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?