I think it is bad risk assessment at it's core (very few people really understand risk assesment anyway), combined with misinformation, but what really drives intellegent people remain anti-vac, even after being presented with the facts, is emotion and subconsious fear of accountability when it comes to something bad happening to their children.
Given two possible outcomes:
A. They decide to vaccinate and their child becomes severely ill due to an allergic reaction, they will blame themselves.
B. They do not vacinnate and their child becomes severely ill from a disease they could have been vaccinated for, they don't see that as being as much their fault.
Even if A is significantly less likely than B and harm to their child is equal, it is still seen as a significantly worse outcome and risk because it will have happend as a direct result of something they chose to do.
Probably most anti-vax beliefs go back to poor understanding and/or missinformation, but among intellegent educated people who are anti-vax (and i've talked to several such people) the main cause always comes down to a cowardly (though perhaps subconsious) avoidance of accountability.