By Milan Studený RNDr, DrSc (auth.), Michael Jordan, Jon Kleinberg, Bernhard Schölkopf, Frank P. Kelly, Ian Witten (eds.)

Conditional independence is a subject that lies among data and synthetic intelligence. Probabilistic Conditional Independence buildings offers the mathematical description of probabilistic conditional independence constructions; the writer makes use of non-graphical equipment in their description, and takes an algebraic approach.

The monograph offers the equipment of structural imsets and supermodular capabilities, and offers with independence implication and equivalence of structural imsets. Motivation, mathematical foundations and parts of software are incorporated, and a coarse evaluate of graphical tools can also be given. specifically, the writer has been cautious to take advantage of appropriate terminology, and offers the paintings as a way to be understood via either statisticians, and via researchers in man made intelligence. the mandatory effortless mathematical notions are recalled in an appendix.

Probabilistic Conditional Independence constructions could be a important new addition to the literature, and should curiosity utilized mathematicians, statisticians, informaticians, machine scientists and probabilists with an curiosity in man made intelligence. The ebook can also curiosity natural mathematicians as open difficulties are included.

Milan Studený is a senior learn employee on the Academy of Sciences of the Czech Republic.

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12. 8 reveals a notable diﬀerence between the Gaussian and discrete case. While in the discrete case a conditional independence statement A ⊥ ⊥ B | C [P ] is equivalent to the collection of requirements A ⊥⊥ B | ∅ [PAB|C ( |z)] for every z ∈ XC with P C (z) > 0, in the Gaussian case it is equivalent to a single requirement A⊥ ⊥ B | ∅ [PAB|C ( |z)] for at least one z ∈ XC , which already implies the same fact for all other z ∈ XC (one uses the conventional choice of “continuous” versions of PAB|C in this case).

1). 5. Let P be a marginally continuous measure over N , µ a dominating measure for P ABC and A, B|C a disjoint triplet over N . Then A ⊥ ⊥ B | C [P ] if and only if P factorizes after D = {AC, BC} relative to µ. e. x ∈ XN . 9) Proof. 4. e. x ∈ XN one has fC (xC ) = 0 ⇒ fBC (xBC ) = 0. 4 (see p. 21), that is, to choose a suitable version f of the density. e. x ∈ XN . e. e. e. e. 10) and the fact that f ↓ABC is µABC -integrable). e. 3). 1 in [81]). 1. Suppose that P is a marginally continuous measure over N and A, B, C, D ⊆ N are pairwise disjoint sets.

3. Suppose that |N | ≥ 2 and A ⊆ N with |A| ≥ 2. Then there exists a discrete (binary) probability measure P over N such that 38 2 Basic Concepts mP (S) = ln 2 0 if A ⊆ S, otherwise. Proof. Put Xi = {0, 1} for i ∈ N and ascribe the probability 21−|N | to every conﬁguration of values [xi ]i∈N with even i∈A xi (remaining conﬁgurations have zero probability). 10. Suppose that |N | ≥ 3, 2 ≤ l ≤ |N | and L ⊆ {S ⊆ N ; |S| = l}. Then there exists a discrete probability measure P over N such that ∀ a, b|K ∈ T (N ) with |abK| = l a⊥ ⊥ b | K [P ] ⇔ abK ∈ L .