Confusing slice sampler

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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 (alpha,beta) to (a,b) in the middle of the example….

Starting with the likelihood/posterior

L(alpha, beta | mathbf{y}) propto prod_{i=1}^n left(dfrac{e^{ alpha +beta x_i }}{1 + e^{ alpha +beta x_i }}right)^{y_i}left(dfrac{1}{1 + e^{ alpha +beta x_i }}right)^{1-y_i}

we use slice sampling to replace each logistic expression with an indicator involving a uniform auxiliary variable

U_i sim mathcal{U}left( 0,dfrac{e^{ y_i(alpha +beta x_i) }}{1 + e^{ alpha +beta x_i }} right)

[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

u_i<text{logit}(alpha+beta x_i) or of u_i<1-text{logit}(alpha+beta x_i),

depending on whether yi=1 or yi=0. The first case produces the equivalent condition

This is how we derive both uniform distributions in alpha 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 y_i. It should be

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