Hi all,

I am currently trying to implement a colleague’s structural reliability code in Stan. The gist of it is as follows: we have a database of successes and failures y and their associated measured predictors **x**. Reliability theory assumes that these outcomes can be explained by a limit state function g(x, \theta), with \theta being vector of model parameters. The limit state function is negative for failures and positive for successes. The idea was to use Bayesian updating to determine the posterior distributions of model parameters.

The original derivation of the likelihood looks, to me, almost identical to a latent variable probit model:

First, they added a model error term to account for an imperfect specification of the limit state function.

g = \hat{g}(x_i, \theta) + \epsilon_i, \epsilon \sim N(0,\sigma_e)

The original data had six predictor variables and used the following form of the limit state:

\hat{g}(x, \theta) = x_1*(1+\theta_1*x_2) +(\theta_2*x_2) + x_3 *(1+\theta_3*x_2) - \theta_4*ln(x_4) - \theta_5*ln(x_5) - \theta_6*ln(x_6) - \theta_7

Pr(y = 0 | x) = Pr (g < 0 | x) = \Phi (-\frac{\hat{g}(x, \theta)}{\sigma_e})

Pr(y = 1 | x) = Pr (g > 0 | x) = \Phi (\frac{\hat{g}(x, \theta)}{\sigma_e})

\ell(\theta, \sigma_e | x) = \prod_{i = 1}^k \Phi (-\frac{\hat{g}(x_i, \theta)}{\sigma_e}) *\prod_{i = k + 1}^n\Phi (\frac{\hat{g}(x_i, \theta)}{\sigma_e}) for k cases of failures and n total cases.

Now for my question. It is my understanding that in the latent variable motivation for probit models the scale parameter is not identifiable because scaling it and the coefficients by the same constant would give the same results. Does this formulation have the same limitation?