the state of the whole system. This quantity is exactly , which gives the chain rule of conditional entropy: The chain rule follows from the above definition of conditional entropy: In general, a chain rule for multiple random variables holds: It has a similar form to chain rule in probability theory, except that addition instead of multiplication is used. Bayes' rule Bayes' rule for conditional entropy states Proof. and . Symmetry entails . Subtracting the two equations implies Bayes' rule. If is conditionally independent of given we have: Other properties For any and : where is the mutual information between and . For independent and : and Although the specific-conditional entropy can be either less or greater than for a given random variate of , can never exceed . Conditional differential entropy Definition The above definition is for discrete random variables. The continuous version of discrete conditional entropy is called conditional differential (or continuous) entropy. Let and be a continuous random variables with a joint probability density function . The differential conditional entropy is defined as Properties In contrast to the conditional entropy for discrete random variables, the conditional differential entropy may be negative. As in the discrete case there is a chain rule for differential entropy: Notice however that this rule may not be true if the involved differential entropies do not exist or are infinite. Joint differential entropy is also used in the definition of the mutual information between continuous random variables: with equality if and only if and are independent. Relation to estimator error The conditional differential entropy yields a lower bound on the expected squared error of an estimator. For any random variable , observation and estimator the following holds: This is related to the uncertainty principle from quantum mechanics. Generalization to quantum theory In quantum information theory, the conditional entropy is generalized to the conditional quantum entropy. The latter can take negative values, unlike its classical counterpart. See also Entropy (information theory) Mutual information Conditional quantum entropy Variation of information Entropy power inequality Likelihood function References Entropy and information Information theory