Algebraic tests of general Gaussian latent tree models
We consider general Gaussian latent tree models in which the observed variables are not restricted to be leaves of the tree. Extending related recent work, we give a full semi-algebraic description of the set of covariance matrices of any such model. In other words, we find polynomial constraints that characterize when a matrix is the covariance matrix of a distribution in a given latent tree model. However, leveraging these constraints to test a given such model is often complicated by the number of constraints being large and by singularities of individual polynomials, which may invalidate standard approximations to relevant probability distributions. Illustrating with the star tree, we propose a new testing methodology that circumvents singularity issues by trading off some statistical estimation efficiency and handles cases with many constraints through recent advances on Gaussian approximation for maxima of sums of high-dimensional random vectors. Our test avoids the need to maximize the possibly multimodal likelihood function of such models and is applicable to models with larger number of variables. These points are illustrated in numerical experiments.
# After authors response  I thank the authors for their detailed answers and explanations, that corrected some of the misconceptions I had.  I acknowledge that the application of the methodology of section 3 to the general framework of section 2 would require some further developments, that might be outside of the scope of this conference paper. Hence, the problem of the discrepancy between the two sections remains for this submission, although both sections are interesting in themselves.  # Summary  This well written and nicely presented work is mainly interested in the algebraic characterization and statistical test of a latent Gaussian tree structure. This problem has received some attention over the last few years, and can be related to the work of Shiers et al. 2016 (or Zwiernik et al. 2016 although with a different focus).  Section 2 focuses on the mathematical derivation of polynomial constraints that characterize variance matrices induced by Gaussian latent tree models. It follows closely the derivations of Shiers et al. 2016, extending them to the case where observations can be anywhere on the tree, including on inner nodes, and not just its leaf nodes. The results are presented in an intuitive way, and the proofs are given in the appendices.  In Section 3, a test statistic is derived, in the simplified case of a star-tree. A link is drawn between this star-tree case and the tetrads problem, as studied in the factor analysis literature. The derivation of the new statistics and its approximate distribution relies on recent work by Chernozhukov et al. 2013 and Zhang and Wu 2017. The statistic is justified by heuristic argument, but the proofs are not given.  In Section 4, some numerical experiments are conducted to study the empirical levels of the proposed test. It is compared with a standard maximum-likelihood test as traditionally used in factor analysis (function `factanal` in base R).  # Main comments  My main concern is about the coherence and unity of the ideas presented. In Section 2, the authors describe with much mathematical details a new framework allowing observations to be at inner nodes of a tree. However, in section 3 and following, only the case of a star tree is considered, with observations only at leaf nodes. Maybe I missed something and I would be happy to be corrected, but from my understanding all the developments in Section 3 could have been made with results already exposed in Shier et al. 2016, their main inspiration for Section 2. Hence, as it stands, section 2 seems somewhat decorrelated from Section 3, and, although the results presented in Section 2 are theoretically interesting in themselves, it is not clear what they bring in the progression of the article.  As a minor note related to the same point, only one reference is given to justify the need of the theoretical developments of section 2 (l.34). Adding more examples could help the reader understanding the pertinence of extending existing results to allow for inner-node measurements.  That being said, I tend to think that both parts of the paper (section 2 and sections 3-4) are valuable advances to the field, and have the potential to be the starting point for further developments. I found the results of section 2 and very well presented. The background, as well as the novelty of the results, are both clearly exposed. Section 3 makes use of novel statistical techniques to build a new test (heuristically justified) that might have the potential to be used, beyond the star tree, to the more general structures presented in section 2.  # Other issues  ## Section 2  While I spent a reasonable amount of time to try to understand the results in themselves, I did not review the proofs in details. I found that the new notions and differences with previous characterisations where well explained, and the intuitions were easy to follow.  Other comments: * Why are the observations limited to nodes with degree less or equal than 2 ?   Is this a constrain inherited from applications, or would the developments of   the paper be wrong otherwise ? From Fig. 2-e, it does not prevent sub-trees   to have observed nodes of degree 3. * I did not see why a pseudo-distance is needed, while a distance seems to be   used in Shiers et al 2016 ? * Cor. 2.2 is linked with Th. 3 in Shiers et al, the same way Lemma 2.1 is   linked with Cor 1. Although the links and differences between Lemma 2.1 and   Cor 1 are clearly mentioned, the relationships between Cor 2.2 and Th 3   remain allusive. An intuitive description of these differences might help the   reader understand the new constraints implied by the new problem. * Fig 2.1 and 2.2 are very helpful. Could it be useful to add as legends    {p, q, r, s} such that {p, q}|{r, s} on Fig 2.2 a-c ?  ## Sections 3 - 4    While section 2 seems theoretically sound, the results in section 3 are only justified heuristically, and heavily rely on cited references. Since I'm not very familiar with these techniques, I only have a few remarks about the form:  * l.141-142: should it be \epsilon_1, …, \epsilon_m and X_1, …, X_m (m instead   of p) ? * The authors compare their test statistics with the one in Shier et al 2016   at the end of Section 3, but then do not include this method in the numerical   experiments of Section 4. Are there arguments, other than computational time   (which could be a legitimate concern) to exclude it ? * Fig. 4.1: Maybe a legend on the rows/columns of the figure could help.   Also, since the interval (0, 0.1) is particularly significant, maybe a visual   aid, such as a vertical line, could help. * l.255-256: the statement "The sizes of our test based on T also align   better with 45° line as the sample sizes grow." is not clear to me in the   context of Setup 1 (which is discussed in this paragraph). In particular, it   does not seem to be the case in the interval of interest (0, 0.1). * Could a quantitative score, such as the area between the empirical curves   and the 45° line, be useful to make more objective statements about the   performances of the methods ? * It might be interesting to get an idea of the computational times involved   by the application of these tests.  ## Reproducibility  The test statistics as well as the numerical experiments are well described, so that it should be possible for a researcher willing to reproduce these analyses to write the necessary code. However, no package, or R script, is provided to make this exercise easier.