Is mutual information symmetric?
Given the definition for mutual information

$$I(X;Y) = \sum_{y \in Y} \sum_{x \in X}
  p(x,y) \log{ \left(\frac{p(x,y)}{p(x)\,p(y)}
  \right) },$$

it follows from rearrangement of the summands

$$I(Y; X) =
  \sum_{x \in X} \sum_{y \in Y}
  p(x,y) \log{ \left(\frac{p(x,y)}{p(x)\,p(y)}
  \right) }.$$

Hence $I(X; Y) = I(Y; X)$.


Edit


  "$I(X;Y)$ measures the average reduction in uncertainty of $X$ that results from knowing $Y$"


If you interpret $H(X) - H(X|Y)$ where $H(X)$ is the marginal entropy of $X$ and $H(X|Y)$ the conditional entropy of $X$ given $Y$ as the reduction in uncertainty, then it is equal to $H(Y) - H(Y|X)$ and $I(X; Y)$ (see Wikipedia).