Guidance on Heirarchical model with group interceptf

Modeling
1

Hi,

I’m having a very difficult time getting a rather simple model to fit. (traces are wandering, but chains not mixing, and no “fuzzy caterpillar” of samples.). No matter what I try (centered, non-centered, diffuse priors, informative priors) I can’t get good posterior samples.

I eventually want to add many. more things to this model, but need to get the basic version working first.

The base model

data {
    int<lower=0> N;
    vector[N] y; 
  
    int<lower=0> J; // Number of groups
    array[N] int group;
}

parameters {
    real<lower=0> a0;
    real<lower=0> sg;

    real<lower=0> a_group_tau;
    vector<lower=0>[J] a_group;
}

model {
    a0  ~ normal(15, 0.1);  // Informative prior taken from data
    sg ~ normal(0, 0.5);    // Informative prior taken from data


    a_group_tau ~ normal(0, 0.1); // Informative
    a_group ~ normal(0, a_group_tau);

    for (n in 1:N) {
    real mu = a0 + a_group[group[n]];
        y[n] ~ normal(mu, sg);
    }
}

Here is the non-centered model:

// non-centered version
// informative priors
data {
    int<lower=0> N;
    vector[N] y; 
  
    int<lower=0> J; // Number of groups
    array[N] int group;
}

parameters {
    real<lower=0> a0;
    real<lower=0> sg;

    real<lower=0> group_scale;
    vector<lower=0>[J] group_eta;
}
transformed parameters {
    vector[J] a_group;
    a_group = group_scale * group_eta;
}

model {
    a0  ~ normal(15, 0.1);  // Informative prior taken from data
    sg ~ normal(0, 0.5);    // Informative prior taken from data


    group_scale ~ normal(0,0.5);
    group_eta   ~ normal(0,1);

    for (n in 1:N) {
    real mu = a0 + a_group[group[n]];
        y[n] ~ normal(mu,  sg);
    }
}

The posterior samples I get look “reasonable”. They’re in the ranges I would expect. But, the R-hat is bad, and trace plots look terrible.

Screenshot 2024-03-28 at 5.26.48 PM
Screenshot 2024-03-28 at 5.26.48 PM1476×954 102 KB

My guess is that somehow the math is oscillating between putting weight into the population mean and the group means.

What I’m looking for is some guidance on how to resolve this. What steps can I take to find better priors, or a better way to parameterize my model. I plan to add a few more random effects, and would love to have a principled workflow on how do that well.

Any and all suggestions would be appreciated!!

2

Your intercept in the model block is v0, but you have no such parameter. You have a parameter a0, that you don’t use. So should v0 be a0? I’m surprised the model runs at all given that v0 is not declared - I would have expected a compile error.

3

Oh, typo in my post. I’ll fix it now.
(The actual model doesn’t have this typo)

Thanks for catching this!