So there’s two ways to handle missing data:

- Multiple Imputation
- Modeling the missing data mechanism

With multiple imputation you generate multiple hypothetical datasets that have all the missing values imputed, you do your inference on each hypothetical dataset separately in Stan, then you combine the results from your posteriors. For this approach I would recommend the mice package in R, and the excellent accompanying book “Flexible Imputation of Missing Data” by Stephen Van Burren.

The second way is to treat your missing data as a random variable to be inferred by including it in your Stan parameters block. Then in the model block you can assign that random variable a distribution, or you can even have it be determined based on a regression using data that we do observed. For example in trauma data, we often have blood tests that are missing because a patient was in too bad of a shape for the clinicians to collect a blood sample. So you can treat the blood test result as a parameter to be inferred and have it be determined using a regression on the injury severity score of the patient.

So in your case it sounds like you have a regression of the form

\mathbb{E} [y_i] = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \beta_3 x_{i1} x_{i2}

where x_{i2} is occasionally missing. In that case you either multiply impute x_{i2} outside of Stan in a package like mice, or you can you can include it in the parameters block and infer it. If you do the latter it’s perfectly fine to have something like the following in Stan

```
data {
real y;
real x1;
}
parameters {
real b0;
real b1;
real b2;
real b3;
real x2;
}
model {
x2 ~ normal(1.2, 2.0);
y ~ normal(b0 + b1*x1 + b2*x2 + b3*x1*x2 , sigma);
}
```

This way every time you sample a new possible value of the missing value, x2, will be drawn and the regression is inferred that way, since the x2 is declared as a parameter. The transformed data block is only calculated once at the beginning of inference, and that same value is used throughout.