Pareto.Entropy has the wrong sign

[Steivan]

The implementation of the Entropy for the Pareto distribution -> (https://github.com/mathnet/mathnet-numerics/blob/master/src/Numerics/Distributions/Pareto.cs) line 167 has the wrong sign:

public double Entropy => Math.Log(_shape/_scale) - (1.0/_shape) - 1.0;

should be:

public double Entropy => Math.Log(_scale/_shape) + (1.0/_shape) + 1.0;

as, e.g., found in https://en.wikipedia.org/wiki/Pareto_distribution

[cdrnet]

Interesting, looks like english and german wikipedia might be contradicting each other on this: https://de.wikipedia.org/wiki/Pareto-Verteilung

[Steivan]

Yes it is an interesting observation, but in this case, the german wikipedia is inconsistent.

The entropy $$H(X)$$ (see, https://en.wikipedia.org/wiki/Entropy_(information_theory) and https://de.wikipedia.org/wiki/Entropie_(Informationstheorie)) is generally defined as:

$$H(X):=-\sum_{x \in X} p(x) \log p(x)$$

i.e., $$H(X) \geq 0$$. This definition is implemented for most distributions in the mathnet library (at least for the ones in the subset I tested), the only exception I found is the Pareto distribution.