the estimation of the minimal allowable stress is treated in a bayesian setting. We do not directly put a prior on the parameter $\beta_0$, but rather on the probability of failure. We consider a prior information of the form: *at a given stress level $s$, we can expect $k$ failures out of $n$ trials.* Denote $\pi_s$ the prior indexed on the stress level $s$. $\pi_s$ models the failure probability at level $s$ and has a Beta distribution given by $$\label{priorP}
\pi_{s} \sim \beta(k,n-k+1).$$

Let $R$ follow an exponential distribution: $\forall s \ge 0,  \psi(s,\beta_0) = p_s = 1 - \exp(- \beta_0 s)$.

It follows $ \forall s,~ \beta_0 = - \frac{1}{s}\log(1- p_s)$.

Define the random variable $\Lambda_s =  - \frac{1}{s}\log(1-\pi_{s})$ which, by definition of $\pi_s$, is distributed as an k-order statistic of a uniform distribution $U_{k,n}$.

The estimation procedure of the CRM is obtained as follows:

**Step 1.** Compute an initial estimator of the parameter $$\Lambda_{s} =  \frac{1}{L} \sum_{l=1}^L - \frac{1}{s}\log(1-\pi_{s}^{l} )$$ with $\pi_{s}^l \sim \beta(k,n-k+1), 1\le l\le L$. Define $$s_{1}^{\ast }:=\underset{s_j\in \{s_{1},...,s_{J}\}}{\text{arginf}}{|( 1 - \exp(- \Lambda_s s_j))-\alpha |}.$$ and perform $J$ trials at level $s_1^\ast$. Denote the observations $Y_{1,j}:={\Large 1}_{R_{1,j}<s_{1}^{\ast }},1\leq j\leq J.$

**Step i.** At iteration $i$, compute 