I’m playing with transformations, currently with a toy model that mirrors what I aim to later do in a larger model. Here is my simple toy model:

x_{i,j} \sim \rm{Normal} (m_{i}, \sigma)

m_{i}=1/\hat m_{i}

\hat m_{i} \sim \rm{Normal}(0,\tau)

\sigma \sim \rm{Half-Cauchy}(2.5)

\tau \sim \rm{Half-Cauchy}(2.5)

The code below includes the determinant of the inverse transformation m_{x,i}=1/m_{y,i}, i.e. -2 log(m_{x,i}), and as far as I can get this to work, I think this seems to be OK. Yet, I get quite poor mixing, especially of \tau, as is demonstrated in the figure below. I’ve tried a variable exchange trick that usually works for hierarchical models, here `m_hat[i]= m_hat_raw[i] * tau`

and then `m_hat_raw[i] ~ normal(0,1)`

, but it doesn’t improve much, so there is presumably something more I should adjust. I’ve used simulated data and, to give Stan a best case scenario, seeded on the values used for simulation. Any suggestions for how to improve mixing in a model like this?

Edit: the figure is from an example with:

N=50

n=10

\tau=0.1

\sigma=1

```
data {
int N;
int n;
real x[N,n];
}
parameters {
real m_hat_raw[N];
real<lower=0> sigma;
real<lower=0> tau;
}
transformed parameters{
real m[N];
real m_hat[N];
for (i in 1:N){
m_hat[i]= m_hat_raw[i] * tau;
m[i]=1/m_hat[i];
}
}
model {
for (i in 1:N){
for (j in 1:n){
x[i,j]~normal(m[i],sigma);
}
// m_hat[i]~normal(0,tau);
m_hat_raw[i] ~ normal(0,1);
target += log(m[i]^-2);
}
tau~cauchy (0,2.5);
sigma~cauchy (0,2.5);
}
```