A Comprehensive Approach to Inferring Functional and Statistical Dependencies
Abstract
Statement of Work:The research tasks undertaken in this proposal are as follows. Task 1: Inferring statistical directional dependence for times series data: We plant to investigate the relationship between directed information graphs (DIGs) and Causal interventional graphs (CIGs). We see to characterize the conditions under which it possible to recover the structure by mere observation. Task 2: Understanding optimal interventions for structure learning: We will investigate whether indeed CIGs provide a unique factorization of the joint, i.e., their Markov equivalence class is a singleton. We will seek optimal interventions when side information about the joint is available and its effect on reducing the complexity of the learning task. Task 3: Limits to learning functional dependencies: Given our newly proposed functional dependency graphs (FDGs), we aim to investigate and identify which classes of dynamics are learnable. We will also seek efficient learning approaches and optimal intervention strategies for the identified classes of dynamics. Task 4: Equipped with general and appropriately defined measures of dependency and their corresponding graphical models CIGs and FDGs, we plan to investigate the relationship between functional and statistical notions of directional dependency. Task 5: Application of our theoretical results to realworld network datasets.Objective:The problem of learning the influence structure has been approached predominantly from two angles: 1) understanding functional dependencies and 2) learning statistical dependencies. While much progress has been made, still fundamental questions remain unanswered for both aforementioned directions. More importantly, very little is known about how the two types of casual influence, functional and statistical, are in general related. The goal of this proposal is to introduce and investigate a framework that not only allows resolving questions of influence in a broader class of models than the state of art, but also provides a unifying theory to relate functional and statistical dependencies beyond linear models and restrictive assumptions on the distribution of noise in the system. Approach:To attain the aforementioned objectives, we propose an interventional framework. Discovering dependence structure by intervention is based on measuring the influence of a variable (potential cause) on another variable (target) in a network through the following processes. The behavior of the target variable is observed when different values are assigned to the potential cause, while other variables~ effects are removed. We define a measure of functional dependency among processes (or variables) in dynamical systems. Using this measure, we define a new type of graphical model, functional dependency graph (FDG), that encodes such dependencies. Further, we define a metric to capture statistical causation among processes (or variables) and define a new type of probabilistic graphical model, causal interventional graph (CIG,) based on this metric. The statistical measure uses a formulation by Dobrushin based onWasserstein distance that allows capturing the effect of interventions on target variables based on changes in measures. FDGs and CIGs show great promise for capturing directional dependencies for large class of models both when ordering among the variable exists (time series data) and when no such order is present (random variables). This is a huge leap forward compared to the state of the art, where inference approaches are tailor made for specific types of functional dependencies (e.g., linear systems), or graphical models (e.g., DAGs), or noise models (e.g., Gaussian noise), etc. Furthermore, we believe that theformulation of our functional and statistical metrics are well suited to investigate the relationship between the two models and existing approaches such as DIGs for analyzing times series data. Overall Merit and ONR Mission/Relevance:A framework for learning and tra
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Sep 23, 2016
- Source ID
- N000141612804
Entities
People
- Negar Kiyavash
Organizations
- Office of Naval Research
- United States Navy
- University of Illinois Urbana–Champaign