By Milan Studeny

Probabilistic Conditional Independence constructions presents the mathematical description of probabilistic conditional independence constructions; the writer makes use of non-graphical equipment in their description, and takes an algebraic technique. The monograph offers the equipment of structural imsets and supermodular capabilities, and bargains with independence implication and equivalence of structural imsets. Motivation, mathematical foundations and components of program are incorporated, and a coarse evaluation of graphical equipment can also be given. particularly, the writer has been cautious to exploit compatible terminology, and provides the paintings with a view to be understood through either statisticians, and through researchers in synthetic intelligence. the required simple mathematical notions are recalled in an appendix.

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**Extra resources for Probabilistic Conditional Independence Structures**

**Sample text**

The reason behind this phenomenon is that the actual number of free parameters characterizing a Gaussian measure over N is, in fact, smaller than the number of parameters characterizing a discrete measure (if |Xi | ≥ 2 for i ∈ N ). Therefore, discrete measures oﬀer a wider variety of induced conditional independence models than Gaussian measures. This is perhaps a surprising fact for those who anticipate that a continuous framework should be wider than a discrete framework. The point is that the “Gaussianity” is quite a restrictive assumption.

N 2 , n 2 . 4 Imsets An imset over N is an integer-valued function on the power set of N , that is, any function u : P(N ) → Z or, alternatively, an element of ZP(N ) . Basic operations with imsets, namely summation, subtraction and multiplication by an integer are deﬁned coordinate-wisely. Analogously, we write u ≤ v for imsets u, v over N if u(S) ≤ v(S) for every S ⊆ N . A multiset is an imset with non-negative values, that is, any function m : P(N ) → Z+ . Any imset u over N can be written as the diﬀerence u = u+ − u− of two multisets over N where u+ is the positive part of u and u− is the negative part of u, deﬁned as follows: u+ (S) = max {u(S), 0} , u− (S) = max {−u(S), 0} for S ⊆ N .

7. 18) gives − |N | 1 |N | · ln(2π) − − · ln(det(Σ)) − 2 2 2 1 = 2 i∈N i∈N − ln (2π) 1 1 − − · ln (σii ) 2 2 2 1 det(Σ) 1 1 ln σii − · ln(det(Σ)) = − · ln = − · ln(det(Γ )) , 2 2 σ 2 i∈N ii which is the fact that was needed to show. On the other hand, a singular Gaussian measure need not be marginally continuous as the following example shows. 5 is not universally valid. 3. There exists a singular Gaussian measure P over N with |N | = 3 such that a⊥ ⊥ b | {c} [P ] and a ⊥ ⊥ b | ∅ [P ] for any distinct a, b, c ∈ N.