Why is my boyfriend so publically affectionate towards me but privately not so much?

He grabs me and has me sit on his lap in front of hi friends and family. He likes to kiss on me while Im on his lap as well or play with my hair. He stares at me and looks into my face while we are in public even more than he does in private. Like in private he is nice but in public he literally just stares at me and tells me how beautiful I am which is flattering but why is he doing it all in public and not so much in private? Also in public he calls me his baby, darling, sweetheart, etc but in private he just says "bae". He has tried to fight a guy for looking at me in oublic. Literally looking at me caused my boyfriend to first makeout with me then stare the guy down, THEN he was about to get up and try to fight until I pulled him away. He has said he loves me in public and kissed me in front of his female friends, he makes sexual insinuations about us in front of his friends, he likes to hold my hand or just touch me in general, but in private he is kind of reserved. He only touches me when we have sex or when he is really in a confessional mood but for the most part in private he just likes to watch TV and talk. He doesn't stare at me lovingly all that much unless were having sex or rare occasions when I make a stupid joke. And I have to beg him to cuddle with me because he gets hot. He once yelled at me for accidentally pulli g his beard while cuddling, and sometimes even after sex he would just drop me right back iff at home with the music turned all the way up so we couldnt talk at all. He seems bipolar, sometimes he loves me and sometimes he acts like he doesn't want to be bothered with me
Why is my boyfriend so publically affectionate towards me but privately not so much?
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