Markov Chains and partial views/Search space
From GusWiki
- does this parametrization give us the full space of 4-state MCs that is a mutual refinement?
- continuous path from the Kronecker product
- X view has 2 basic constraints (rows add to 1) and 2 coarsening equations (1 for each state)
- Y view likewise
- joint matrix has 4 constraints
- total of 12 constraints
- Suppose each view has $k$ states.
- each view gives us $k^2$ constraints from coarsening ($k$ constraints for each state)
- thus the joint matrix has $(k^2)^2$ entries
- $k^4 - 3 k^2$
generalizing more:
- Suppose view $X$ has $p$ states and view $Y$ has $q$ states
- each state of view $X$ has $q$ constraints, thus view $X$ has $pq$ constraints
- likewise, view $Y$ has $pq$ constraints
- joint matrix has $pq$ basic constraints ($pq$ rows in joint matrix)
- joint matrix has $(pq)^2$ entries
- thus, we have $(pq)^2 - 3pq$ degrees of freedom
The property of being a mutual refinement of $X$ and $Y$ is defined by satisfying linear constraints.
Each linear constraint defines a linear subspace.
The intersection of any two linear subspaces is either a linear subspace or empty.
Hence the set of all mutual refinements is either a linear vector space or empty.
It is not empty, since Kronecker product is in the set.
Since every linear subspace is simply connected, every mutual refinement can be reached by a continuous path (in which every point is a mutual refinement) from the Kronecker products.
Therefore, we define a parameterization of the subspace of mutual refinements, starting with the Kronecker product as the origin.
