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, $\mathrm {I_{ss}}$, is defined by a Kullback–Leibler divergence,

$\mathrm {I_{ss}} (X;Y=y)\equiv \mathrm {D_{KL}} [P_{X|y}\parallel P_{X}]$ .

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,

$\mathrm {I_{ss}} (XZ;Y=y)=\mathrm {I_{ss}} (X;Y=y)+\mathrm {I_{ss}} (Z;Y=y|X)$.

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, $\mathrm {I_{si}}$, is defined by a difference of entropies,

$\mathrm {I_{si}} (X;Y=y)\equiv \mathrm {H} (X)-\mathrm {H} (X|Y=y)$ .

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

$\mathrm {I_{si}} (X;YZ=yz)=\mathrm {I_{si}} (X;Y=y)+\mathrm {I_{si}} (X;Z=z|Y=y)$ .

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 $\mathrm {I_{ss}}$ and $\mathrm {I_{si}}$ are equivalent.

State-Specific-Information

The state-specific information, $\mathrm {I_{ssi}}$, is a synonym for the Pointwise mutual information.

References

• Timme, Nicholas; Lapish, Christopher (2018). "A Tutorial for Information Theory in Neuroscience". eNeuro. 5 (3). doi:10.1523/ENEURO.0052-18.2018. PMC 6131830.
• Deweese, Michael; Meister, Markus (1999). "How to measure the information gained from one symbol". Network: Computation in Neural Systems. 10 (4): 325–40. CiteSeerX 10.1.1.553.8013. doi:10.1088/0954-898X/10/4/303. PMID 10695762.
• Butts, Daniel (2003). "How much information is associated with a particular stimulus?". Network: Computation in Neural Systems. 14 (2): 177–87. doi:10.1088/0954-898X/14/2/301. PMID 12790180.
• Itti, Laurent; Baldi, Pierre (2009). "Bayesian surprise attracts human attention". Vision Research. 49 (10): 1295–1306. doi:10.1016/j.visres.2008.09.007. PMC 2782645.