A Branch-and-Bound Algorithm for MDL Learning Bayesian Networks

This paper extends the work in [Suzuki, 1996] and presents an efficient depth-first branch-and-bound algorithm for learning Bayesian network structures, based on the minimum description length (MDL) principle, for a given (consistent) variable ordering. The algorithm exhaustively searches through all network structures and guarantees to find the network with the best MDL score. Preliminary experiments show that the algorithm is efficient, and that the time complexity grows slowly with the sample size. The algorithm is useful for empirically studying both the performance of suboptimal heuristic search algorithms and the adequacy of the MDL principle in learning Bayesian networks.Comment: Appears in Proceedings of the Sixteenth Conference on Uncertainty in Artificial Intelligence (UAI2000

Generating Markov Equivalent Maximal Ancestral Graphs by Single Edge Replacement

Maximal ancestral graphs (MAGs) are used to encode conditional independence relations in DAG models with hidden variables. Different MAGs may represent the same set of conditional independences and are called Markov equivalent. This paper considers MAGs without undirected edges and shows conditions under which an arrow in a MAG can be reversed or interchanged with a bi-directed edge so as to yield a Markov equivalent MAG.Comment: Appears in Proceedings of the Twenty-First Conference on Uncertainty in Artificial Intelligence (UAI2005

Identifying Dynamic Sequential Plans

We address the problem of identifying dynamic sequential plans in the framework of causal Bayesian networks, and show that the problem is reduced to identifying causal effects, for which there are complete identi cation algorithms available in the literature.Comment: Appears in Proceedings of the Twenty-Fourth Conference on Uncertainty in Artificial Intelligence (UAI2008

Identifying Conditional Causal Effects

This paper concerns the assessment of the effects of actions from a combination of nonexperimental data and causal assumptions encoded in the form of a directed acyclic graph in which some variables are presumed to be unobserved. We provide a procedure that systematically identifies cause effects between two sets of variables conditioned on some other variables, in time polynomial in the number of variables in the graph. The identifiable conditional causal effects are expressed in terms of the observed joint distribution.Comment: Appears in Proceedings of the Twentieth Conference on Uncertainty in Artificial Intelligence (UAI2004

A Criterion for Parameter Identification in Structural Equation Models

This paper deals with the problem of identifying direct causal effects in recursive linear structural equation models. The paper establishes a sufficient criterion for identifying individual causal effects and provides a procedure computing identified causal effects in terms of observed covariance matrix.Comment: Appears in Proceedings of the Twenty-Third Conference on Uncertainty in Artificial Intelligence (UAI2007

Boundedness and exponential convergence of a chemotaxis model for tumor invasion

We revisit the following chemotaxis system modeling tumor invasion \begin{equation*} \begin{cases} u_t=\Delta u-\nabla \cdot(u\nabla v),& x\in\Omega, t>0,\\ v_t=\Delta v+wz,& x\in\Omega, t>0,\\ w_t=-wz,& x\in\Omega, t>0,\\ z_t=\Delta z-z+u, & x\in\Omega, t>0,\\ \end{cases} \end{equation*} in a smooth bounded domain $\Omega \subset \mathbb{R}^n(n\geq 1)$ with homogeneous Neumann boundary and initial conditions. This model was recently proposed by Fujie et al. \cite{FIY14} as a model for tumor invasion with the role of extracellular matrix incorporated, and was analyzed by Fujie et al. \cite{FIWY16}, showing the uniform boundedness and convergence for $n\leq 3$. In this work, we first show that the $L^\infty$-boundedness of the system can be reduced to the boundedness of $\|u(\cdot,t)\|_{L^{\frac{n}{4}+\epsilon}(\Omega)}$ for some $\epsilon>0$ alone, and then, for $n\geq 4$, if the initial data $\|u_0\|_{L^{\frac{n}{4}}}$, $\|z_0\|_{L^\frac{n}{2}}$ and $\|\nabla v_0 \|_{L^n}$ are sufficiently small, we are able to establish the $L^\infty$-boundedness of the system. Furthermore, we show that boundedness implies exponential convergence with explicit convergence rate, which resolves the open problem left in \cite{FIWY16}.Comment: 15pages, Submmitte

Local Markov Property for Models Satisfying Composition Axiom

The local Markov condition for a DAG to be an independence map of a probability distribution is well known. For DAGs with latent variables, represented as bi-directed edges in the graph, the local Markov property may invoke exponential number of conditional independencies. This paper shows that the number of conditional independence relations required may be reduced if the probability distributions satisfy the composition axiom. In certain types of graphs, only linear number of conditional independencies are required. The result has applications in testing linear structural equation models with correlated errors.Comment: Appears in Proceedings of the Twenty-First Conference on Uncertainty in Artificial Intelligence (UAI2005

Probabilities of Causation: Bounds and Identification

This paper deals with the problem of estimating the probability that one event was a cause of another in a given scenario. Using structural-semantical definitions of the probabilities of necessary or sufficient causation (or both), we show how to optimally bound these quantities from data obtained in experimental and observational studies, making minimal assumptions concerning the data-generating process. In particular, we strengthen the results of Pearl (1999) by weakening the data-generation assumptions and deriving theoretically sharp bounds on the probabilities of causation. These results delineate precisely how empirical data can be used both in settling questions of attribution and in solving attribution-related problems of decision making.Comment: Appears in Proceedings of the Sixteenth Conference on Uncertainty in Artificial Intelligence (UAI2000

Causal Discovery from Changes

We propose a new method of discovering causal structures, based on the detection of local, spontaneous changes in the underlying data-generating model. We analyze the classes of structures that are equivalent relative to a stream of distributions produced by local changes, and devise algorithms that output graphical representations of these equivalence classes. We present experimental results, using simulated data, and examine the errors associated with detection of changes and recovery of structures.Comment: Appears in Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence (UAI2001

Polynomial Constraints in Causal Bayesian Networks

We use the implicitization procedure to generate polynomial equality constraints on the set of distributions induced by local interventions on variables governed by a causal Bayesian network with hidden variables. We show how we may reduce the complexity of the implicitization problem and make the problem tractable in certain causal Bayesian networks. We also show some preliminary results on the algebraic structure of polynomial constraints. The results have applications in distinguishing between causal models and in testing causal models with combined observational and experimental data.Comment: Appears in Proceedings of the Twenty-Third Conference on Uncertainty in Artificial Intelligence (UAI2007