I heard someone pronounce UI as yooey. I guess this comes from the pronunciation of GUI, which is gooey. How common is this compared to yoo-eye and user-interface?
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I've heard "user interface" used quite a lot, but never "yooey" - I'd say the latter was nonstandard.
Personally I'd pronounce it "yoo eye" in technical circles, and "user interface" in less technical circles.
UI is pronounced as "you I"
I work in IT industry and have never came across a different pronunciation.
Youey is informal, and is acceptable for relaxed chat or friends. But for a technical audience you should probably stick to expanding to User Interface for the first use and 'You Eye' afterwards.
This helps people who are not technical (who may not have come across the term before) and also people who don't speak English as a first language, where the initials U and I may have very different pronunciation. ('Uh ee'?)
 I do this all the time, and my friends find it endearing, or annoying, I'm never sure which. I also say 'wee-fee" for WIFI.  Sorry for not using ASCII IPA.