jmc

Connecting the midpoints of the sides of a square in order forms a square that has half the area of the original square. So, the square formed by connecting the midpoints of $ABCD$ has area $\frac{1}{2}\cdot 4^2=8$ square inches. The smallest square in the diagram is formed by connecting the midpoints of this square with area 8, so the smallest square has area $\frac{1}{2}\cdot 8=4$ square inches, leaving $8-4=\boxed{4}$ square inches of shaded area.