I think you've assumed a couple of fallacies here. First of all, I'm not so sure belief is a matter of choice. It's more a matter of perception. Second, Belief isn't a matter of "faith." It's a matter of perception. Perception really doesn't have much to do with choice.
Unexamined perception doesn't have much to do with choice.
As noted in another thread, beliefs are composed ideas or thoughts, formulated to identify and/or understand something.
A belief could be formulated with what may, rather easily, appear as low level awareness. Like in my night dreams, I may believe President Obama needs me to fly to Washington to accept an award for greatest truth teller in the nation. I wake up to realize I'm not actually up for that award (though I should be). I can tell myself, rather easily, that I had no choice in that belief, but that assertion comes from, I would say, lack of self examination, lack of awareness of who I am, or who I was then (in the dream).
It seems to me that we think concepts like "belief" and "opinion" stand on their own, and are 'completely different' from knowledge or faith. I believe all these are thoughts and are derived from a knowledge we still have, pretend not to have, think we are working toward, searching for, currently lacking, and that belief(s) are wholly distinct from what is rather universal.
Belief is more like a thread of a enormously large tapestry, where some beliefs are made of 'material' that dissolves over time, and fades into larger pattern.