Can NUTS help me to find a set of parameters with largest likelihood if my model is identifiable only up to rearrangement?

One of the parameter is a matrix \mathcal{A}. Let’s say matrix A_{1} is a possible value of \mathcal{A}.

I can form a new matrix A_{2} by the following two steps:

- swap i^{\text{th}} and j^{\text{th}} row of A_{1} to form B
- swap i^{\text{th}} and j^{\text{th}} column of B to form A_{2}

Likelihood of A_{1} is same as likelihood of A_{2}.

I’m using a Dirichlet prior for \mathcal{A}.