  Zellner (1988) modeled statistical inference in terms of information
processing and postulated the Information Conservation Principle (ICP) between
the input and output of the information processing block, showing that this
yielded Bayesian inference as the optimum information processing rule.
Recently, Alemi (2019) reviewed Zellner's work in the context of machine
learning and showed that the ICP could be seen as a special case of a more
general optimum information processing criterion, namely the Predictive
Information Bottleneck Objective. However, Alemi modeled the model training
step in machine learning as using training and test data sets only, and did not
account for the use of a validation data set during training. The present note
is an attempt to extend Alemi's information processing formulation of machine
learning, and the predictive information bottleneck objective for model
training, to the widely-used scenario where training utilizes not only a
training but also a validation data set.
