Monte Carlo trials

Since our task is to probabilistically track targets, sequential Monte Carlo methods is a natural choice. As a first heuristic, we perform simple trials such that each trial starts with the initial distribution of targets. These targets are propagated through the event observations by querying a Monte Carlo simulator. During each trial, the outcome of simulation may contradict an observation, in which case the trial is simply discarded. After a certain number of successful trials are completed, the final target distribution is obtained. For example, the mean of the number of targets in a shadow at $t = t_f$ is simply the average of the number of targets in that shadow over all successful runs. For simulations in this paper, we require $1000$ successful trials. Note that since the particular Monte Carlo simulation we perform in this paper do not depend on data, its result is probabilistically correct and therefore can serve as baselines for verifying results from other algorithms.