Suppose that my application's users are asked to give their opinion about the probability that a statement is true. They are presented with a slider widget that goes from 0% to 100%, but let's say the slider goes from 0 to 1. Due to the discrete nature of computing, there are a finite number of possibilities for their answer. Perhaps it is even preferred to limit the number of ticks on the slider to, say, 101 (including a tick for 0), because it's unlikely that the precision of our estimates for the probability that a statement is true allows for, say, an estimate like 0.9888888834.
Suppose that I want to construct a latent variable representation of this ordinal data, and the latent variable is the probability that a statement is true. Well, standard probit regression would have me assume that the latent variable is normally distributed. But the latent variable can only be approximately normally distributed because the latent variable is a probability of truth, which is constrained to being between 0 and 1. So my question is whether it is worth it to investigate other probability distributions for the latent variable, or if the normal case is good enough for most purposes. And for a more specific question, I wonder if I could represent it with a logit link.