  Most general-purpose classification methods, such as support-vector machine
(SVM) and random forest (RF), fail to account for an unusual characteristic of
astronomical data: known measurement error uncertainties. In astronomical data,
this information is often given in the data but discarded because popular
machine learning classifiers cannot incorporate it. We propose a
simulation-based approach that incorporates heteroscedastic measurement error
into existing classification method to better quantify uncertainty in
classification. The proposed method first simulates perturbed realizations of
the data from a Bayesian posterior predictive distribution of a Gaussian
measurement error model. Then, a chosen classifier is fit to each simulation.
The variation across the simulations naturally reflects the uncertainty
propagated from the measurement errors in both labeled and unlabeled data sets.
We demonstrate the use of this approach via two numerical studies. The first is
a thorough simulation study applying the proposed procedure to SVM and RF,
which are well-known hard and soft classifiers, respectively. The second study
is a realistic classification problem of identifying high-$z$ $(2.9 \leq z \leq
5.1)$ quasar candidates from photometric data. The data are from merged
catalogs of the Sloan Digital Sky Survey, the $Spitzer$ IRAC Equatorial Survey,
and the $Spitzer$-HETDEX Exploratory Large-Area Survey. The proposed approach
reveals that out of 11,847 high-$z$ quasar candidates identified by a random
forest without incorporating measurement error, 3,146 are potential
misclassifications with measurement error. Additionally, out of $1.85$ million
objects not identified as high-$z$ quasars without measurement error, 936 can
be considered new candidates with measurement error.
