A polar bear's guard hairs contain no white pigment. They are transparent and hollow, and the apparent whiteness comes from light scattering off the internal structure and the interfaces within the shaft, the same reason snow and foam appear white. The skin underneath is black.
The fibre-optic story emerged in the 1970s: the suggestion that the hollow hairs channel ultraviolet and visible light down to the black skin where it is absorbed as heat. Optical measurements published in the 1980s found the hairs to be poor light guides, absorbing and scattering rather than transmitting along their length. Insulation in a polar bear is straightforwardly conventional, from trapped air in the dense underfur, the guard coat and a thick blubber layer.
One practical consequence of the transparency survives. Polar bears photograph nearly black in ultraviolet, and hollow hairs can accumulate algae in warm, humid captive conditions, which is why some zoo bears have appeared distinctly green.