Non-persistent collection of marked point process data in space and time occurs in many remote sensing applications, resulting in known intervals of time where events are unobserved. Self-exciting marked point processes (i.e., Hawkes processes) have been shown to be effective in these situations to understand the underlying process. Recent estimation procedures assume a full history of the data and are not capable of accounting for regions of space and/or time where data are missing. A Bayesian estimation procedure for self-exciting marked point processes is developed to naturally incorporate the missing data mechanism probabilistically. Accounting for the missing data improves estimation and prediction of the process. This is demonstrated through simulation and an application to real conflict monitoring data from the Global Terrorism Database where records over significant time periods are missing.