"It was collaborative. He's a really pop video director and a filmmaker. He did 'I Am Legend' and I'm a huge Will Smith fan, so I knew he could execute the video in a way that I could give him all my weirdest, most psychotic ideas, but it would come across to and be relevant to the public."
bathhouse: a building for bathing, sometimes equipped with swimming pools, medical baths, etc.
sauna: a bath that uses dry heat to induce perspiration, and in which steam is produced by pouring water on heated stones.
Check your sources, babe. They're essentially the same thing. Francis and Britney called it both a bathhouse and a sauna in the making the video anyways, so your "point" is void.