Correcting Parameters of Events Based on the Entropy of Microlensing Ensemble

Abstract

We entertain the idea that robust theoretical expectations can become a tool in removing hidden observational or data-reduction biases. We illustrate this approach for a specific problem associated with gravitational microlensing. Using the fact that a group is more than just a collection of individuals, we derive formulae for correcting the distribution of the dimensionless impact parameters of events, u_min. We refer to the case when undetected biases in the u_min distribution can be alleviated by multiplication of impact parameters of all events by a common constant factor. We show that in this case the general maximum likelihood problem of solving an infinite number of equations reduces to two constraints, and we find an analytic solution. Under the above assumptions, this solution represents a state in which the ``entropy'' of a microlensing ensemble is at its maximum, that is, the distribution of u_min resembles a specific, theoretically expected, box-like distribution to the highest possible extent. We also show that this technique does not allow one to correct the parameters of individual events on the event by event basis independently from each other.