Okay, would this also work:
theta_prior = pm.Normal('theta_prior',mu=0,sd=10)
theta = pm.Normal('theta',mu=theta_prior,sd=4,observed=2)
for the priors?
For the likelihood, the use case would be (for example) if we had 3 time series: t1, t2 and t3.
Then we would have coefficients on each: theta1, theta2 and theta3. We want to fit each to their subsequent data, but we also know that theta`` and theta2havetheta3` as a prior.
So we would fit (in Stan):
target += t1 (fitting theta1)
target += t2 (fitting theta2)
target += t3 (fitting theta3)
and the prior has
theta1 ~ normal(theta3,1)
theta2 ~ normal(theta2,1)
or something similar.
So the likelihoood is fitting mutliple things simultaneously.