Truncated Poisson vs ordered multionomial + beta regression I want to model a random variable that takes values between 0 and 1, and where 0 and 1 are included. Zero and one are possible values of the random variable and occur frequently. I have several covariates available
I think I have (at least) two options:
(a) Use an ordered logit for which part of the interval we are in: (1) 0, (2) the interval (0,1) and (3) the point 1. Afterwards I  can use a beta regression model for part (2).
(b) Discretize the interval [0,1] into say 10 equally spaced "intervals" 0,(0,0.1),[0.1,0.2),[0.2,0.3),[0.3,0.4),[0.5,0.6),[0.6,0.7),[0.7,0.8),[0.8,0.9),[0.9,1) 1 and use truncated Poisson, where I truncate at value 10.
Do you see any other options? What do you think about the options?

Instead of (a) you also see practitioners using a multinomial logit (rather than ordered logit) for the first step.
In (b) you would typically censor the distribution rather than truncate it. Also, I wouldn't use a distribution for discrete data but a distribution for continuous data, e.g., a normal distribution (censored at both 0 and 1).
Additionally, there are other similar extensions like the "ordered beta regression" proposed by Robert Kubinec in Political Analysis, see: doi:10.1017/pan.2022.20.
