallicity, binary fraction and IMF affect our inferences. 
Deriving star formation histories: Inverting HR diagrams through a   variational calculus maximum likelihood method