Great Inagua is the pirate hideout island in Assassin's Creed IV: Black Flag.
Edward Kenway uses Great Inagua as a base after acquiring the island. The hideout includes facilities that can be upgraded, and it serves as a safe location for managing parts of Edward's pirate operation.
Great Inagua is based on the real island of Great Inagua in the Bahamas, although the game presents a fictionalized version within its Caribbean world. Black Flag also features Nassau, a major pirate settlement and an important location in the story.
Nassau is therefore a plausible mix-up, but it is not Edward's personal hideout island. Tortuga is another famous Caribbean pirate location, yet it is not the answer to this specific Black Flag question.