Liquids soak into marble really well, and a cool way to think about ink or dye and paint removal is that they can be soaked up in the reverse. Marble surfaces have pores, and similar to skin, a reaction pulling out -- for example a sponge or clay mask lifting out blackheads and impurities in facial skin -- is kind of what happens with marble stains.
Applying a layer of poultice, or a thick powder including a mixing agent called whiting and a low to medium concentration of hydrogen peroxide, will actually lift the stain from the marble. It may be best to leave the mixture to settle into the stain for 12 hours or more and even to cover it with plastic wrap to keep moisture seeping in before pulling up and sloughing off the stain pigments.
If the stain is still visible after treating, repeat this process until it lifts and rinse with warm water and gentle soap to remove any residue from the porous surface and edges.