The Kullback-Leibler Divergence is defined as $$K(f:g) = \int \left(\log \frac{f(x)}{g(x)} \right) \ dF(x)$$ It measures the distance between two distributions $f$ and $g$. Why would this be better than the Euclidean distance in some situations?

I am learning machine learning and encountered KL divergence: $$ \int p(x) \log\left(\frac{p(x)}{q(x)}\right) \, \text{d}x $$ I understand that this measure calculates the difference between two probability distributions. I have written down the formula as follows (where $p(x)$ is the true distri...

For discrete probability distributions $P,Q$, the Kullback-Leibler divergence of $Q$ from $P$ is defined to be $$D_{\mathrm{KL}} ( P \mathop{\|} Q ) = \sum_i P(i) \ln \left( \frac{P(i)}{Q(i)} \right).$$ Wikipedia's article on Kullback–Leibler divergence states The Kullback–Leibler divergence...

Let us consider the following two probability distributions P Q 0.01 0.002 0.02 0.004 0.03 0.006 0.04 0.008 0.05 0.01 0.06 0.012 0.07 0.014 0.08 0.016 0.64 0.928 I have have calculated the Kullback-Leibler divergence which is equal $0.492820258$, I want to know...

I'm looking at the NIPS 2003 paper A Kullback-Leibler divergence based kernel for SVM classification in multimedia applications. The authors suggest using the following kernel function for two distributions $p$ and $q$: $$ k(p,q)= \exp (-a (D_{KL}(p,q) + D_{KL}(q,p))) $$ where $a>0$ and $D_{KL}$ ...

I have two discrete random variables $X$ and $Y$ and their distributions have different support. Assume $X$ and $Y$ can both take on the same number of values. Let's say $X \in \{10,13,15,17,19\}$ and $Y \in \{12,14,16,18,20\}$. I would like to use the Kullback-Leibler divergence but it requires ...

So, I was looking at the paper by Andrzej CICHOCKI and in the preliminary and notation part where they define some identities, the following identity is also given: $$\mathrm {tr}(P)-\log \det(P) \geq n$$ where $P$ may be any positive definite matrix. I was wondering how one would prove the a...

Background Information For the Gamma Function in the real plane, we have the following definition Definition. $\displaystyle \Gamma(x) := \int_0^{\infty} e^{-t} t^{x-1} dt \quad (x>0)$ and from this definition we can derive the functional identity Functional Identity. $\Gamma(x+1) = x \Gamma(x...

Background Information For the Gamma Function in the real plane, we have the following definition Definition. $\displaystyle \Gamma(x) := \int_0^{\infty} e^{-t} t^{x-1} dt \quad (x>0)$ and from this definition we can derive the functional identity Functional Identity. $\Gamma(x+1) = x \Gamma(x...