Editing Entropy (information theory) (section)

===Limiting density of discrete points===
{{Main|Limiting density of discrete points}}

It turns out as a result that, unlike the Shannon entropy, the differential entropy is ''not'' in general a good measure of uncertainty or information. For example, the differential entropy can be negative; also it is not invariant under continuous co-ordinate transformations. This problem may be illustrated by a change of units when {{math|''x''}} is a dimensioned variable. {{math|''f''(''x'')}} will then have the units of {{math|1/''x''}}. The argument of the logarithm must be dimensionless, otherwise it is improper, so that the differential entropy as given above will be improper. If {{math|''&Delta;''}} is some "standard" value of {{math|''x''}} (i.e. "bin size") and therefore has the same units, then a modified differential entropy may be written in proper form as:
<math display="block" display="block">\Eta=\int_{-\infty}^\infty f(x) \log(f(x)\,\Delta)\,dx ,</math>
and the result will be the same for any choice of units for {{math|''x''}}. In fact, the limit of discrete entropy as <math> N \rightarrow \infty </math> would also include a term of <math> \log(N)</math>, which would in general be infinite. This is expected: continuous variables would typically have infinite entropy when discretized. The [[limiting density of discrete points]] is really a measure of how much easier a distribution is to describe than a distribution that is uniform over its quantization scheme.