The general idea of this work is clearly in a direction of interest to the community and the results look strong. However, there are a few aspects to this work that I find quite unclear. If they were clearer this work would have much more potential for impact. In particular, it is not clear enough if this work uses a truly'end-to-end' approach (as stated for one of the contributions). For example on line 46 it is stated that: "We show step by step how approximations are made to use an end-to-end learning CNN for implementing such CRF model."

We develop a tree-based reparameterization framework that pro(cid:173) vides a new conceptual view of a large class of iterative algorithms for computing approximate marginals in graphs with cycles. It includes belief propagation (BP), which can be reformulated as a very local form of reparameterization. More generally, we consider algorithms that perform exact computations over spanning trees of the full graph. On the practical side, we find that such tree reparameterization (TRP) algorithms have convergence properties superior to BP. The reparameterization perspective also provides a number of theoretical insights into approximate inference, in(cid:173) cluding a new characterization of fixed points; and an invariance intrinsic to TRP /BP.

Probabilistic graphical models use local factors to represent dependence among sets of variables. For many problem domains, for instance climatology and epidemiology, in addition to local dependencies, we may also wish to model heavy-tailed statistics, where extreme deviations should not be treated as outliers. Specifying such distributions using graphical models for probability density functions (PDFs) generally lead to intractable inference and learning. Cumulative distribution networks (CDNs) provide a means to tractably specify multivariate heavy-tailed models as a product of cumulative distribution functions (CDFs). Currently, algorithms for inference and learning, which correspond to computing mixed derivatives, are exact only for tree-structured graphs.

Probabilistic graphical models use local factors to represent dependence among sets of variables. For many problem domains, for instance climatology and epidemiology, in addition to local dependencies, we may also wish to model heavy-tailed statistics, where extreme deviations should not be treated as outliers. Specifying such distributions using graphical models for probability density functions (PDFs) generally lead to intractable inference and learning. Cumulative distribution networks (CDNs) provide a means to tractably specify multivariate heavy-tailed models as a product of cumulative distribution functions (CDFs). Currently, algorithms for inference and learning, which correspond to computing mixed derivatives, are exact only for tree-structured graphs.

We present the embedded trees algorithm, an iterative technique for estimation of Gaussian processes defined on arbitrary graphs. By exactly solving a series of modified problems on embedded spanning trees, it computes the conditional means with an efficiency comparable to or better than other techniques. Unlike other methods, the embedded trees algorithm also computes exact error covariances. The error covariance computation is most efficient for graphs in which removing a small number of edges reveals an embedded tree. In this context, we demonstrate that sparse loopy graphs can provide a significant increase in modeling power relative to trees, with only a minor increase in estimation complexity.

We present the embedded trees algorithm, an iterative technique for estimation of Gaussian processes defined on arbitrary graphs. By exactly solving a series of modified problems on embedded spanning trees, it computes the conditional means with an efficiency comparable to or better than other techniques. Unlike other methods, the embedded trees algorithm also computes exact error covariances. The error covariance computation is most efficient for graphs in which removing a small number of edges reveals an embedded tree. In this context, we demonstrate that sparse loopy graphs can provide a significant increase in modeling power relative to trees, with only a minor increase in estimation complexity.

We present the embedded trees algorithm, an iterative technique for estimation of Gaussian processes defined on arbitrary graphs. By exactly solving a series of modified problems on embedded spanning trees,it computes the conditional means with an efficiency comparable to or better than other techniques. Unlike other methods, theembedded trees algorithm also computes exact error covariances. Theerror covariance computation is most efficient for graphs in which removing a small number of edges reveals an embedded tree.In this context, we demonstrate that sparse loopy graphs can provide a significant increase in modeling power relative totrees, with only a minor increase in estimation complexity. 1 Introduction Graphical models are an invaluable tool for defining and manipulating probability distributions. In modeling stochastic processes with graphical models, two basic problems arise: (i) specifying a class of graphs with which to model or approximate the process; and (ii) determining efficient techniques for statistical inference.