__ Summary and Contributions__: * Extends the framework / algorithm presented in [CR14] with the idea of EVAL
approximators. Gives a new algorithm for approximating the distance between two
probability distributions and gives theoretical guarantees for sample complexity.
* Applies this framework to four settings: Bayesian networks, Ising models,
multivariate Gaussians, causal models, extending or improving upon existing
results.

__ Strengths__: The central idea is conceptually simple, which makes it adaptable to a variety of
settings. The claims in this paper are supported by theoretical proofs. The topic
addressed (distance learning / testing) is highly applicable to the broader
Statistics / ML community.

__ Weaknesses__: The novelty of this paper over [CR14], upon which the central result of this paper
is based, is not immediately clear to me. However, this may simply be because I am
not as familiar with the surrounding research. I discuss this in greater detail
in the "Relation to prior work" section of my review. The paper would benefit from a
more thorough discussion of the differences between the two papers, and how
Algorithm 1 enables results that were not possible using the ideas presented in
[CR14].
The paper would also benefit from empirical studies of its performance in practice.

__ Correctness__: The central results of the paper are supported by proofs in the Appendix. No
empirical / simulation studies are given.

__ Clarity__: The ideas are presented clearly.
The paper overall could benefit from restructuring,
in my opinion: for example, moving Section 2.6 (Previous work) into or just following
the introduction; giving the main conceptual result its own section; and putting in
another sections the implications of the conceptual result for the various illustrative
models.

__ Relation to Prior Work__: The main conceptual contribution (Algorithm 1, the use of EVAL approximators)
is a direct extension of [CR14], as mentioned by the authors in Section 2.1 and at
the end of Section 2.6. This paper would benefit from a more thorough discussion
of of the limitations of [CR14] in contrast to the paper under consideration.
In particular, it would be useful to understand why the results that flow from
Algorithm 1 would not have been directly attainable through the framework
proposed in [CR14].
In terms of the results that flow from Algorithm 1, the authors lay out clearly the
limitations of past work in testing for the models considered (Bayesian etworks,
Ising models, multivariate Gaussians, etc.)

__ Reproducibility__: Yes

__ Additional Feedback__: *** UPDATE AFTER REBUTTAL ***
I thank the authors for addressing my concerns regarding the novelty of this
paper, especially over [CR14]. I am glad to hear that the authors intend to
include their extended comparison with [CR14] in the paper itself (ideally, as
part of Section 2.6). I adjusted my overall score accordingly.
Typo that I spotted: line 61 - should "class of" be "classes of"?

__ Summary and Contributions__: This paper considers estimating the statistical distance between two generative models given sample access from the two models. The general framework proceeds by first learning the model, and then a simple unbiased estimator of the statistical distance can be applied with the learnt model for distance estimation. The framework is applied to bayesian networks, Ising models, gaussian distributions and causal models. In particular, a new learning algorithm for bayesian networks on a known DAG G is introduced.

__ Strengths__: First computational and sample efficient distance approximation algorithm for a variety of structured high dimensional distributions.

__ Weaknesses__: The approach is rather straightforward, and the technical novelty is not clear.

__ Correctness__: Yes

__ Clarity__: Yes

__ Relation to Prior Work__: Yes

__ Reproducibility__: Yes

__ Additional Feedback__: Post rebuttal comment:
After discussion with other reviewers, I think the paper is somewhat between 6 and 7. Therefore I choose not change my score.

__ Summary and Contributions__: The paper is concerned with the problem of total variation distance estimation in a variety of prominent high dimensional structured distributions. The principle contribution is a scheme centered around the use of EVAL approximators.
A (beta, gamma) EVAL approximator for a law P is a function E_P with an associated distribution \hat{P} such that TV(P, \hat{P}) \le \beta, and for any x, |E_P(x)/P(x) - 1| \le \gamma. (0,0) EVAL queries were first used in CR14 to develop strong testing/distance approximation schema. Thm 2.3 of this paper extends this is (\epsilon, \epsilon) EVAL queries, showing that O(\epsilon^{-2}) samples from a distribution P and O(\epsilon^{-2}) queries to such EVAL approximators for two laws P and Q allows efficient estimation of TV(P,Q) to error O(\epsilon).
This allows the authors to construct efficient distance approximation (and tolerant testing) schema by developing EVAL approximators. This is pursued in a variety of settings by utilising recently developed methods in the literature that learn efficient approximations to distributions in the relevant classes.

__ Strengths__: I find the method underlying Thm 2.3 to be clever, simple, and flexible. The resulting bounds on sample/time complexity of distance approximation and tolerant testing are novel, and are developed for high dimensional families relevant to the recent literature.

__ Weaknesses__: To me the main weakness of the paper is insufficient discussion of how tight these results are.
I'll focus on sample complexity. The main discussion regarding this (starting line 141) argues that since for completely unstructured distributions and for product distributions on binary cubes the sample complexities of learning are not too separated from that of distance approximation. However, to me these are natural edge cases of high dimensional models, and I don't find it implausible that something different can occur in the bulk of the problem.
This matters because of features such as the exponential dependence on width in the case of Ising models that occur in the derived bounds, which don't appear in non-tolerant testing. (Of course, in this case, this could be due to the fact that the learnt \hat{P} in KM has |\hat{P}(x)/P(x) - 1| \le \epsilon for every x, which is much stronger than TV(\hat{P}, P) < \epsilon.)
I think clear discussion of if the individual bounds of secs 2.2-2.5 are (expected to be) tight or not would go a far way in strengthening this aspect of the paper.

__ Correctness__: Yes.

__ Clarity__: I found the paper well written and easy to read. The problem is contextualised well, and the main idea is well explained. I also appreciate the effort taken to ground the discussion of each of the models studied.

__ Relation to Prior Work__: Yes. To the best of my knowledge, most relevant work is discussed, and the differences well delineated.

__ Reproducibility__: Yes

__ Additional Feedback__: --- Section 2.1 and 2.2-2.5 and 2.6 seem to me to be thematically separate sections. The first sets up the underlying method, and context, the next four can be viewed as applications of this when coupled with learning methods, and the final is clearly separate. Reorganising along these lines should smoothen the presentation.
--- I think the argument of Corollary D.3 is simple and short enough to be pushed to the main text.
--- for Ising models - [1] describes the minimax rate of learning an Ising model in total variation given the underlying graph of the model. In KM17, the sample complexity of estimating the graph is much smaller than that of approximation in the sense used in Thm C.1 (there's no n^8). Can these be daisy chained to improve the sample costs of Thm. 2.5?
[1]: arxiv.org/abs/1806.06887