Most embarrassingly, Liaosa Xu from Virginia Tech sent the following email almost a month ago and I forgot to reply:
I have a question regarding your example 7.11 in your book Introducing Monte Carlo Methods with R. To further decompose the uniform simulation by sampling a and b step by step, how you determine the upper bound for sampling of a? I don’t know why, for all y(i)=0, we need a+bx(i)>- log(u(i)/(1-u(i))). It seems that for y(i)=0, we get 0>log(u(i)/(1-u(i))). Thanks a lot for your clarification.
There is nothing wrong with our resolution of the logit simulation problem but I acknowledge the way we wrote it is most confusing! Especially when switching from to in the middle of the example….
Starting with the likelihood/posterior
we use slice sampling to replace each logistic expression with an indicator involving a uniform auxiliary variable
[which is the first formula at the top of page 220.] Now, when considering the joint distribution of
we only get a product of indicators. Either indicators that
or of ,
depending on whether yi=1 or yi=0. The first case produces the equivalent condition
This is how we derive both uniform distributions in and $beta$.
What is both a typo and potentially confusing is the second formula in page 220, where we mention the uniform over the set.
instead of the . It should be