Algebra for logistic regression slice sampler I am having some difficulties when trying to do a little bit of algebra from Example 7.11 from the book "Introducing Monte Carlo Methods with R: Robert & Casella"
The example relates to logistic regression using a slice sampler. The model is (omitting $i$ index)
$$y \sim Bern(p(x)), \qquad p(x) = \frac{\exp(a + bx)}{1 + \exp(a + bx)}$$
And assuming flat priors on (a,b) the posterior distribution is given by
$$ p(a,b) = \prod_{i=1}^n  \frac{\exp(y_i(a + bx_i))}{1 + \exp(a + bx_i)}$$
The first step of the slice sampler is given as
$$ U_i \sim\mathcal U \Bigg(0, \frac{\exp(y_i(a + bx_i))}{1 + \exp(a + bx_i)}\Bigg) $$
and the second as
$$ \mathcal U\left\{ (a,b): y_i(a+bx_i) > \log \frac{u_i}{1-u_i}\quad i=1,\ldots,n \right\} $$
I am struggling with this second step. I cannot get these values if I work backwards:
\begin{align}
y(a+bx) &> \log u/(1-u) \\
\exp(y(a+bx)) &> u/(1-u) \\
(1-u)*\exp(y(a+bx)) &> u \\
\exp(y(a+bx)) &> u + u\exp(y(a+bx)) \\
\frac{\exp(y(a+bx))}{1 + \exp(y(a+bx))} &> u \\
\end{align}
The final term on the left-hand side does not equal the posterior distribution,whic I belive should be of the form $\{x: f(x) \geq U_i\}$. Where is my mistake please?
 A: Sorry that this section proves a challenge! (I corrected the question to indicate that the Uniform on $(a,b)$ is imposing the inequality for all $i$'s, as this is (too) implicit in the book.) There is alas a typo in the book when defining the set (second displayed formula on p.220), not a mistake in your reasoning!
The constraint on $(a,b)$ is thus that for all $1\le i\le n$,
$$\frac{\exp(y_i(a + bx_i))}{1 + \exp(a + bx_i)}\ge u_i$$
hence when $y_i=0$
$$u_i^{-1}\ge 1 + \exp(a + bx_i)$$
or
$$\log\frac{1-u_i}{u_i}\ge a + bx_i$$
and when $y_i=1$
$$1-u_i\ge \frac1{1 + \exp(a + bx_i)}$$
or
$$\log\frac{u_i}{1-u_i}\le a + bx_i$$
Meaning that
$$ \max_{y_i=1}\left\{ \log\frac{u_i}{1-u_i}-bx_i\right\} \le a \le \min_{y_i=0} \left\{ \log\frac{1-u_i}{u_i}-bx_i\right\}$$
and the same for $b$. Thus the slice distributions (and the R code below) remain valid.
Apologies!!! I went back to our (LaTeX) source file for this chapter on Gibbs samplers and found that we had written

...variables. Generating a uniform distribution over the set $$
  \bigcap_{i=1} \left\{ (a,b)\,:\ (-1)^{y_i}(a+bx_i) >
  \log\frac{u_i}{1-u_i} \right\} $$ being rather unwieldy, we can
further decompose the uniform simulation by consecutively simulating...

Which is correct. For some reason I can neither fathom nor remember, both the intersection and power terms did not make it to the printed versions... (Including the French and Japanese translations!)
