This is a silly question, but I have tried different reparametrizations and priors without any success.

I have an ordered data vector as an input. This ordered data vector is scaled by an unknown parameter `theta`

. When I give Stan the true value of `theta`

, I can correctly infer the parameters `delta`

and `T`

associated with the distribution of the unscaled data vector(divided by `theta`

).

```
data{
int<lower=2> n;
vector<lower = 0.0>[n] sorted_vector_scaled_by_theta;
}
parameters{
real<lower=0.001> theta;
real<lower=0> hyper_parameter;
real<lower=0> delta;
real<lower=0>T;
}
model{
theta~exponential(10);
//log_theta= log(theta);
//log_theta ~ uniform(1, 1000);
//theta~lognormal(20, 100);
//log_theta~student_t(3,1,3);
//target += -log_theta;
hyper_parameter~gamma(0.001, 0.001);
delta~exponential(hyper_parameter);
target +=log_likelihood(n, delta, T, (1.0/theta)* sorted_vector_scaled_by_theta);
}
```

Where log_likelihood is custom likelihood function of sorted_vector_scaled_by_theta given delta, n and T.

I have tried different priors for theta: uniform, log uniform, log normal,… but none of those can recover the true values of delta and T. Can anyone help me with this modeling problem?.

P:D: I have thought to model ` log(sorted_vector_scaled_by_theta)`

instead and then every element is shifted in the same value:` log(theta)`