  Surprising events trigger measurable brain activity and influence human
behavior by affecting learning, memory, and decision-making. Currently there
is, however, no consensus on the definition of surprise. Here we identify 18
mathematical definitions of surprise in a unifying framework. We first propose
a technical classification of these definitions into three groups based on
their dependence on an agent's belief, show how they relate to each other, and
prove under what conditions they are indistinguishable. Going beyond this
technical analysis, we propose a taxonomy of surprise definitions and classify
them into four conceptual categories based on the quantity they measure: (i)
'prediction surprise' measures a mismatch between a prediction and an
observation; (ii) 'change-point detection surprise' measures the probability of
a change in the environment; (iii) 'confidence-corrected surprise' explicitly
accounts for the effect of confidence; and (iv) 'information gain surprise'
measures the belief-update upon a new observation. The taxonomy poses the
foundation for principled studies of the functional roles and physiological
signatures of surprise in the brain.
