In information theory, state-dependent information is the generic name given to the family of state-dependent measures that in expectation converge to the mutual information.
State-dependent informations often appear in neuroscience applications.
Let and be random variables and be a state within . The state-dependent information between a random variable and a state is written as . There are currently three known varieties of state-dependent information: specific-surprise, specific-information, and state-specific-information.
The specific-surprise, , is defined by a KullbackâÂÂLeibler divergence,
As a special case of the chain-rule for Kullback-Liebler divergerences, specific-surprise follows the chain-rule for variables. Using as a random variable, this is specifically,
Intuitively, specific-surprise is thought of as âÂÂhow much did my beliefs about change upon learning that âÂÂ? Which is zero when thereâÂÂs no change. It is nonnegative. Specific-surprise has also been called âÂÂBayesian SurpriseâÂÂ.
The specific-information, , is defined by a difference of entropies,
Specific-information follows the chain-rule for states. Using a state as a state of random variable , this is specifically,
Specific-information is interpreted as "how did the uncertainty about change upon learning ?" This can be in the positive or negative. When follows a uniform distribution, the and are equivalent.
The state-specific information, , is a synonym for the Pointwise mutual information.