I am working through a proof in this document
http://ee.tamu.edu/~georghiades/courses/ftp647/Chapter7.pdf
for Theorem 3 (The entropy of a multivariate Gaussian distribution):
Let X = (X1, X2, · · · , Xn) be jointly Gaussian distributed with mean µ and covariance matrix K(N(µ, K)). Then,
$$ h(X) = \frac{1}{2}log(2\pi e)^n|K| $$
The only doubt I have with the proof is the beginning:
$$ h(X) = -E[ln f(x)] $$
In the beginning of the document (Definition 1):
The differential entropy of a continuous random variable X, denoted by h(X), is defined as
$$ h(X) = -\int_S f(x) log f(x) dx\ = E[-log f(X)] $$
I understand that by using the property of logs:
$$ log_e (x) = \frac{log(x)}{log(e)} $$
and therefore
$$ log(x) = log_e(x) log(e) $$
substituting in h(X):
$$ h(X) = -log(e) \int_S f(x) ln f(x) dx\ = -log(e) E[ln f(X)] $$
Where then is the missing log(e) in the beginning of the proof???
Thanks in advance!
In $h(X) = -E[lnf(x)]$ the entropy is in nats, the units of entropy when you work with natural logarithm.
In $h(X) = -E[logf(x)]$ the entropy is in bits, the units of entropy when you work with base 2 logarithm.
$1 $nat = log$_2(e)$ bits, so...