A Maximum Likelihood Method for Fitting Color Magnitude Diagrams

In their paper, Naylor and Jeffries presented a method of fitting clusters to an isochrone in a color- magnitude diagram using Bayesian statistics in such a way as to have numerical backing for a fit and to get useful error bounds. For each star, they multiply the star’s Gaussian distribution together with the density of the isochrone, then integrating over all space to obtain the probability that the star came from that isochrone. Tau squared, a very similar quantity to chi squared, is negative two times the sum of the logarithms of the probabilities. A good isochrone fit minimizes tau squared. Minimizing tau squared is the same as maximizing the likelihood that the data came from the given isochrone, since tau squared is a modification of the log of the likelihood. The likelihood of a fit given the data is proportional to the likelihood of the data given the fit by Bayes’s Theorem, so maximizing one is maximizing the other.

Past this theory, several additional factors enter into actual computation. For example, one standardizes the number of stars per magnitude range so that the initial mass function is not a parameter in the fit. Error bounds were found by a bootstrap method when the cluster was simulated (a set of different inputs according to a Gaussian were inputted and the result examined), and integrating over tau squared space gave error bounds in a real cluster. To account for the interaction between B error and B-V error, the color magnitude diagram must be translated into a magnitude-magnitude diagram to compute the probabilities, yet this only happens for the one step.

They worked this method of fitting both on simulated data sets and an actual cluster, and both agreed with the back of the book answer.

There are still multiple things I do not understand in the paper, but they are more details of implementation and terminology. How is the normalization over magnitude ranges accomplished, and why does that not give unfair wait to higher magnitudes? What is the theta and gradient discussed on the seventh page? Though the paper says the method accounts for binary systems, how does it do that?