Reliability of using a joint density of states reconstructed with a histogram

We perform simulations of the two dimensional Ising model on a square lattice by using the Wang-Landau sampling method with two different transition probabilities and obtain two kinds of joint densities of states g ( E,M ): one is exact and the other is reconstructed with a histogram H R ( E,M ). We compare the thermodynamic quantities estimated by using the two different joint densities of states, and observe that they become different as the temperature is decreased.