7 research outputs found
Recovering Causal Structures from Low-Order Conditional Independencies
One of the common obstacles for learning causal models from data is that
high-order conditional independence (CI) relationships between random variables
are difficult to estimate. Since CI tests with conditioning sets of low order
can be performed accurately even for a small number of observations, a
reasonable approach to determine casual structures is to base merely on the
low-order CIs. Recent research has confirmed that, e.g. in the case of sparse
true causal models, structures learned even from zero- and first-order
conditional independencies yield good approximations of the models. However, a
challenging task here is to provide methods that faithfully explain a given set
of low-order CIs. In this paper, we propose an algorithm which, for a given set
of conditional independencies of order less or equal to , where is a
small fixed number, computes a faithful graphical representation of the given
set. Our results complete and generalize the previous work on learning from
pairwise marginal independencies. Moreover, they enable to improve upon the 0-1
graph model which, e.g. is heavily used in the estimation of genome networks
Algebraic Statistics in Practice: Applications to Networks
Algebraic statistics uses tools from algebra (especially from multilinear
algebra, commutative algebra and computational algebra), geometry and
combinatorics to provide insight into knotty problems in mathematical
statistics. In this survey we illustrate this on three problems related to
networks, namely network models for relational data, causal structure discovery
and phylogenetics. For each problem we give an overview of recent results in
algebraic statistics with emphasis on the statistical achievements made
possible by these tools and their practical relevance for applications to other
scientific disciplines
Counting Markov equivalence classes by number of immoralities
Two directed acyclic graphs (DAGs) are called Markov equivalent if and only if they have the same underlying undirected graph (i.e. skeleton) and the same set of immoralities. When using observational data alone and typical identifiability assumptions, such as faithfulness, a DAG model can only be determined up to Markov equivalence. Therefore, it is desirable to understand the size and number of Markov equivalence classes (MECs) combinatorially. In this paper, we address this enumerative question using a pair of generating functions that encode the number and size of MECs on a skeleton G, and in doing so we connect this problem to classical problems in combinatorial optimization. The first generating function is a graph polynomial that counts the number of MECs on G by their number of immoralities. Using connections to the independent set problem, we show that computing a DAG on G with the maximum possible number of immoralities is NP-hard. The second generating function counts the MECs on G according to their size. Via computer enumeration, we show that this generating function is distinct for every connected graph on p nodes for all p < 10.National Science Foundation (U.S.) (DMS-1606407)United States. Defense Advanced Research Projects Agency (W911NF-16-1-0551)National Science Foundation (U.S.) (1651995)United States. Office of Naval Research (N00014-17-1-2147