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

and the second case the equivalent condition

alpha+beta x_i < - log(u_i/(1-u_i))

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.

This set is missing (a) an intersection sign before the curly bracket and (b) a (1-)^y_i instead of the y_i. It should be

Filed under: Books, R, Statistics Tagged: auxiliary variables, Introducing Monte Carlo Methods with R, logistic regression, slice sampling

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