February 14, 2020 | Written by: Debarun Bhattacharjya and Karthikeyan Shanmugam
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Real-world decision making often involves situations and systems whose uncertain and inter-dependent variables interact in a complex and dynamic way. Additionally, many scenarios are influenced by external events that affect how system variables evolve.
Consider the following examples:
- Health – a diabetic patient’s blood glucose level and mental well-being are influenced by insulin intake, meals and physical activity.
- Finance – stock prices within an industry can be affected by natural disasters or trade deals.
- Social impact – social services, such as counseling sessions and classes, impact a person’s level of education, employment, and well-being.
To address these complex scenarios for decision making, together with colleagues at the IBM T. J. Watson Research Center, we have developed a new dynamic, probabilistic graphical model called Event-driven Continuous Time Bayesian Networks (ECTBNs) . We applied the ECTBN model and its companion learning algorithm to a unique dataset provided by CityLink Center, a non-profit organization based in Cincinnati, Ohio, that provides social services to help people progress out of poverty.
An ECTBN model includes a graph with two kinds of variables — event labels and state (or system) variables — such as those shown in Figure 1, as well as model parameters. The graph may include cycles and even self-loops for event labels, as well as arcs from event labels to state variables. Importantly, there is an assumption that state variables cannot influence event label arrivals. Model parameters include a set of conditional intensity matrices for state variables and conditional intensity rates for event labels.
Figure 1: Illustrative ECTBN graph representing a dynamic process involving 4 state variables (X1 through X4) and 3 event labels (E1 through E3)
We partnered with CityLink Center to apply and validate the ECTBN model for modeling the impact of social services. CityLink helps individuals progress out of poverty by providing integrated social services and tracking their improvement along multiple dimensions (referred to as “outcome areas”), such as educational attainment, employment and wage levels. CityLink coordinates multiple agencies that provide one-on-one and group classes aimed at helping those in the program. One of the key questions for decision makers to guide future policy is: what is the effect of classes (modeled as events) on the outcome area scores (modeled as state variables with discrete states)? We trained the ECTBN model to help answer this question.
The ECTBN model discovered how some events (classes offered) affected the state transitions (score changes across various dimensions). For example, the one-on-one financial education class caused transitions in scores for the financial education dimension, and the window parameters provided information about time-to-effect. A sample graph obtained by applying the algorithm on a section of the processed data is shown in Figure 2.
Figure 2: Sample ECTBN graph of CityLink data
The ECTBN graph captures the effect of historical arrivals of different event labels on the state variables of a system, thereby combining the abilities of two model families — continuous time Bayesian networks (CTBNs)  and graphical event models (GEMs) [3,4]. Continuous time Bayesian networks are graphical representations of conditional Markov processes, where a state variable’s transitions are determined by the current state of its parents in some underlying graph.
On the other hand, graphical event models are a family of models for event streams, representing a marked point process; here, the arrival rate for a particular type of event at any time depends on the historical occurrences of its parents. By incorporating events, ECTBNs fundamentally extend the capabilities of CTBNs, allowing for the non-Markovian effects of event arrivals in the otherwise Markovian system.
The algorithm for training an ECTBN from joint temporal data with event arrivals and state variable transitions requires further assumptions on the historical dependence in the model. We make the following proximal (or recency) assumption : a state variable’s transition rate depends only on whether its parent event labels have occurred at least once in some recent time window. Through a theoretical analysis, we show that an ECTBN represents a continuous time stochastic process which can be viewed as an ensemble of CTBNs; this perspective is then used to prove that our algorithm is asymptotically consistent, i.e. recovers the true underlying graph with sufficient data. Experiments with synthetic data show that the learner also performs adequately with limited data, with reasonable precision on the task of structure recovery.
An ECTBN model can capture complex temporal dynamics through an interpretable graphical representation, providing insights that are otherwise difficult to obtain from other graphical models. Potential directions for future work include models that generalize further and incorporate even more complex historical dependencies among events and state variables, as well as a decision-making framework using parameters obtained from structure learning to optimally guide system evolution to achieve a desired goal.
 Bhattacharjya, D. Shanmugam, K., Gao, T., Mattei, N., Varshney K. R., and Subramanian, D., 2020, Event-driven continuous time Bayesian networks, Forthcoming in AAAI.
 Nodelman, U., Shelton, C. R., and Koller, D. 2002. Continuous time Bayesian networks. In UAI, 378–378.
 Didelez, V. 2008. Graphical models for marked point processes based on local independence. Journal of the Royal Statistical Society Series B, 70 (1):245–264.
 Gunawardana, A., and Meek, C. 2016. Universal models of multivariate temporal point processes. In AISTATS, 556–563.
 Bhattacharjya, D., Subramanian, D., and Gao, T. 2018. Proximal graphical event models. In NeurIPS, 8147-8156.