ThmDex – An index of mathematical definitions, results, and conjectures.

Formulation 0

Let $X, Y \in \text{Random}(\mathcal{X})$ each be a D5723: Simple random variable.
Let $\log_a$ be the D866: Standard real logarithm function in base $a \in (0, \infty) \setminus \{ 1 \}$.

Then

(1)	\begin{equation} D_a(X \parallel Y) \geq 0 \end{equation}
(2)	\begin{equation} D_a(X \parallel Y) = 0 \quad \iff \quad \forall \, x \in \mathcal{X} : \mathbb{P}(X = x) = \mathbb{P}(Y = x) \end{equation}

Formulation 1

Let $X, Y \in \text{Random}(\mathcal{X})$ each be a D5723: Simple random variable.
Let $\log_a$ be the D866: Standard real logarithm function in base $a \in (0, \infty) \setminus \{ 1 \}$.

Then

(1)	\begin{equation} \sum_{x \in \mathcal{X}} \mathbb{P}(X = x) \log_a \frac{\mathbb{P}(X = x)}{\mathbb{P}(Y = x)} \geq 0 \end{equation}
(2)	\begin{equation} \sum_{x \in \mathcal{X}} \mathbb{P}(X = x) \log_a \frac{\mathbb{P}(X = x)}{\mathbb{P}(Y = x)} = 0 \quad \iff \quad \forall \, x \in \mathcal{X} : \mathbb{P}(X = x) = \mathbb{P}(Y = x) \end{equation}

Formulation 2

Let $X, Y \in \text{Random}(\mathcal{X})$ each be a D5723: Simple random variable such that

(i)	\begin{equation} \forall \, x \in \mathcal{X} : p(x) : = \mathbb{P}(X = x) \end{equation}
(ii)	\begin{equation} \forall \, x \in \mathcal{X} : q(x) : = \mathbb{P}(Y = x) \end{equation}

Let $\log_a$ be the D866: Standard real logarithm function in base $a \in (0, \infty) \setminus \{ 1 \}$.

Then

(1)	\begin{equation} \sum_{x \in \mathcal{X}} p(x) \log_a \frac{p(x)}{q(x)} \geq 0 \end{equation}
(2)	\begin{equation} \sum_{x \in \mathcal{X}} p(x) \log_a \frac{p(x)}{q(x)} = 0 \quad \iff \quad \forall \, x \in \mathcal{X} : p(x) = q(x) \end{equation}