Many writers use believe when they really mean think. For the most part, you think with your brain, believe with your soul and feel with your senses. So: “The police think they have the right person in custody,” not believe. Other times, this isn’t so clear-cut, as in “He believes he saw aliens at the Missouri game” or “She felt her husband needed more compassion.” So always think about the proper use of these words. See Bremner.
A word of caution: We never truly know what other people believe, feel or think. We can’t read their minds or see into their souls. (If you can, don’t waste your time in journalism. Head straight to the casino.) We know only what they say. So much of the time, we need attribution when we say that someone believes, feels or thinks something, as in:
“The police said they thought they had the right person in custody.”
“She believes in aliens, sea monsters and the wisdom of politicians, she said.”
“Her classmates said they felt queasy whenever they saw her.”
Don’t overdo it, though. If we insist on attributing every such word or sentence, we risk looking silly.