Source-linked AI summary
Graphical models for marked point processes based on local independence
Vanessa Didelez
TL;DR
The paper addresses graphical modeling for time-dependent event data and statistical inference for local independence graphs. It proposes graphs for these dependencies and introduces δ-separation, which informs independencies preserved after marginalising, while facilitating reasoning and simplifying calculations.
Problem
Graphical models for truly time-dependent data and statistical inference for local independence graphs remain important issues.
Method
The paper proposes graphs representing local independencies and introduces δ-separation to characterize independencies preserved after marginalising.
Results
δ-separation informs independencies preserved after marginalising, while the framework facilitates reasoning about complex dependencies and simplifies calculations by reducing dimensionality.
Takeaways & Limitations
The framework supports reasoning about complex dependencies, especially with unobservable information, and reduces computational dimensionality.
Takeaways & Limitations
Without further assumptions, arrows in local independence graphs do not necessarily represent causal dependencies.
Abstract
from arXiv · showhide
A new class of graphical models capturing the dependence structure of events that occur in time is proposed. The graphs represent so-called local independences, meaning that the intensities of certain types of events are independent of some (but not necessarily all) events in the past. This dynamic concept of independence is asymmetric, similar to Granger non-causality, so that the corresponding local independence graphs differ considerably from classical graphical models. Hence a new notion of graph separation, called delta-separation, is introduced and implications for the underlying model as well as for likelihood inference are explored. Benefits regarding facilitation of reasoning about and understanding of dynamic dependencies as well as computational simplifications are discussed.
1 Introduction
The paper develops graphical models for local independence in multivariate event-history data, where event intensities can depend asymmetrically on selected past events. It extends local-independence concepts to multiple processes, introduces δ-separation, and derives implications for marginal independencies and likelihood factorisation.
- Motivation: Graphical models for truly time-dependent data such as event histories remain less developed than models for cross-sectional or classical multivariate data.Existing approaches do not capture how the present or future depends on, or is affected by, the past.
- Interpretation: Local independence is dynamic and asymmetric, with a close analogy to Granger non-causality.The resulting graphs therefore differ substantially from classical graphical models based on symmetric conditional-independence structures.
- Contribution: The paper proposes graphs representing local independence structures in event-history data.Local independence means that, given specific past events, a future event’s intensity is independent of other past events.
- Contribution: The framework extends earlier bivariate local-independence ideas to more than two processes and permits conditioning on the past of other processes.This generalisation describes dynamic dependencies in multivariate processes.
- Graphical implications: δ-separation identifies independencies that remain after marginalising over some processes.The paper also investigates how likelihoods factorise for a given local-independence graph.
2 Local independence for marked point processes
This section formalizes marked point processes and local independence through counting processes, histories, filtrations, and intensities. It defines local independence for bivariate and multivariate processes as invariance of a target intensity when selected past-event information is removed, subject to stated assumptions.
- Marked point processes: A marked point process represents event times and event types through a finite mark space and associated counting processes.The multivariate counting process collects the mark-specific counting processes, and subprocesses correspond to subsets of marks.
- Histories and intensities: The process history is represented by filtrations, which record event information up to time t for the whole process or selected subprocesses.Whole-process and subprocess filtrations support comparing intensities based on different past-information sets.
- Histories and intensities: Under absolute continuity and predictability assumptions, each counting process has an intensity that provides the short-term prediction of its next behaviour.The intensity is based on the chosen filtration and is linked to the compensator through the Doob–Meyer decomposition.
- Assumptions: The no-jumps-at-the-same-time assumption excludes simultaneous jumps among the counting processes.It is imposed so dependencies can be attributed to the past rather than common innovations; simultaneous events can instead be represented as a new mark.
- Local independence: Bivariate local independence means that one process’s intensity remains unchanged when the past of another process is omitted from the conditioning history.The relation is directional, so one process may be locally independent of another without the reverse also holding.
- Local independence: Multivariate local independence conditions the intensity of subprocess B on the past of C while requiring independence from the past of A.The notation distinguishes conditional, marginal, and dependent relations among disjoint process subsets.
3 Local independence graphs
Local independence graphs encode which event intensities depend on past subprocess histories, using directed edges and asymmetric dynamic independence. Their Markov properties and δ-separation support graph-based conditional-independence reasoning, dimension reduction, and likelihood factorisation.
- Definition: Classical conditional-independence graphs cannot express the skin-disease example because the two occurrence times are dependent despite asymmetric local independence.The example concerns menopause and skin disease.
- Definition: A local independence graph uses directed edges between marks to represent local dependence, with absent edges encoding conditional local independence from the remaining processes.The graph is defined for a multivariate counting process associated with a marked point process.
- δ-separation: The global dynamic Markov property uses δ-separation to identify conditional local independencies and reduce the conditioning set to a relevant subset.Under regularity conditions, pairwise, local, and global dynamic Markov properties are equivalent.
- Dynamic Markov properties: The local dynamic Markov property restricts each mark intensity to the history of its closure, consisting of the mark and its parents.This identifies the immediately relevant past information for a single mark.
- Dynamic Markov properties: Under conditional measurable separability, the home-visits example permits the visits process to be locally independent of hospitalisation and health status while the patient remains alive.The assumption may fail if health status is determined only by past hospitalisation information.
- δ-separation: Ignoring an underlying health process can create spurious local dependence of survival on home visits, even when visits do not directly affect survival.The moral graph exposes why conditioning or marginalising processes requires care.
- Likelihood factorisation: The mark-specific likelihood based on the whole past remains unchanged when information is restricted to the histories of a mark and its parents.This closure-based reduction leads to a likelihood factorisation analogous to DAG factorisation.
- δ-separation: In the chemotherapy example, tumour-size history separates surgery history from toxic reactions before death, yielding a dynamic conditional-independence statement.The claim concerns whether and when surgery and toxic reactions occurred.
4 Discussion and conclusions
The discussion describes how local independence graphs support reasoning about dynamic dependencies and reduce computational dimensionality, while distinguishing local dependence from causation and identifying open inference problems.
- Graphical representations allow algebraic manipulation of dependency relations and can replace some explicit formulae for intensities.
- Reading relations among subprocesses facilitates reasoning about complex dependencies, especially when information is unobservable.
- The graphical representation simplifies calculations by reducing dimensionality.
- Local independence graph arrows do not necessarily represent causal dependencies because observation is not equivalent to intervention.
- Causal inference from these graphs requires further assumptions, and developing analogous results remains a topic for further research.
- Statistical inference includes quantifying dependency strength under a given graph and searching for the graph from data, with broader graphical simplifications still requiring research.
A Appendix
The appendix develops additional properties of local independence and δ-separation to support the proof of Theorem 3.5.
- The appendix proves Theorem 3.5 using further results on local independence and δ-separation.
- These properties are examined along the lines of graphoid axioms generalized to the asymmetric case.
A.1 Properties of local independence
The appendix establishes algebraic properties of local independence, including decomposition, weak union, contraction, intersection, and conditions for right decomposition.
- Local independence satisfies left redundancy, left decomposition, left weak union, left contraction, and right weak union under the stated definitions.
- The filtration used for the intensity process is generated at least by the process itself, supporting the stated right-intersection argument.
- Left redundancy, left decomposition, and left contraction imply A → / B | C ⇔ A\C → / B | C.
- Local independence implies A → / B\C | C when A → / B | C, but the converse does not generally hold.
- Under assumption (6), left intersection holds: if A → / B | C and C → / B | A, then (A ∪ C) → / B | (A ∩ C).
- Right decomposition requires additional conditions involving local independence relations among A, B, C, and D.
A.2 Properties of δ–separation
This section formalizes δ–separation through allowed trails and shows that it satisfies key properties of local independence. It also establishes an equivalent graph-reading procedure and right-decomposition results.
- Definition and equivalent characterization: The empty set δ–separates A from B when A and B are unconnected in the relevant graph.
- Properties: δ–separation satisfies local-independence properties including left redundancy, decomposition, weak union, contraction, and modified intersection properties.The paper writes the corresponding relation as A irδB|C.
- Properties: Under conditions analogous to Proposition A.3, δ–separation has a right-decomposition property, including a special case identified with Lemma 4.11 in Didelez (2006).
- Graphical characterization: An equivalent way to read δ–separation from a local independence graph is introduced to support familiarity with d–separation and simplify parts of the proof of Theorem 3.5.
- Definition and equivalent characterization: δ–separation is defined using allowed trails whose blocking by C characterizes separation between disjoint vertex sets.The equivalent condition states that C δ–separates A from B exactly when all allowed trails from A to B are blocked by C.
A.3 Proof of Theorem 3.5
The proof of Theorem 3.5 derives local independence from δ–separation by induction and separation arguments. It handles nonpartitioning and nonancestral cases using the relevant contraction, union, decomposition, and intersection properties.
- Proof structure: The proof accounts for the asymmetry of local independence, making this equivalence more involved than analogous undirected conditional-independence arguments.
- Inductive proof: The proof uses backward induction on the number of vertices in the separating set C.The base case has singleton A and B when |C| = |V| − 2; smaller separating sets are handled recursively.
- Partition case: For partitions of V, left and right weak union, decomposition, contraction, and intersection reduce setwise claims to smaller subsets.
- Nonpartition cases: When A ∪ B ∪ C is ancestral, vertices outside this set are handled by showing their allowed trails to B are blocked by A ∪ C.
- Nonpartition cases: The more complicated case uses right decomposition for local independence and δ–separation to manage an additional vertex γ and vertices in C.The argument derives the needed separations by ruling out an unblocked path between A and γ.
A.4 Proof of Theorem 3.6
The proof of Theorem 3.6 derives conditional independence from graph separation by restricting to ancestors and factorizing the marked point-process likelihood. The resulting factorization corresponds to an undirected graph Markov property.
- Ancestor restriction: The likelihood can discard events outside An(A ∪ B ∪ C) because intensities for ancestral nodes do not depend on those events.
- Likelihood factorization: After ancestor restriction, the likelihood factors into terms depending only on the cliques of the moralized ancestral graph.The sets cl(k) are rearranged into maximal fully connected node sets.
- Markov implication: This clique factorization corresponds to the factorization property of undirected graphs and implies their global Markov property.
- Markov implication: When C δ–separates A from B in the local independence graph, the corresponding conditional independence holds in the moralized ancestral graph.