 # Graphical structure of conditional independencies in determinantal point processes

Determinantal point process have recently been used as models in machine learning and this has raised questions regarding the characterizations of conditional independence. In this paper we investigate characterizations of conditional independence. We describe some conditional independencies through the conditions on the kernel of a determinantal point process, and show many can be obtained using the graph induced by a kernel of the L-ensemble.

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## 1. Introduction to the model

In this paper will be a positive semi-definite matrix. Let , . We call a random subset of a determinantal point process if the following holds

 Pr(A⊂Y)=det(KA),

and by definition . (Where .)

Basically, we have a set of points, and we pick a random subset

of them. We model the probability that all the points in the subset

were chosen by .

Instead of modeling with the kernel , in practice a determinantal point process is modeled as an -ensemble. The process is called the -ensemble with the kernel if

 Pr(Y=A)=det(LA)det(L+I),

where is a positive semi-definite matrix.

###### Theorem 1.

An -ensemble with kernel is a DPP with the kernel

 K=L(L+I)−1=I−(L+I)−1.