I have a more general query about some confusion ive been having lately deciding what the bayesian predictions are in my model. Suppose I have a model, $y_{j} = \mu_{j} + e_{j}$, (for which i have observed values $y_j$). I ran a bayesian approach, for which i have priors on $e_j$ and $\mu_j$. My data is distributed $y_j \sim N(\mu_j,e_j)$. If i were to simulate from the posterior distribution for $y_j$ (the posterior predictive distribution, conditioned on the data) using random number generator in RStan during the run, then is it correct to assume that these are my predictions for $y_j$. Or are my predicted values in fact the posterior predicted value for $\mu_j$ ? So, say if i wanted to plot observed vs predicted values, which quantity of the two (posterior mean for $\mu_j$ or posterior predictive distribution for $y_j$ from a random number generator $N(\mu_j,e_j)$) would make more sense to plot?
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When simulating, you can expect any values that are possible under the model. When making predictions you are usually interested in most likely values given the model. So it depends if you want to explore some possible contractual scenarios, or get the prediction right away (then you would take posterior mean, median, mode etc. depending on your needs). In this thread you can find example of predictions vs simulation from regression model.
