  Presence/absence data and presence-only data are the two customary sources
for learning about species distributions over a region. We illuminate the
fundamental modeling differences between the two types of data. Most simply,
locations are considered as fixed under presence/absence data; locations are
random under presence-only data. The definition of "probability of presence" is
incompatible between the two. So, we take issue with modeling strategies in the
literature which ignore this incompatibility, which assume that
presence/absence modeling can be induced from presence-only specifications and
therefore, that fusion of presence-only and presence/absence data sources is
routine. We argue that presence/absence data should be modeled at point level.
That is, we need to specify a surface which provides the probability of
presence at any location in the region. A realization from this surface is a
binary map yielding the results of Bernoulli trials across all locations.
Presence-only data should be modeled as a point pattern driven by specification
of an intensity function. We further argue that, with just presence/absence
data, preferential sampling, using a shared process perspective, can improve
our estimated presence/absence surface and prediction of presence. We also
argue that preferential sampling can enable a probabilistically coherent fusion
of the two data types. We illustrate with two real datasets, one
presence/absence, one presence-only for invasive species presence in New
England in the United States. We demonstrate that potential bias in sampling
locations can affect inference with regard to presence/absence and show that
inference can be improved with preferential sampling ideas. We also provide a
probabilistically coherent fusion of the two datasets to again improve
inference with regard to presence/absence.
