In information theory, the limiting density of discrete points is an adjustment to the formula of Claude Shannon for differential entropy.

It was formulated by Edwin Thompson Jaynes to address defects in the initial definition of differential entropy.

Definition

Shannon originally wrote down the following formula for the entropy of a continuous distribution, known as differential entropy:

Unlike Shannon's formula for the discrete entropy, however, this is not the result of any derivation (Shannon simply replaced the summation symbol in the discrete version with an integral), and it lacks many of the properties that make the discrete entropy a useful measure of uncertainty. In particular, it is not invariant under a change of variables, which means that the argument to the logarithm is not dimensionless as required.

Jaynes argued that the formula for the continuous entropy should be derived by taking the limit of increasingly dense discrete distributions. called the "invariant measure":

From the above invariant measure, Jaynes derived the following formula for continuous entropy, which he argued should be taken as the correct formula:

is omitted, as that would typically not be finite. So the actual common definition is

term should be omitted, one could write

represents a uniform density over the quantized continuous space used in the Riemann sum..

is actually the (negative) Kullback–Leibler divergence.

are transformed in the same way. (This motivates the name "invariant measure" for m term) would always be non-positive, because a KL divergence would always be non-negative.

:

References

Further reading