I am an engineering student and grappling with some statistical concepts for my research study. I would like to get some suggestions on how to tackle this problem properly.
Problem description (see https://doi.org/10.1115/1.2204969 for more details): Let the reliability $R = \Pr( g(X) > 0 \mid d_k)$ where $\Pr( )$ is the probability, $g( )$ is some function (limit state), $X$ are the random variables and d are deterministic variables or 'observed quantities'. Now I want to infer the distribution of R when several values of $d_k$ are observed. I used the Bayesian inference such that
$$f(r\mid d_k) \propto f(d_k\mid r) \times f(r)$$
where a binomial likelihood is used for $f(d_k\mid r)$ is used and a uniform (i.e. beta(1,1,) ) is used for $f(r)$ and the posterior can be easily derived using the Beta-Binomial conjugate pair. My question is if instead the reliability $R$ is expressed as an interval i.e., $R_L < \Pr( g(X) > 0 | d_k) < R_U$ where the reliability is only known within an interval with lower bound $R_L$ and upper bound $R_U$. Thus I want to know the new distribution of this interval using Bayesian inference:
$$f(r_L, r_U\mid d_k) \propto f(d_k\mid r_L, r_U) \times f(r_L, r_U)$$
Thus, my question is how I would set my prior, likelihood, and posterior distribution for this case. If you have some textbooks or readings as reference for a similar problem, kindly share it to me.
