What is the item response theory (IRT) response probability?

IRT can be confusing and one aspect that is especially ill-defined the response probability, Pr(y=1|theta). Here, I work through my conceptualization of what the probability represents and an alternative way of thinking about it.

[ArgumentumAdHominem]

I think it is important to emphasize where the randomness comes from. If a person is asked to answer the same question with an interval of 1minute apart, there is a very high chance they would answer the question exactly the same way. The most of randomness comes from the fact that our model is not a perfect description of reality, and there are other hidden parameters about the person that we do not observe. So, given that hidden variable H, the probability P[y=1|theta, H] would be deterministically 0 or 1. However, due to lack of knowledge, we are forced to integrate over this hidden variable and consider the marginal distribution. To do that we need a prior P[H] which is the distribution of hidden variable values across the population, which is exactly the randomness that is acquired by the marginal distribution. Its exactly what you said in the second part