Paper ID: 1103
Title: Square Root Graphical Models: Multivariate Generalizations of Univariate Exponential Families that Permit Positive Dependencies

Review #1
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Summary of the paper (Summarize the main claims/contributions of the paper.): The authors propose a method to include negative correlations in multivariate extensions of univariate distributions where traditional copula methods can introduce issues like non-identifiability. They give inference methods and apply the results to airport delay data.

Clarity - Justification: This paper was easy to follow. It addressed relevant methods, potential problems, and implementation details.

Significance - Justification: The significance in this paper is not in the area where the methodology is applied (multivariate extensions of univariate distributions where standard copula methods have issues) as that area is really narrow. Rather, this paper introduces a nice trick (using the square root of the sufficient statistic for correlation) that could be more widely applied to semi-parametric model construction. 

Detailed comments. (Explain the basis for your ratings while providing constructive feedback.): This is a really nice paper and I do not have too many specific comments. I am curious about the computation time associated with parameter estimation and how this compares to the more restricted case. Is this something that is feasible for a larger parameter matrix?

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Review #2
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Summary of the paper (Summarize the main claims/contributions of the paper.): This paper addresses the problem of estimating exponential family MRFs that allow for positive conditional dependencies among the variables. Previous method for estimating MRFs via univariate exponential family distributions do not in general allow for positive dependencies, and workarounds for Poisson MRFs are also problematic. In order to fix this problem, the paper proposes a new class of model that replaces sufficient statistics in the previous model with their square root.

Clarity - Justification: The paper is well organized and is generally well written.

Significance - Justification: The paper is based on a simple but novel idea of using a square root of the sufficient statistics. The results are of interest to the community. Multiple recent papers have tried to solve this problem of multivariate generalizations of univariate exponential families with less success.  

Detailed comments. (Explain the basis for your ratings while providing constructive feedback.): The experimental results were somewhat weak -- only one one real dataset. While the results on that one dataset seem to make sense, without a synthetic experiment it's hard to know whether it's really picking up true patterns.  Also, fitting square root graphical models appears computationally expensive. It would be nice to see at least some results about how long it took to run, to get a sense of whether the proposed method is practical without further computational improvements.  Overall, however, I think this paper has a novel approach for a significant problem.

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Review #3
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Summary of the paper (Summarize the main claims/contributions of the paper.): The paper introduces square root graphical models, which are a multivariate extension of univariate exponential families with non-negative sufficient statistics. Unlike previous approaches, this enables the multivariate distribution to capture both positive and negative dependencies across dimensions.

Clarity - Justification: The paper clearly presents the current approaches, the context of the problems underyling multivariate extensions of exponential families, and the context under which their proposed approach lies in. 

Significance - Justification: I'm not familiar with the field; but their simple solution for extending univariate exponential families, and their contrasts to previous related work, is convincing that this is a worthy contribution. I imagine the intuition behind node conditional specifications vs radial conditional specifications could be useful in other domains as well.

Detailed comments. (Explain the basis for your ratings while providing constructive feedback.): The methodology is clearly addressed and the examples of particular exponential families are convincing that this is a good approach to the domainating quadratic term in Equation 1.  The explanation regarding alternatives to explicit multivariate exponential families using copulas make sense, and at some level there must be a compromise in the choice of the explicit exponential family distribution as done with the square rooting of the sufficient statistics. However, it could be helpful to explain whether this pursuit is worth it then; traditionally in Bayesian analysis, one poses a hierarchical model with a prior that, when marginalized out, correlates the univariate parameters associated to data points (not necessarily mixtures). Is marginal likelihood maximization as in these approaches more difficult than the parameter estimation techniques taken here? In general, why consider multivariate extensions of this form?  The experiments taken compare to baseline approaches and are convincing that this is a novel extension.

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