Multiple Dirichlet sampling under linear constraints

1

Note sure if PYMC is the best place for this question as I am not sure that it can be done by sampling, but here it is:

Suppose that I have the following:

  • Three Dirichlet distributions of dimension k: distr1, distr2, distr3
  • A vector A of dimension 3 and a vector B of dimension k.

Does anyone know about a way to find 3 probability vectors p1, p2 and p3 such that:

  1. p1, p2 and p3 are in an area of high probability for distr1, distr2, distr3 respectively
  2. They satisfy the linear constraint A[0]*p1 + A[1]*p2 + A[2]*p3 = B

(You can assume that the set of probability vectors p1, p2, p3 satisfying the linear constraint is not empty)

Intuitively the idea is to sample the probability vectors from each Dirichlet probability distributions, but shift the sampling in directions so that we get closer to the linear constraints being satisfied.

Any help or direction appreciated in case you have ever seen a similar case of constrained problem

thank you!

2

Are you talking about something like this:

with pm.Model():
  p1 = pm.Dirichlet("p1", a1)
  p2 = pm.Dirichlet("p2", a2)
  p3 = pm.Dirichlet("p3", a3)
  sigma = pm.HalfNormal("sigma", 1)

  pm.Normal("likelihood", A[0]*p1 + A[1]*p2 + A[2]*p3, sigma, observed=B)

In this case a1,a2,a3 represent the concentration parameters of distr1, distr2, distr3 which sets the priors for your vectors p1,p2,p3. p1,p2,p3 are then optimized within this constraint such that the linear condition is satisfied with some noise. I don’t quite understand what you mean by “shift the sampling”, but Bayesian method is on some sense doing something like that, finding best p1, p2, p3 given the “Dirichlet constraints”. Or am I misunderstanding this question?