Why No HUD?
Developers are always looking for new ways to immerse the player into the game they are playing. Lately the concept of No HUD has become a big thing, making the player focus on what they're playing rather than the number in their health bar or ammo counter. Some ways of limiting of removing the HUD includes using a regenerating health system. This will make the screen turn a different colour or flash red when the player is low on health, allowing them to be informed on the current situation without having a health bar.
Notable HUDless games
The early Resident Evil games were some of the earliest popular 3D games without a traditional HUD. This was in order to increase the immersion and tension of the game's survival horror atmosphere, and to make it look more like a film rather than a game (along with the pre-rendered backgrounds and fixed camera angles). In addition, in Resident Evil 2, the player character's reactions and movements were used to indicate health rather than a HUD.
Another game that is noted for having no traditional HUD is Dead Space. EA Redwood Shores made sure that the player is immersed in this horror game by removing all traditional elements of a HUD. Your character Isaac Clarke has a special space suit which displays everything the player would need to know in-game. A health bar is located on the spine, an ammo counter is projected above each gun when you go into aiming mode, and inventory is shown as a in game hologram that Isaac is also seeing.
Heavy Rain goes for more of a interactive story element for its gameplay and having a HUD would supposedly take the player out of the experience. So all in-game prompts show up in game around the character based on how they are feeling. If a character is nervous about a situation their thoughts will be floating around them in a sporadic manner, but if they are calm the prompts are in plain sight and easy to see. Unfortunately, in practice, this sometimes made it difficult to find and hit the prompts, and next to impossible to focus on the events taking place in-game.