I’m working on a model of advice taking. The full model is fairly complex, so I’m going to start with a simplified version and can include the more complicated version if it comes to it.

The basic idea is I am testing how people take advice based on imputed values for their subjective prior estimates. When everything is observed (no missing data) the model is as follows

b_2 = (1 - w)b_1 + (w)a

Where b_1 is a person’s prior belief, a is the advice they receive, b_2 is their posterior belief, and w is a weighting parameter.

In my version of this model, b_1 is missing data that is imputed. For each person, there are 25 cases where b_{1,x} is observed and 25 cases where b_{1,y} = is missing (I’m mostly interested in the missing cases). I have a separate sample where all 50 items are observed, and because they are relatively highly correlated, I can fit an imputation model

\pmb{b_{1,y}} = \pmb{\alpha Z} + \pmb{b_{1,x}\beta} + \pmb{\epsilon}

I then estimate a measurement model for each person’s advice-taking tendency for the items where their prior beliefs were unobserved, which is given a hierarchical prior for an overall population-level w parameter. In Stan the code looks like this

```
data{
int<lower=0> N_obs; //N for fully observed sample
int<lower=0> N_mis; //N for partially observed sample
int<lower=0> K_obs; //Number of items always observed
int<lower=0> K_mis; //Number of pitems partially observed
matrix[N_obs, K_mis] b1_y_obs; //Prior beliefs for partially observed items and fully observed sample
matrix[N_obs, K_obs] b1_x_obs; //Prior beliefs for fully observed items and fully observed sample
matrix[N_mis, K_obs] b1_x_mis; //Prior beliefs for fully observed items and partially observed sample
matrix[N_mis, K_mis] advice_mis; //advice values for partially observed sample
matrix[N_mis, K_mis] b2_mis; //posterior beliefs for partially observed sample
}
parameters{
//imputation model pars
matrix[K_mis, K_obs] b; //coefficients for imputation model
vector[K_mis] b0; //intercepts for imputation model
vector<lower=0>[K_mis] sd_imputation; //standardized SD for imputation
matrix[N_mis, K_mis] y_mis; //imputed values
//measurement model pars
real w; //population parameter for advice weight
real<lower = 0> sd_w; //variance of judge level advice weight parameters
vector[N_mis] j_w; //judge level advice weight parameters
reall<lower = 0> sd_post; //error variance for advice weighting model
}
model {
target += normal_lpdf(j_w | w, sd_w); //hierarchical prior for judge level parameters
//(other priors leftt out for simplicity)
for (k in 1:K_mis){
//imputation model
mu_obs[,k] = b0[k] + (b[k,] * b1_x_obs')';
mu_mis[,k] = b0[k] + (b[k,] * b1_x_mis')';
target += normal_lpdf(b1_y_obs[,k] | mu_obs[,k], sd_imputation[k]);
target += normal_lpdf(b1_y_mis[,k] | mu_mis[,k], sd_imputation[k]);
//measurement model
for (n in 1:N_mis){
target += normal_lpdf(b2_mis[n,k] | (1 - j_w[n]) * b1_y_mis[n,k] + j_w[n] * advice_mis[n,k], sd_post);//,
}
}
```

I can get this model to work just fine. The problem is I **also** want to include a version that references the absolute difference between the imputed prior beliefs and the advice, because there’s a large literature on how the distance between prior and advice influences advice weighting (see things like “egocentric discounting”). The new model looks something like this (I don’t have enough data to fit the extra parameters hierarchically, particularly in the complex version of the model, but this data is experimental and the judges are exchangeable):

```
parameters{
//imputation model pars
matrix[K_mis, K_obs] b; //coefficients for imputation model
vector[K_mis] b0; //intercepts for imputation model
vector<lower=0>[K_mis] sd_imputation; //standardized SD for imputation
matrix[N_mis, K_mis] y_mis; //imputed values
//measurement model pars
real w; //population parameter for advice weight intercept
real <lower = 0> sd_w; //variance of judge level advice weight intercepts
vector[N_mis] j_w; //judge level advice weight intercept
real<lower = 0> sd_post; //error variance for advice weighting model
real bdif; //slope for difference between imputed prior belief and advice
}
transformed parameters {
abs_dif = fabs(b1_y_mis - advice_mis);
matrix[N_mis, K_mis] w_adv;
for (k in 1:K_mis){
for (n in 1:N_mis){
w_adv[n,k] = j_w[n] + bdif * abs_dif[n,k];
}
}
}
model {
target += normal_lpdf(j_w | w, sd_w); //hierarchical prior for judge level parameters
//(other priors leftt out for simplicity)
for (k in 1:K_mis){
//imputation model
mu_obs[,k] = b0[k] + (b[k,] * b1_x_obs')';
mu_mis[,k] = b0[k] + (b[k,] * b1_x_mis')';
target += normal_lpdf(b1_y_obs[,k] | mu_obs[,k], sd_imputation[k]);
target += normal_lpdf(b1_y_mis[,k] | mu_mis[,k], sd_imputation[k]);
//measurement model
for (n in 1:N_mis){
target += normal_lpdf(b2_mis[n,k] | (1 - w_adv[n,k]) * b1_y_mis[n,k] + w_adv[n,k] * advice_mis[n,k], sd_post);//
}
}
}
```

The problem, as far as I can tell, is that the imputed missing values, b1_y_mis, are now showing up in the measurement model more than once, because w_adv is also a function of b1_y_mis. Because the likelihood of this model is also informative for estimating b1_y_mis, the model is not identified (at least, I think this is the problem, and it appears to be a reasonable explanation for what’s happening when I check the pairs plots). Ideally, the likelihood of the measurement model would not influence b1_y_mis, only the imputation model.

This thread seemed to suggest one approach would be to avoid the “fully Bayesian” estimation of doing everything in a single joint likelihood, and instead do something more like multiple imputaition, where random values are first extracted from the imputation model, then a measurement model is fit separately on those to account for the uncertainty in the estimates, which seems feasible, though like it could be slow and time consuming to implement (again especially because the “true” model I’m working with is quite a bit more complex than this, though the problem shows up when I try this version as well).

But this post and reply seemed to be more along the lines of what I originally had in mind, though the follow-up thread seemed to lead somewhere quite different.

Appreciate any advice anyone has, and happy to provide more info! Just figured this would be a good starting point to avoid (hopefully) unnecessary complexity.