  Many important classification problems, such as object classification, speech
recognition, and machine translation, have been tackled by the supervised
learning paradigm in the past, where training corpora of parallel input-output
pairs are required with high cost. To remove the need for the parallel training
corpora has practical significance for real-world applications, and it is one
of the main goals of unsupervised learning. Recently, encouraging progress in
unsupervised learning for solving such classification problems has been made
and the nature of the challenges has been clarified. In this article, we review
this progress and disseminate a class of promising new methods to facilitate
understanding the methods for machine learning researchers. In particular, we
emphasize the key information that enables the success of unsupervised learning
- the sequential statistics as the distributional prior in the labels.
Exploitation of such sequential statistics makes it possible to estimate
parameters of classifiers without the need of paired input-output data.
  In this paper, we first introduce the concept of Caesar Cipher and its
decryption, which motivated the construction of the novel loss function for
unsupervised learning we use throughout the paper. Then we use a simple but
representative binary classification task as an example to derive and describe
the unsupervised learning algorithm in a step-by-step, easy-to-understand
fashion. We include two cases, one with Bigram language model as the sequential
statistics for use in unsupervised parameter estimation, and another with a
simpler Unigram language model. For both cases, detailed derivation steps for
the learning algorithm are included. Further, a summary table compares
computational steps of the two cases in executing the unsupervised learning
algorithm for learning binary classifiers.
