Hi,

A question about interpretation - predictive distribution versus missing data. Assume a series of exchangeable experiments (w/ data **y_1** and **n_1** below) inform a parameter (the logit-probability **mu_1**). In another series of exchangeable experiments, another (different) parameter (the logit-probability **mu_2**) can be estimated from data ( **y_2** and **n_2**).

A third(!) series of exchangeable experiments were performed where only data on a function of those two initial parameters are quantified but quantities directly informing the initial parameters themselves weren’t. I use data from 3rd series of experiments to improve estimation of the two first quantities, given finite number of experiments (in an evidence synthesis context).

My question: the synthesis of these different datasets requires predicting the unobserved quantities (**ytilde_1** and **ytilde_2**) for the 3rd set of experiments where these quantities are not observed (but a function of their values is). The sampling of **ytilde_1** and **ytilde_2** in the model block here – can it be interpreted as posterior predictive simulations (even if not in generated quantities block and not sampling using _rng) or would the correct interpretation be as missing data? (noting the information flow from **ytilde_1** and **ytilde_2** to other parameters - which is the goal and clear from running the model). It would be great to know your thoughts (sorry for long - possibly unclear - description)

data{

int N_1; // number of experiments - informing parameter 1

int N_2; // number of experiments - informing parameter 2

int N_3; // number of experiments - informing combination of parameters 1 and 2

int y_1[N_1];

int y_2[N_2];

int n_1[N_1];

int n_2[N_2];

int n_3[N_3];

int y_3[N_3];

}

parameters{

real mu_1;

real mu_2;

real<lower=0> tau_1;

real<lower=0> tau_2;

vector[N_1] mu_study_1;

vector[N_2] mu_study_2;

vector[N_3] ytilde_1;

vector[N_3] ytilde_2;

}

transformed parameters{

vector[N_3] p_3 = inv_logit(ytilde_1).*inv_logit(ytilde_2); // Function of quantities derived from two initial parameters

}

model{

// Priors

mu_1 ~ normal(0, 5);

mu_2 ~ normal(0, 5);

tau_1 ~ normal(0, 1);

tau_2 ~ normal(0, 1);

```
//Hierarchical Priors
mu_study_1 ~ normal(mu_1, tau_1);
mu_study_2 ~ normal(mu_2, tau_2);
// Likelihood
y_1 ~ binomial_logit(n_1, mu_study_1);
y_2 ~ binomial_logit(n_2, mu_study_2);
// Predicting unobservable values of first two parameters in 3rd set of experiments
ytilde_1 ~ normal(mu_1, tau_1);
ytilde_2 ~ normal(mu_2, tau_2);
// Likelihood (3rd set)
y_3 ~ binomial(n_3, p_3);
```

}