How to add (soft?) constraints to the model?

Questions
1

Sorry for a newbie question. I have a simple model like so:

model = pm.Model()
with model:

  # Priors for unknown model parameters
  a = pm.Normal('a', mu=0, sd=10, shape=(4, 1))
  b = pm.Normal('b', mu=0, sd=10, shape=(1, 4))
  c = pm.Normal('offset', mu=0, sd=10)

  # Expected probabilities (4 x 4) 
  p = sigmoid(a + b + c)

  # Likelihood (sampling distribution) of observations, data, N_data = (4 x 4) 
  pm.Binomial('L', n=N_data, p=p, observed=data_successes)

Now I want to obtain parameters a, b and c via MCMC but I want to incorporate additional information like:

x0 = x1* f(a[0], b[1], c) + x2 * f(a[0], b[2], c)  + x3 * f(a[0], b[3], c) 
x4 = x5* f(a[1], b[0], c) + x6 * f(a[1], b[2], c)  + x7 * f(a[1], b[3], c) 
....

Where [x0, x1, …] are known (f is just a sigmoid), a, b and c are results of MCMC so they’re not “hard” constraints per se. Is this possible?

2

Depending on the function f(), it might not be easy to do. One approach is to add the constrain using pm.Potential, but I am not sure it applies here.

3

The function is sigmoid. I wonder how to incorporate this additional information I have to help with the learning… I’m not sure if this could be integrated into the prior either

4

It would be much easier if we have a solver for algebraic equation, but we dont…

5

Could you explain what you mean by a solver for algebraic equation & how would it help?

6

You can find more information in the stan manual: http://mc-stan.org/users/documentation/. See part 47. In brief it allows you to put down the system of equation in your model as constrain without specifically work out how to express the solution.