Azaria Paz

TCS Journal 2024 Journal Article

The n-trigles, a new variant in the SameSum family

• Azaria Paz

A SameSum graph is a graph whose vertices are inscribed with positive integers which are arranged in such a way that there are several subsets of the inscribed integers which sum to the same constant. An example of a SameSum graph is a Magic Square: the triplets of integers along the rows, along the columns and along the two diagonals of a magic square sum to the same magic constant of that specific square. Some recent results of the author of this article present several groups of SameSum graphs providing some additional examples. This article introduces the n-trigles, a new variant in the SameSum family, as a planar connected graph based on triangles only. The n-trigles are defined, their versatile properties are described and several examples are shown.

UAI Conference 2010 Conference Paper

Confounding Equivalence in Causal Inference

• Judea Pearl
• Azaria Paz

The paper provides a simple test for deciding, from a given causal diagram, whether two sets of variables have the same bias-reducing potential under adjustment. The test requires that one of the following two conditions holds: either (1) both sets are admissible (i.e., satisfy the back-door criterion) or (2) the Markov boundaries surrounding the manipulated variable(s) are identical in both sets. Applications to covariate selection and model testing are discussed.

TCS Journal 1994 Journal Article

An algorithm for finding a shortest vector in a two-dimensional modular lattice

• Mody Lempel
• Azaria Paz

Let 0 < a, b < d be integers with a ≠ b. The lattice Ld (a, b) is the set of all multiples of the vector (a, b) modulo d. An algorithm is presented for finding a shortest vector in Ld (a, b). The complexity of the algorithm is shown to be logarithmic in the size of d when the number of arithmetical operations is counted.

UAI Conference 1994 Conference Paper

On Testing Whether an Embedded Bayesian Network Represents a Probability Model

• Dan Geiger
• Azaria Paz
• Judea Pearl

Testing the validity of probabilistic models containing unmeasured (hidden) variables is shown to be a hard task. We show that the task of testing whether models are structurally incompatible with the data at hand, requires an exponential number of independence evaluations, each of the form: "X is conditionally independent of Y, given Z." In contrast, a linear number of such evaluations is required to test a standard Bayesian network (one per vertex). On the positive side, we show that if a network with hidden variables G has a tree skeleton, checking whether G represents a given probability model P requires the polynomial number of such independence evaluations. Moreover, we provide an algorithm that efficiently constructs a tree-structured Bayesian network (with hidden variables) that represents P if such a network exists, and further recognizes when such a network does not exist.

I&C Journal 1991 Journal Article

Axioms and algorithms for inferences involving probabilistic independence

• Dan Geiger
• Azaria Paz
• Judea Pearl

This paper offers an axiomatic characterization of the probabilistic relation “X is independent of Y (written (X, Y))”, where X and Y are two disjoint sets of variables. Four axioms for (X, Y) are presented and shown to be complete. Based on these axioms, a polynomial membership algorithm is developed to decide whether any given independence statement (X, Y) logically follows from a set Σ of such statements, i. e. , whether (X, Y) holds in every probability distribution that satisfies Σ. The complexity of the algorithm is O(|Σ| · k 2 + |Σ| · n), where |Σ| is the number of given statements, n is the number of variables in Σ ∪ {(X, Y)}, and k is the number of variables in (X, Y).

ECAI Conference 1987 Conference Paper

Graphoids: Graph-Based Logic for Reasoning about Relevance Relations or When would x tell you more about y if you already know z?

• Judea Pearl
• Azaria Paz