  A critical step in data analysis for many different types of experiments is
the identification of features with theoretically defined shapes in
N-dimensional datasets; examples of this process include finding peaks in
multi-dimensional molecular spectra or emitters in fluorescence microscopy
images. Identifying such features involves determining if the overall shape of
the data is consistent with an expected shape, however, it is generally unclear
how to quantitatively make this determination. In practice, many analysis
methods employ subjective, heuristic approaches, which complicates the
validation of any ensuing results - especially as the amount and dimensionality
of the data increase. Here, we present a probabilistic solution to this problem
by using Bayes' rule to calculate the probability that the data has any one of
several potential shapes. This probabilistic approach may be used to
objectively compare how well different theories describe a dataset, identify
changes between datasets, and detect features within data using a corollary
method called Bayesian Inference-based Template Search (BITS); several
proof-of-principle examples are provided. Altogether, this mathematical
framework serves as an automated 'engine' capable of computationally executing
analysis decisions currently made by visual inspection across the sciences.
