  Errors in measurements are key to weighting the value of data, but are often
neglected in Machine Learning (ML). We show how Convolutional Neural Networks
(CNNs) are able to learn about the context and patterns of signal and noise,
leading to improvements in the performance of classification methods. We
construct a model whereby two classes of objects follow an underlying Gaussian
distribution, and where the features (the input data) have varying, but known,
levels of noise. This model mimics the nature of scientific data sets, where
the noises arise as realizations of some random processes whose underlying
distributions are known. The classification of these objects can then be
performed using standard statistical techniques (e.g., least-squares
minimization or Markov-Chain Monte Carlo), as well as ML techniques. This
allows us to take advantage of a maximum likelihood approach to object
classification, and to measure the amount by which the ML methods are
incorporating the information in the input data uncertainties. We show that,
when each data point is subject to different levels of noise (i.e., noises with
different distribution functions), that information can be learned by the CNNs,
raising the ML performance to at least the same level of the least-squares
method -- and sometimes even surpassing it. Furthermore, we show that, with
varying noise levels, the confidence of the ML classifiers serves as a proxy
for the underlying cumulative distribution function, but only if the
information about specific input data uncertainties is provided to the CNNs.
