I definitely believe that viewing a selection of posts and pages on facebook that a person has "liked" can determine at least part that person's personality profile. It may not account for or determine all of someone's personality but it can at least help to judge them in the realm of where their true interests lay.
Facebook has overstepped its bounds and shows its hubris with the attempt to predict personalities through monitoring of the users' Facebook pages. This is a facade and all setup to try and regain its once former leadership position within the social media world. Facebook can no more predict a personality through a Facebook page than a carnival seer can predict future events.
On one hand, as advertisers would very much like to believe, what a person 'likes' on Facebook can tell you a certain amount of information about that person. After all, much can be revealed through a person's interests, and even political beliefs. But as for personality, I do not believe Facebook's algorithm is this powerful. There are simply too many gaps, too many things not addressed by 'likes', for this to be an accurate measure.
Facebook likes cannot predict personalities. One main reason is people sometimes accidentally click "like" and not realize it which would skew any data collected to help with the prediction. Also, ones personality online is very different than ones in real life. One might act differently and say different things to someone online versus if they were face to face with that person which makes me think they would "like" different things online than what their true personality would portray.