I agree with you that you have to choose to believe. Some people no matter what experiences they have; no matter what they see, hear, or feel simply refuse to believe. I see the Lord's creations around me and I choose to believe in their creator. I have tried God's commandments and found them good and know they they have enriched my life and brought me and my family great happiness. I choose to believe in the giver of these guides to happiness. I have had many, many personal experiences where the Lord has helped and guided me. I trust my own experiences and choose to believe in a God who loves me. My brother once said to me that if there really was a God, then He should show Himself to him and he would believe.
Here is something to think about from the Book of Mormon - Alma 32:17 to 19
17 Yea, there are many who do say: If thou wilt show unto us a sign from heaven, then we shall know of a surety; then we shall believe.
18 Now I ask, is this faith? Behold, I say unto you, Nay; for if a man knoweth a thing he hath no cause to believe, for he knoweth it.
19 And now, how much more cursed is he that knoweth the will of God and doeth it not, than he that only believeth, or only hath cause to believe, and falleth into transgression?
I have wondered if my brother wouldn't believe his own senses, and other people's experiences (prophets etc), then would be really believe? Would he have the spiritual strength to live the commandments? If God appeared to him would he just called it a "delusion"? There have been many people who have seen angels and not been changed. You are right that belief is a choice. To me it is a choice to see and hear what is real. It is a choice to accept humbly that I need help in life and be grateful when I receive it. It is a choice to learn and grow from my own and others experiences. It is a choice to turn to the source of all light and love. It is a choice to believe. A choice that is confirmed and verified by the Holy Spirit by a sense of peace and assurance of right.