My intuition is that belief is not a choice. When someone asks me if I believe that the sky is blue, I don’t decide to believe the sky is blue and then say “yes, I believe the sky is blue”. Instead, I introspect and notice that yes, I believe the sky is blue, and then say “yes, I believe the sky is blue”.
As a less trivial example, if in an argument I am asked if I believe the thing I was originally arguing for, I introspect and consider whether or not I still believe it after thinking about what the other person has said. This process doesn’t feel like making a choice. If the other person has convinced me, then I notice that I feel like I am noticing that I have been convinced. It doesn’t feel like it is up to me whether or not to be convinced, it just happens.
Of course, it isn’t as simple as this though. In an arugment, there can often come a point where I am “on the edge” about a question — based on what I’ve thought about so far, I could see my belief going either way. But not strongly. At this point, its useful to consider the idea of credence, which is the degree of beleif. When I say that I believe something, what I usually mean is that I have a reasonably high (certainly more than 50%) credence in it. So if I could see the argument convincing me either way, my credence is very close to 50%. Under this view, perhaps my belief is not really in question yet, because it doesn’t seem like my credence will swing significantly enough towards on direction or the other in order to become strong enough to be called a belief, without further consideration. The counter to this would be that I could choose to swing my credence significantly enough to yield a belief.
A common sticking point here and in the free will debate is on the meaning of “choice”. One approach is to try to define “choose” as a fundamental physical action that humans (at least) can do with their minds. This approach leads quickly to the conclusion that there really is no such thing as choice because there could not be some basic physical action that minds can do that isn’t explainable in terms of just the more basic physical processes that clearly arent under the human mind’s control (some are deterministic, some are stochastic). While this is sometimes used as an argument against the existence of (libertarian) free will, it is actually a reductio ad absurdum of the approach itself. This shouldn’t be an argument about the definition of words, it should be about the actual thing that we refer to when we say “choice”. Choice is a social construct (as opposed to a physical one) that is used to refer to actual actions all the time, so it certainly exists. To refute its existence is to deny that people can communicate meaning with language. Note, however, that I am not claiming that everyone’s conceptions about the properties of choice are true — I am only asserting that “choice” refers to something real, even if its users would also make false claims about the things being referred to. I.e. “choice” means whatever kind of thing it is that is referred to when people talk about choice.
With this consideration, come back to the concern about choosing to believe. It does sometimes appear that people end up believing what they believe regardless of their desire to believe otherwise. Even if I try to believe the sky is green, I cannot. At the same time, it also sometimes appear that people choose to believe what they believe. Often this happens by them making a series of choices that leave them in a circumstance where they are strongly influenced to believe something, and sometimes even a person appears to just decide to believe something on the spot by an act of will. Even in these latter situations, usually there are some reasosn for a believe to form beyond the pure willing of it. But even with that considered, there is undeniably a choice involved in swinging the credence one way rather that another.
In conclusion, beliefs are influenced by a combination of factors some of which are chosen and some of which are not. There can often be many more non-chosen factors than are recognized by the chooser, but undeniably there can still be choice involved.