State-dependent information

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 $X$ and $Y$ be random variables and $y$ be a state within $Y$. The state-dependent information between a random variable $X$ and a state $Y=y$ is written as $I(X;Y=y)$. There are currently three known varieties of state-dependent information: specific-surprise, specific-information, and state-specific-information.

Specific-Surprise

The specific-surprise, ${\mathrm{I}}_{\mathrm{s}\mathrm{s}}$, is defined by a Kullback–Leibler divergence,

${\mathrm{I}}_{\mathrm{s}\mathrm{s}}(X;Y=y)\equiv {\mathrm{D}}_{\mathrm{K}\mathrm{L}}[P_X\parallel P_{X|y}]$ .

As a special case of the chain-rule for Kullback-Liebler divergerences, specific-surprise follows the chain-rule for variables. Using $Z$ as a random variable, this is specifically,

${\mathrm{I}}_{\mathrm{s}\mathrm{s}}(XZ;Y=y)={\mathrm{I}}_{\mathrm{s}\mathrm{s}}(X;Y=y)+{\mathrm{I}}_{\mathrm{s}\mathrm{s}}(Z;Y=y|X)$.

Intuitively, specific-surprise is thought of as “how much did my beliefs about $X$ change upon learning that $Y=y$”? Which is zero when there’s no change. It is nonnegative. Specific-surprise has also been called “Bayesian Surprise”.

Specific-Information

The specific-information, ${\mathrm{I}}_{\mathrm{s}\mathrm{i}}$, is defined by a difference of entropies,

${\mathrm{I}}_{\mathrm{s}\mathrm{i}}(X;Y=y)\equiv \mathrm{H}(X)-\mathrm{H}(X|Y=y)$ .

Specific-information follows the chain-rule for states. Using a state $z\in Z$ as a state of random variable $Z$, this is specifically,

${\mathrm{I}}_{\mathrm{s}\mathrm{i}}(X;YZ=yz)={\mathrm{I}}_{\mathrm{s}\mathrm{i}}(X;Y=y)+{\mathrm{I}}_{\mathrm{s}\mathrm{i}}(X;Z=z|Y=y)$ .

Specific-information is interpreted as "how did the uncertainty about $X$ change upon learning $Y=y$?" This can be in the positive or negative. When $X$ follows a uniform distribution, the ${\mathrm{I}}_{\mathrm{s}\mathrm{s}}$ and ${\mathrm{I}}_{\mathrm{s}\mathrm{i}}$ are equivalent.

State-Specific-Information

The state-specific information, ${\mathrm{I}}_{\mathrm{s}\mathrm{s}\mathrm{i}}$, is a synonym for the Pointwise mutual information.