P 1124p 3112

Only 2 node pairs are not connected by an edge; the corresponding model is

Each such conditional independence statement can be translated into a system of quadratic polynomials in r{Yi>Y2>Y3>Y4}. The representation of independence statements as polynomial equations is an implicit representation of the toric model, where the common zero set of the polynomials represents the model.

For binary alphabets £, the following eight quadric forms compose the set Qmg representing the probability distribution from the example above.

P0001P1000 — P0000P1001 j P0001P0010 — P0000P0011 j P0011P1010 — P0010P1011j P0101P0110 — P0100P0111j P0101P1100 — P0100P1101j P1001P1010 — P1000P1011j P0111P1110 — P0110P1111 j P1101P1110 — P1100P1111-

This set specifies a model which is the intersection of the zero set of the invariants with the 15-dimensional probability simplex A with coordinates . The non-negative points on the variety defined by this set of polynomials represent probability distributions which satisfy the conditional independence statements in Mg.

The same quadric polynomials can also be determined by the vectors in the kernel of the parametric model matrix Ag. Each quadric form corresponds to a vector in the kernel of Ag. The columns of Ag are indexed by the m states in the state space, and the rows represent the potential values, indexed by the d model parameters, with a separate potential value for each possible assignment on the clique. Each column in Ag represents a possible assignment of the graph:

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