I am having trouble building some intuition about joint entropy. $H(X,Y)$ = uncertainty in the joint distribution $p(x,y)$; $H(X)$ = uncertainty in $p_x(x)$; $H(Y)$ = uncertainty in $p_y(y)$.
If H(X) is high then the distribution is more uncertain and if you know the outcome of such a distribution then you have more information! So H(X) also quantifies information.
Now we can show $H(X,Y) \leq H(X) + H(Y)$
But if you know $p(x,y)$ you can get $p_x(x)$ and $p_y(y)$ so in some sense $p(x,y)$ has more information than both $p_x(x)$ and $p_y(y)$, so shouldn't the uncertainty related to p(x,y) be more that the sum of the individual uncertainties?