Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$X : \Omega \to \mathcal{X}$ and $Y : \Omega \to \mathcal{Y}$ are each a D5723: Simple random variable on $P$

Let $\log_a$ be the D866: Standard real logarithm function in base $a \in (0, \infty)$.
The mutual information of $(X, Y)$ in base $a$ is the D4767: Unsigned real number \begin{equation} I(X; Y) : = \sum_{x \in \mathcal{X}} \sum_{y \in \mathcal{Y}} \mathbb{P}(X = x, Y = y) \log_a \frac{\mathbb{P}(X = x, Y = y)}{\mathbb{P}(X = x) \mathbb{P}(Y = y)} \end{equation}

Let $X \in \text{Random}(\mathcal{X})$ and $Y \in \text{Random}(\mathcal{Y})$ 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 \, y \in \mathcal{Y} : p(y) : = \mathbb{P}(Y = y) \end{equation}
(iii)	\begin{equation} \forall \, x \in \mathcal{X}, y \in \mathcal{Y} : p(x, y) : = \mathbb{P}(X = x, Y = y) \end{equation}

Let $\log_a$ be the D866: Standard real logarithm function in base $a \in (0, \infty)$.
The mutual information of $(X, Y)$ in base $a$ is the D4767: Unsigned real number \begin{equation} I(X; Y) : = \sum_{x \in \mathcal{X}} \sum_{y \in \mathcal{Y}} p(x, y) \log_a \frac{p(x, y)}{p(x) p(y)} \end{equation}