Confusing slice sampler

May 18, 2010

(This article was first published on Xi'an's Og » R, and kindly contributed to R-bloggers)

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

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

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.

left{ (a,b),: y_i(a+bx_i) > logdfrac{u_i}{1-u_i} right}

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

displaystyle{bigcap_{i=1}^n} left{ (a,b),: (-1)^{y_i}(a+bx_i) > logdfrac{u_i}{1-u_i} right}

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

To leave a comment for the author, please follow the link and comment on their blog: Xi'an's Og » R. offers daily e-mail updates about R news and tutorials on topics such as: Data science, Big Data, R jobs, visualization (ggplot2, Boxplots, maps, animation), programming (RStudio, Sweave, LaTeX, SQL, Eclipse, git, hadoop, Web Scraping) statistics (regression, PCA, time series, trading) and more...

If you got this far, why not subscribe for updates from the site? Choose your flavor: e-mail, twitter, RSS, or facebook...

Tags: , , , , , ,

Comments are closed.


Never miss an update!
Subscribe to R-bloggers to receive
e-mails with the latest R posts.
(You will not see this message again.)

Click here to close (This popup will not appear again)