The canonical probability distribution describes a system in thermal equilibrium with an infinite heat bath. When the bath is finite the distribution is modified. These modifications can be derived by truncating a Taylor-series expansion of the entropy of the heat bath, but their form depends on the expansion parameter chosen. We consider two such expansions, which yield supercanonical (i.e., higher-order canonical) distributions of exponential and power-law form. The latter is identical in form to the "Tsallis distribution," which is therefore a valid asymptotic approximation for an arbitrary finite heat bath, but bears no intrinsic relation to Tsallis entropy.