This is a motivating example that demonstrates the problem with existing methods and how our methods will improve density estimates. (a) and (b) give the valid region to be considered and the true density for the example. Figure (c) gives the 4000 events sampled from the true density. (d) and (e) show two of the current methods used. (f), (g), and (h) show how our methods will produce better estimates. The color scale represents the relative probability of an event occurring in a given pixel. The images are 80 pixels by 80 pixels. (a) Valid region, (b) True density, (c) 4,000 events, (d) Kernel density estimate, (e) TV MPLE, (f) Our modified TV MPLE method, (g) Our weighted H1 MPLE method (h) Our weighted TV MPLE method.