Running models "backwards" to get predictions

Forgive me if these are dumb questions - I’m still learning. Here I’m modeling the data generating process for a disease.

In my dataset, I have health metrics (just 1 here for simplicity) and the disease state (true or false). If a person (sample) has the disease, the health metric changes. Otherwise the metric stays at normal levels:

with pm.Model() as model:
    p_disease = pm.Uniform('p_disease', lower=0, upper=1)
    disease = pm.Bernoulli('disease', p=p_disease, observed=has_disease)  

    μ_metric1 = pm.Normal('μ_metric1', 0, sigma=10)
    ρ_metric1 = pm.Normal('ρ_metric1', 1.5, sigma=2)
    σ_metric1 = pm.HalfCauchy('σ_metric1', 1)
    data_metric1 = pm.Data('data_metric1', X_train)
    metric1 = pm.Normal('metric1', 
        # If there is disease, the metric changes. If not, the metric is at normal levels.
        tt.switch(tt.eq(disease,1),ρ_metric1*μ_metric1, μ_metric1), 

    trace = pm.sample()

I have a few questions in order of priority:

  1. Is it possible to predict whether someone has a disease (binary classification) using this model, given health metric data? Am I formulating the wrong model? Note: I have another model that uses logistic regression that works correctly, but I’m curious about this particular model.
  2. If I put the observed variable has_disease into its own pm.Data(), I run into optimization issues. What’s going wrong?
  3. Is there a way to handle missing data and still get predictions? pm.Data() doesn’t like it, and I use it during the prediction step, which looks like this:
with model:
    pm.set_data({"data_metric1": X_test})
    predictions = pm.sample_posterior_predictive(trace)

Thanks in advance!