Implicit Brain-Computer Interfaces (BCI) adapt system settings subtly based on real time measures of brain activation without the user's explicit awareness. For example, measures of the user's cognitive profile might drive a system that alters the timing of notifications in order to minimize user interruption. Here, we consider new avenues for implicit BCI based on recent discoveries in cognitive neuroscience and conduct a series of experiments using BCI's principal non-invasive brain sensors, fNIRS and EEG. We show how Bayesian and systems neuroscience formulations explain the difference in performance of machine learning algorithms trained on brain data in different conditions. These new formulations posit that the brain aims to minimize its long-term surprisal of sensory data and organizes its calculations on two anti-correlated networks. We consider how to use real-time input that portrays a user along these dimensions in designing Bidirectional BCIs, which are Implicit BCIs that aim to optimize the user's state by modulating computer output based on feedback from a brain monitor. We introduce Entropic Brain-Computer Interfacing as a type of Bidirectional BCI which uses physiological measurements of information theoretical dimensions of the user's state to evaluate the digital flow of information to the user's brain, tweaking this output in a feedback loop to the user's benefit.