Inferring correlated distributions: boosted top jets

Introduction, page 2, line 2 from the bottom added text "...obtained from re-clustering the constituent particles of the jet with a smaller R.. "

Section 2.3, page 8, 2nd and 3rd paragraph from above added text " Additionally, we compute the Kullback-Leibler (KL) divergence between the posterior and the data distributions both for each class and for the complete case where no labels are used. The latter case in particular allows us to get a broader picture which does not rely on knowledge of “true” parameters. In all cases, a smaller divergence signals a better agreement. Although the KL divergences provide useful metrics, they are not rigorous statistical comparisons between models. Depending on the access to the true parameters, one could use an array of more involved but powerful techniques. If the true parameters are known, posterior credible intervals could be computed and a coverage study could be performed by bootstrapping the dataset. If one does not wish to rely on access to the true parameters, a posterior predictive check could be performed, which also involves additional samplings under the learned posterior model. Thus, more rigorous tests will in general involve ad- ditional, expensive simulations. In this work, we find that the cheaper metrics provide enough information to justify the need for correlations and validate the specific modeling proposed, without the necessity of more expensive computa- tions. However, future implementations may and probably will need to rely on these well-established suite of metrics to ensure that the learned probabilistic model is appropriate"

Section 3.1, page 10, last paragraph added text "Although the choice of Σ is subjective, it is not arbitrary: the prior needs to encode the trust in the Monte Carlo simulations while providing an adequate range of allowed discrepancies. Because in this work we have access to the true parameter values, we can validate the choice of Σ simply by observing that the resulting posterior can be likelihood-driven and centered around these true values for large enough number of events. When applying this method to data where the true values are unknown, a data-driven prior validation should be implemented. One possibility is to perform prior predictive checks [51] to ensure that the chosen Σ produces realistic datasets. Another possibility is to build a hierarchical model where Σ is promoted to a random variable and marginalized over (see Ref. [52], Chapter 5), reducing the prior dependence at the expense of increased model uncertainty. Since the goal of this work is to validate the introduction of correlations, we leave a more detailed prior exploration for future work."

Section 3.1, page 11, 1st paragraph added two new references [51] and [52]

Section 3.1, page 11, last paragraph of this section added text "... , although we have not explicitly explored the lowest sensitivity of the tagger since we expect fairly enriched samples to be available from t¯t production."

Section 3.1.1, page 13, line 4 from the bottom added text "...with the final π1 ∈ [0, 0.5]."

Section 3.1.3, page 16, line 5 of the 1st paragraph of this section added text "For all cases, we observe how if the number of events is too small the prior dominates and biases the inferred probability. The number of events needed to overcome the prior bias depends on the choice of Σ."

Section 3.1.3, page 16, line 3 of the 2nd paragraph of this section added text "... one should note that the tight prior still biases all distributions even for 10^5 events while the loose prior..."

Section 3.1.1, page 13, Figure 5. changed the scale on the x-axis of the right panels

Section 3.1.2, page 15, Figure 7. changed the scale of the x-axis of the right panels

We updated the repository providing the requirements list.