If you want to stream BioShock directly from your console, you're out of luck. Last night, I tried to stream BioShock: The Collection from my PS4 but found that the entire game was blocked from broadcast. No audio, no video. A similar issue exists on the Xbox One as well. A post on 2K's support site states: "Unfortunately, streaming BioShock: The Collection isn't an option right now."
Kotaku has reached out to 2K but has yet to hear a reply. It's not entirely clear why the game is being blocked. Some games like Metal Gear Solid: Ground Zeroes block crucial cutscenes, but blocking a whole game is pretty extreme. There are ways around this, including streaming directly off capture card software, but if you want to stream from integrated console features, expect a lot of this: