  Exoplanet catalogs produced by surveys suffer from a lack of completeness
(not every planet is detected) and less than perfect reliability (not every
planet in the catalog is a true planet), particularly near the survey's
detection limit. Exoplanet occurrence rate studies based on such a catalog must
be corrected for completeness and reliability. The final Kepler data release,
DR25, features a uniformly vetted planet candidate catalog and data products
that facilitate corrections. We present a new probabilistic approach to the
characterization of Kepler completeness and reliability, making full use of the
Kepler DR25 products. We illustrate the impact of completeness and reliability
corrections with a Poisson-likelihood occurrence rate method, using a recent
stellar properties catalog that incorporates Gaia stellar radii and essentially
uniform treatment of the stellar population. Correcting for reliability has a
significant impact: the exoplanet occurrence rate for orbital period and radius
within 20% of Earth's around GK dwarf stars, corrected for reliability, is
0.015+0.011-0.007, whereas not correcting results in 0.034+0.018-0.012 -
correcting for reliability reduces this occurrence rate by more than a factor
of two. We further show that using Gaia-based vs. DR25 stellar properties
impacts the same occurrence rate by a factor of two. We critically examine the
the DR25 catalog and the assumptions behind our occurrence rate method. We
propose several ways in which confidence in both the Kepler catalog and
occurrence rate calculations can be improved. This work provides an example of
how the community can use the DR25 completeness and reliability products.
