Tracking Time-varying Graphical Structure
Structure learning algorithms for graphical models have focused almost exclusively on stable environments in which the underlying generative process does not change; that is, they assume that the generating model is globally stationary. In real-world environments, however, such changes often occur without warning or signal. Real-world data often come from generating models that are only locally stationary. In this paper, we present LoSST, a novel, heuristic structure learning algorithm that tracks changes in graphical model structure or parameters in a dynamic, real-time manner. We show by simulation that the algorithm performs comparably to batch-mode learning when the generating graphical structure is globally stationary, and significantly better when it is only locally stationary.
Paper Summary: The paper tackles an 
      interesting and under explored problem of tracking changes in the 
      underlying dependency structure between variables in a stream (e.g. time 
      based) of data points. The author focus on a efficient 
      (computational/memory wise) solution for the simpler SEM models. They 
      develop a new algorithm for this task, named LoSST. LoSST uses a standard 
      batch learning algorithm for the underlying structure learning (PC by 
      Spirtes et al, 2000) and tracks several statistics from the stream of data 
      to decide whether to initiate new structure learning. The tracked 
      statistics involve the distance from the weighted mean of each variable, 
      the correlation matrix (eq. 1,2) and the weighted sample size. For the 
      sample weight there is a weighting scheme for each new point based on how 
      "surprising" the new data point is (Eq. 3,4). Each point's Mahalanobis 
      distance (MD) is computed and the estimated p-values for the MDs by a 
      Hotelling's T2 distribution is combined using Liptak's weighted pooling of 
      p-values to compute an overall "significance" statistic \rho. This stat is 
      then used in a noisy OR configuration with the previous step to decide 
      wether to start a new structure learning. The authors compare to vanilla 
      PC batch learning on synthetic data sets where either a change in params 
      or a change of structure is introduced after a fixed set of datapoints, 
      testing for different number of variables and max degree configurations. 
      Finally, they perform a short analysis of real life data from US price 
      index with 529 data points (every month, starting at 1967) and six 
      variables. They show the algorithm detects some known changes/trends in 
      the US economy while ignore short lived changes. Pros: 1. The 
      paper tackles an interesting and under explored problem. As the author 
      point out there is a lot of room for additional development of practical 
      solutions for these and other models, as well as matching theory and 
      theoretical guarantees. 2. The algorithm is novel and 
      practical/efficient. It makes creative use of a host of different methods 
      for this. 3. Experiments seem adequate trying to track the effect of 
      the various params and include an interesting real life data set. 4. 
      Authors give a good review about previous work on this problem. 5. 
      Overall well written. Cons: 1. The biggest lack in the 
      paper with respect to this interesting problem is that of theory or any 
      guarantees under some assumptions. Convergence and diligence are like 
      opposing forces as the authors point out (p. 4) but it seems like some 
      form of guarantees under simplifying conditions should be plausible. At 
      least mention as another direction for future work. 2. Similarly, the 
      solution suggested by the authors, LoSST, is heavy on the heuristic/hack 
      side. That's OK and enables the authors to get a practical solution, but 
      the decisions involved should be more readably noted. Since tracking the 
      mean and correlation matrix is straightforward, these involve mostly 
      decisions on reweighing, effective sample size, and when/how to re learn. 
      The authors make creative use of some methods under conditions that 
      probably void guarantees of these methods. For example, they estimate the 
      MD p-value using Hotelling's T2 distribution. They set the sample size N 
      to be the effective sample size S^r. However, that distribution was 
      developed for complete counts from a given distribution, not for 
      partial/re weighted counts. If they actually sampled from these 
      nets/structures with these partial weights would the distribution follow 
      this form? Similarly, Liptak's weighted p-values was developed for meta 
      analysis using independent tests with different sample sizes. At least if 
      the methods are used outside their guarantees/context the authors should 
      clearly state that, so readers don't get the wrong 
      impression/understanding. Other comments: - The authors 
      make creative use of footnotes to outline proofs, probably beyond what 
      Leslie Lamport or the conference organizers intended. Consider using supp 
      for that and other missing details. - Figures are way too small, with 
      unnecessary space between them, missing labels, missing details in the 
      captions, legends etc. Description of the experiments is not clear enough 
      (e.g. top paragraph p. 7 is cryptic). - The explanation why LoSST does 
      worse on edge addition when only the params are changed is not 
      satisfactory. - With regards to the above, it seems desirable that 
      sparking params update and structure updates should be decoupled.