The problem articulated and addressed by the paper is important, the techniques introduces are very general in their applicability, and the experimental results successfully demonstrate the utility of the techniques to addressing the problem. The fixed-point MCMC approach is very conceptually compelling, although it lacked theoretical analysis and rigor, and appeared to be not that significant to achieving good results (the results suggest a single iteration may be sufficient). After reading the highly suggestive section on the Markov chain, the theoretical results appeared underwhelming. A rigorous analysis of the Markov chain would have made the paper more self-contained and increased its technical strength. Alternatively, focusing on the single-step stabilization operator case, and obtaining more general theoretical results within that restricted setting, would have made the paper more self-contained and cohesive.

Optimization of the link structure for PR is not a new topic. Apart from papers mentioned in Related work, there are also those not reviewed, including "PageRank Optimization by Edge Selection" by Csaji et al., "Maximizing PageRank with New Backlinks" by Olsen, "PageRank Optimization in Polynomial Time by Stochastic Shortest Path Reformulation" by Csaji et al. **The novelty** of the study is questionable. The probability of reaching the target state \sigma can be viewed as the state's stationary probability for the graph, where the added edges are directed to the state \sigma and the matrix of transition probabilities is raised to an appropriate power. This observation does not immediately reduce the problem of the paper to a known task, however, it may partially explain the similarity between the theoretical part and the works of Olsen, where the stationary probability is maximized. In particular, Section 4 resembles the work "Maximizing PageRank with New Backlinks" (not cited in the paper), where M. Olsen considered a reduction of a Markov chain optimization problem to the independent set problem, which is equivalent to the vertex cover problem. Theorems 5.1, 5.3 are reasonable, but very simple and resemble Lemmas 1,2 from [15].

This paper considers the problem of developing flexible choice models that are not constrained to satisfy traditional, restrictive choice axioms (such as Luce's axiom of independence of irrelevant attributes, IIA), but that can be tractably inferred from data. A (discrete) choice model over n items specifies probabilities of the form p(i,S) Prob( i chosen from S) for each subset of items S \subseteq [n] and each item i \in S. One of the most widely used models of discrete choice is the multinomial logit (MNL) choice model, which can be inferred efficiently from data but which is constrained to satisfy IIA and other restrictive assumptions. The paper proposes a new Markov chain based model of discrete choice that is parametrized by a (n x n) pairwise selection probability matrix. The model avoids several of the earlier restrictive assumptions, but is shown to satisfy an interesting property termed contractibility, which in turn also implies a reasonable property of uniform expansion. Parameter estimation in the model is done by maximum likelihood (the log-likelihood function is non-concave in general, but the experiments suggest that good parameters are learned).

The paper explains well how computing RSA using estimates of regression weights can result in a biased similarity matrix. However, in many cases in neuroscience, the RSA is computed directly on the patterns of activity, and not the estimates of regression weights beta. This diminishes the relevance of this paper to the neuroscience field. The authors very briefly address this alternate way of computing RSA in lines 123-128. It is unclear how this alternative RSA computation is biased if it does not depend on a proxy for beta estimates, and needs to be addressed further.

The paper is well-written and clear. The proposed idea is interesting. I have the following comments/questions: 1) Does the Liptschiz assumption hold here with a probability or is it assumed to always hold? 2) Figure 1: should it be \bar{s}_2 instead of s_2 in the caption? The use of bar for non-sets is confusing. I do not see the need for the last intersection in Equation 4. 4) When you repeatedly apply Equation 4, the number of states that satisfy the safety constraint shrinks because you use Liptschiz in the worst scenario sense.

The authors motivate the proposed model with the setting in which items have "true" but unobserved labels/ratings and the observed labels/ratings given by evaluators are potentially incorrect. This differs from the very common problem in recommendation systems or collaborative filtering where evaluators provide their subjective ratings but there is not assumed to be any "true" rating (e.g., users of Netflix giving 1-5 star ratings to movies). This seems like a common but underexplored setting that is worthy of further study within machine learning. The authors are also right to highlight interpretability as a desired aspect of any machine learning solution that may yield post-hoc insights into common human biases and thus suggest corrective measures. This paper does a good job of motivating the proposed model and situating it within the crowdsourcing and human annotation literature.

If my understanding is correct, Theorem 1 of the authors does not quite apply to their algorithm ADD-1/2-Smoothed Low-Rank. Instead, it applies to the non-computable algorithm where they assume that they have a minimizer of the objective function in Theorem 3. It is not clear if the alternating optimization algorithm proposed in the paper is guaranteed to converge to a minimizer of the objective in Theorem 3. If this is true, the authors should mention this before stating Theorem 1 to avoid misleading the reader. The "discounting" seems important from the Experiments section but this is not described in the main paper. If this is so important, the authors should make room for this in the main paper. The main results (Theorem 1 and 2) are not so surprising given that this is almost a parametric estimation problem with mk parameters (so the rates should be km/n).

The paper is globally sound and makes a new contribution to an important topic. However some technicalities need to be addressed and a revised version should be encouraged. Major remarks: - There is a confusion on whether the Markov chains under consideration are supposed to be stationary or not. Indeed, the concept of t-trail either requires that the Markov chains under study are stationary or one should specify that all the trails start with same initial distribution, i.e. these trails are observations of (X_1,X_2,X_3) and not (X_s, X_{s 1},X_{s 2}) for some s. I first understood that you adopt the second approach (as you count the parameters of initial distributions as free parameters) but in the real data experiments, you take many (3001)-trails and break them into 3000 overlapping 3-trails (by the way, you need a 3002-trail to obtain these).

Technically the paper is very strong. The results presented by the authors are, to the best of my knowledge, novel and significant. However my main criticism of the paper is that the presentation is very esoteric. The is clear already in the introduction where the authors fail to explain some of the basic notation that is central to the remaining of the paper, see (1)-(3) below. This continues throughout the paper making it hard to read for non-experts in the field, see e.g.

The proposed model is novel and practical, as seen from the experimental result. It is rare to see a Bayesian nonparametric model being applied to large data as it is generally not very scalable. It is a feat to see this model applied to data with high dimensions (9000 dimensions with millions of events). I am interested to know how much time is spent for training? It would be good to also present the computational time (say in the supplementary material).