There are two kinds of evidence that the psychological realm is causally deterministic. First, there is indirect evidence. Everything that happens in the mind is a function of what happens in the brain. The brain is physical. The physical world is deterministic (at least to a high degree of approximation). Therefore, the mind is deterministic.
Second, there is direct evidence taken from our knowledge of psychology itself. When we know a person’s motives and beliefs, we can predict their behavior infallibly. People are predictable to the extent that they are known. Prediction is possible only where there is causal determinism. (If I push the elevator button, I can predict that the elevator will come if, and only if, the causal mechanisms linking the two events are intact.) Therefore, since people are predictable to the extent that they are known, they are causally determined to the extent that they are known. And given that they are causally determined to the extent that they are known, it stands to reason (and would be very arbitrary to deny) that they are causally determined through and through.