Confusing slice sampler

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 y_i=1 or y_i=0. The first case produces the equivalent condition

and the second case 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.

This set is missing (a) an intersection sign before the curly bracket and (b) a instead of the . It should be