In the effort to characterize the masses, radii, and atmospheres of
potentially habitable exoplanets, there is an urgent need to find examples of
such planets transiting nearby M dwarfs. The MEarth Project is an ongoing
effort to do so, as a ground-based photometric survey designed to detect
exoplanets as small as 2 Earth radii transiting mid-to-late M dwarfs within 33
pc of the Sun. Unfortunately, identifying transits of such planets in
photometric monitoring is complicated both by the intrinsic stellar variability
that is common among these stars and by the nocturnal cadence, atmospheric
variations, and instrumental systematics that often plague Earth-bound
observatories. Here we summarize the properties of MEarth data gathered so far,
and we present a new framework to detect shallow exoplanet transits in wiggly
and irregularly-spaced light curves. In contrast to previous methods that clean
trends from light curves before searching for transits, this framework assesses
the significance of individual transits simultaneously while modeling
variability, systematics, and the photometric quality of individual nights. Our
Method for Including Starspots and Systematics in the Marginalized Probability
of a Lone Eclipse (MISS MarPLE) uses a computationally efficient semi-Bayesian
approach to explore the vast probability space spanned by the many parameters
of this model, naturally incorporating the uncertainties in these parameters
into its evaluation of candidate events. We show how to combine individual
transits processed by MISS MarPLE into periodic transiting planet candidates
and compare our results to the popular Box-fitting Least Squares (BLS) method
with simulations. By applying MISS MarPLE to observations from the MEarth
Project, we demonstrate the utility of this framework for robustly assessing
the false alarm probability of transit signals in real data. [slightly
abridged]Comment: accepted to the Astronomical Journal, 21 pages, 12 figure