text: * Relative entropy is always non-negative,  a result known as Gibbs' inequality, with  equals zero if and only if  as measures.
  In particular, if  and , then  -almost everywhere. The entropy  thus sets a minimum value for the cross-entropy , the expected number of bits required when using a code based on  rather than ; and the Kullback–Leibler divergence therefore represents the expected number of extra bits that must be transmitted to identify a value  drawn from , if a code is used corresponding to the probability distribution , rather than the "true" distribution .
  * No upper-bound exists for the general case. However, it is shown that if  and  are two discrete probability distributions built by distributing the same discrete quantity, then the maximum value of  can be calculated.
  * Relative entropy remains well-defined for continuous distributions, and furthermore is invariant under parameter transformations. For example, if a transformation is made from variable  to variable , then, since  and  where  is the absolute value of the derivative or more generally of the Jacobian, the relative entropy may be rewritten: 
  * The Taylor expansion is .