Why scale of outcome affect model result

[serenejiang]

I wrote a model with Stan and it works well if I scaled my outcome variables to have mean zero and standard deviation 1 (with All Pareto k estimates are good < 5); however, if I do not standardized my outcome, then the model is performing poorly (all Pareto k diagnostic values are very bad). I do not understand why the range of outcome matters so much. Is this related to the distribution of prior distribution, which were normal (0, 1) or cauchy (0, 1)?

[mitzimorris]

there’s a discussion on a previous thread:

Standardizing predictors and outputs in a hierarchical model

Hello everyone, First of all thanks for the StanCon—it was a great experience. Here is a question that I have been discussing there and many suggested that I post it on the Forum. My question is about Manual Section 28.11 Standardizing Predictors and Outputs (v. 2.17.0), where it is suggested that we standardize our inputs as part of optimization. The model written is a very simple one predictor population variable model of y_{n} = alpha + beta * x_{n} + epsilon_{n}. I was wondering how to ba…

it would help clarify the problem if you could include your model (if it’s simple and you can share), or a simplified version which has the essential behavior here.

[serenejiang]

I read that post earlier, but still unclear about what to do with my case.
Below is a simplified version of my model. I am wondering how I can fit my model well without pre-standardizing my response Y.

model {
 // prior distributions
  sigma_eps ~ cauchy(0, 1); 
  theta_mu ~ normal(0, 1);

  for(k in 1:K){
    Theta[, k] ~ normal(0, 1);
  }

  // posterior draws
  for(n in 1:N){
    alpha[n] ~ multi_normal(zero_k, diag_matrix(rep_vector(1,K)));
    Y[n] ~ multi_normal(B[n]*theta_mu + B[n] * Theta * alpha[n], diag_matrix(rep_vector(sigma_eps^2, V[n])));
  }
} 

[serenejiang]

Forgot to mention that V, B, and Y come from real data.