Commuting birth-and-death processes Caroline Uhler Department of Statistics UC Berkeley (joint work with Steven N. Evans and Bernd Sturmfels) MSRI Workshop on Algebraic Statistics December 18, 2008
Birth-and-death processes Simple Markov chain on Applications: - modeling populations - queueing theory
Birth-and-death processes Transition matrix is tri-diagonal: is easily diagonalizable using spectral decomposition. is straightforward to compute.
Birth-and-death processes in dim 2 Markov chain on Transition probabilities: with (assume absorbing state)
Main goal Problem: Higher dimensional birth-and-death processes are not timereversible in general. So 1-dim theory does not extend. Goal: 1) Find a class of birth-and-death processes for which diagonalization via spectral decomposition is feasible (like in 1 dim). 2) Identify and understand the constraints defining this class.
A property Commuting Spectral theorem for commuting self-adjoint matrices: Let be a set of self-adjoint matrices satisfying Then can be diagonalized simultaneously. Example: 2-dimensional birth-and-death process Decompose then is diagonalizable. ( : horizontal, : vertical). If Click to add title
Example: 2x1 grid
A Commuting property for dim 2 The probability of going from one corner of a square in the grid to the diagonally opposite corner of the square in 2 steps is the same for both paths.
Example: 2x1 grid If then the commuting relations are equivalent to and
Parametrization of 2 x 1 example Rank constraints imply the following parametrization of the commuting variety
Parametrization Theorem: Suppose that for all edge-connected in Then the transition matrices for each coordinate direction commute iff for some constants when The constants constants and and that satisfy for some are unique up to a common multiple, and the are unique.
Identifying independent constraints The set of constraints is redundant if for all edge-connected Take linear algebra approach to identify independent set of constraints. Note that taking logarithms yields constraints of the form
Constraint matrix Let denote the constraint matrix that has columns indexed by neighbors and one row for each constraint. Example: 2 x 1 grid Remark: has format 4(# squares in grid) x 2(# edges in grid):
Design matrix Let denote the design matrix that has columns indexed by neighbors and one row for each parameter. Example: 2 x 1 grid Remark: has format (# parameters) x 2(# edges in grid):
Identifying independent constraints Corollary: The two vector spaces spanned by the rows of the matrix and the rows of the matrix are orthogonal complements. Lemma: The rank of the matrix is one less than the number of rows: Theorem: The rank of the constraint matrix is:
The toric ideal Definition: An integer matrix of rank is unimodular if all its non-zero minors have the same absolute value. Theorem: (Sturmfels) Let be any unimodular matrix. Then every reduced Gröbner bases of the toric ideal consists of differences of squarefree monomials. Moreover, the following three sets coincide: the union of all reduced Gröbner bases, the set of circuits, and the Graver basis of
Unimodularity Theorem: The design matrix format or is unimodular if and only if the grid has for some Outline of the proof: grid: grid: are non-squarefree circuits. grid: show that Graver basis is squarefree using 4ti2. grid: proof using matroid theory.
Minimal non-unimodular examples grid: grid:
Boundary components What can we say if nearest neighbor transition probabilities are allowed to be zero? Perform primary decomposition of ideal generated by the four constraints over each square in the grid in polynomial ring :
Primary decomposition of 2 x 1 example The ideal is the intersection of 11 prime ideals:
Binomial primary decomposition Primary decomposition is hard: grid example: Singular: no memory. Bernd & Singular: enough memory. is the intersection of 199 prime ideals. is the intersection of 135 prime ideals. Implementation of binomial primary decomposition in Singular is desirable. Conjecture: The binomial ideal is radical.
Paper appeared on the arxiv this week: Evans, Sturmfels, U. Commuting birth-and-death processes. arxiv:0812.2724v1 T k n ha! u yo