  Future space telescopes will directly image extrasolar planets at visible
wavelengths. Time-resolved reflected light from an exoplanet encodes
information about atmospheric and surface inhomogeneities. Previous research
has shown that the light curve of an exoplanet can be inverted to obtain a
low-resolution map of the planet, as well as constraints on its spin
orientation. Estimating the uncertainty on 2D albedo maps has so far remained
elusive. Here we present exocartographer, a flexible open-source Bayesian
framework for solving the exo-cartography inverse problem. The map is
parameterized with equal-area HEALPix pixels. For a fiducial map resolution of
192 pixels, a four-parameter Gaussian process describing the spatial scale of
albedo variations, and two unknown planetary spin parameters, exocartographer
explores a 198-dimensional parameter space. To test the code, we produce a
light curve for a cloudless Earth in a face-on orbit with a 90$^\circ$
obliquity. We produce synthetic white light observations of the planet: 5
epochs of observations throughout the planet's orbit, each consisting of 24
hourly observations with a photometric uncertainty of $1\%$ (120 data). We
retrieve an albedo map and$-$for the first time$-$its uncertainties, along with
spin constraints. The albedo map is recognizably of Earth, with typical
uncertainty of $30\%$. The retrieved characteristic length scale is 88$\pm 7
^\circ$, or 9800 km. The obliquity is recovered with a $1-\sigma$ uncertainty
of $0.8^\circ$. Despite the uncertainty in the retrieved albedo map, we
robustly identify a high albedo region (the Sahara desert) and a large
low-albedo region (the Pacific Ocean).
