The authors propose an interesting framework for bias prediction and correction. The framework models the bias in climate model outputs, which may have resulted from any approximations that were made in modeling the behavior of the system. The novelty of the bias correction framework comes from its ability to model spatial association among different locations and uncertainty estimation. The climate model outputs, observed data, and the bias are all modeled as Gaussian processes. This enables bias prediction and uncertainty estimation in these separate components, providing additional utility to the framework in real-life decision-making. The framework is evaluated on simulated data.

Strengths:

1. The paper provides a clear overview of the methodology. Figure 11 is especially interesting and useful in understanding the capability of the bias correction framework.
2. Experiments under different data distributions are very helpful in understanding the benefits of the bias correction framework.
3. The supplementary information is useful. Section A1, especially, explained the GP modeling framework very well.

Concerns:

1. The experiments and results are only on simulated data. While the experiments are beneficial in understanding how the model would perform under different data distributions, the benefits of the framework on real observed / climate model data have not been validated.
2. The clarity of the paper can be improved. Addressing these concerns may make it easier to follow the discussion in the paper.
1. In several places, the term “model” can be clarified. For instance, if it is the climate model or the GP model that is being discussed.
2. Phrases like “spatially varying parameter of the PDF for the climate model output” are a little unclear. Does this refer to the hyper-parameters from the GP model?
3. Figures 3 and 5 have not been discussed as part of the main results/discussion. It is unclear how these figures factor into the main discussion.
4. It is unclear if the authors are separating the train and test set to evaluate the model performance for the GP models. Training on test set may result in overfitting and high R2 scores.
5. The abstract and introduction talk about several potential benefits of the model. In the current structuring, it is a little difficult to follow along which statements correspond with validated claims in the paper and which statements are just potential benefits / future directions. More emphasis on statements that support the results presented in the result section may strengthen the case for the proposed framework.
6. Similarly, in the discussion section, it may be helpful to separate the potential directions / future work in a separate paragraph as opposed to the

Questions:

1. Why is the uncertainty not being plotted for the phi z case in Figures 7 and 9?
2. Is the model performance evaluated on the train set itself?
3. How is the “robustness” of uncertainty propagation being evaluated?
4. There may be concerns relating to the computational complexity of the framework. Since the three terms are modeled as exact GPs, the time and space complexity will grow quickly for larger datasets. Are there any variants/modifications to the framework that are being studied to overcome these challenges?
5. Since simulated data is being used, are the model performances averaged over several repetitions?
6. Does the proposed framework use quantile mapping along with GP modeling for bias correction?
7. How many iterations were used for the MCMC sampling?

Other Comments:

1. The paper can benefit from more discussion on how uncertainty estimates are impacted by bias correction.
2. It is unclear to what extent the quantile mapping mitigates the bias and to what extent the proposed GP framework is useful in bias correction.
3. In the abstract, it is suggested that the model provides value addition that is more than that of “alternative approaches.” A comparative analysis with the alternative approaches may help substantiate this claim.
4. Appendix equations A1 to A11 were really helpful and can potentially be included in the main section

Review comments on the “Bias Correction of Climate Models using a Bayesian Hierarchical Model” submitted to Geoscientific Model Development by Carter et al.

In the submitted manuscript, the authors proposed a Bayesian Hierarchical framework for bias correction of the climate model outputs. According to the authors, the proposed framework has the advantage of conserving the spatial covariance structure of the model bias. Further, the proposed technique can also consider the uncertainties during the bias correction. In the submitted manuscript, the authors spent many words explaining the concept of the proposed approach. The proposed technique, a hierarchical shared latent generating process, was tested on synthetic data against two baseline techniques: a non-hierarchical, shared latent generating process, and a non-hierarchical, non-shared latent generating process. The results showed that the proposed technique does deliver better performance than the baseline techniques in terms of preserving the spatial covariance structure of the climate model and allowing the propagation of uncertainty. Per the author's statements, the novelty of this paper lies in applying a Bayesian hierarchical model to mitigate some of the limitations of pre-existed bias correction techniques. Despite this enhancement, the reviewer believes the submitted manuscript should undergo at least one round of major revision before potential publication for reasons listed below:

1. Manuscript Structure: The reviewer recommends a comprehensive revision of the submitted manuscript to adhere to a conventional structure, encompassing introduction, data, methodology, results, discussion, and conclusion. The current manuscript structure leads to repetitive content, with similar discussions occurring in sections such as 2.3 and later in section 3, particularly regarding the limitations of pre-existing bias correction approaches. To improve continuity, the reviewer suggests maintaining focus within each section, as the current nested content creates a discontinuous logic flow that may be challenging for readers to follow.

1. Novelty: The reviewer observes that the novelty of the submitted manuscript is not adequately emphasized. In the submitted manuscript, only a handful of bias correction related work were mentioned in the introduction section. However, none of the cited works applied the Bayesian-related technique for bias correction. The reviewer believes that there should be more literature available that are related to this study. Notably, the authors themselves acknowledge the foundation of the submitted work on Lima et al. (2021) in section 3. However, the reviewer is left questioning the relationship between Lima et al. (2021) and the current study. To mitigate potential confusion and provide a more comprehensive background, the reviewer recommends introducing this information more explicitly. This comment echoes concerns outlined in detail in Comment #1.

1. Experiment design: Asides from comment#1 where it is suggested the author re-organize the manuscript, the reviewer suggests conducting experiments with the proposed technique using real climate model output. Experiments on climate model output could better highlight the strength and further verify the effectiveness of the proposed techniques.

1. Methodology: It appears to the reviewer that the methodology of the submitted study should be better introduced and clearly explained. It appears to the reviewer that important information is missing which prevents the audience from following and telling the scientific and technical values of the proposed bias correction approach. For instance, to the reviewer’s understanding the non-hierarchical single process model and the non-hierarchical shared process model are the benchmarks to compare with the proposed hierarchical shared process Bayes model. However, within the entire methodology section (i.e., Sections 2 and 3), the definition of “single process model” is not clearly explained (only described with one sentence in line 331). Then later in the result section, the term “single process model” suddenly appeared which caused the reviewer to be confused. The reviewer went to the Appendix A to look for an answer but did success as the formulation of the “single hierarchical model” seems undescribed. The reviewer also finds it hard to understand why the link function is necessary to transform the parameter space of the standard deviation to the same as the sample space (Section 3.2). Is that a common practice for Bayesian models? The reviewer thinks a more detailed explanation will be helpful.  

Minor comment:

The time series plot in Figure 11 exclusively illustrates the performance of the proposed Bayesian framework. The reviewer suggests including the performance of benchmark techniques in the same figure to facilitate comparisons. 

Major comments:

1. I definitely agree with other two reviewers that the manuscript is wordy and lengthy in its current shape. Specifically, there are some technical assumptions that are more or less mentioned in every single section (e.g. sum of GP is still a GP, bias is assumed to be independent from the in-situ observations, etc). I suggest authors revise their manuscript such that there can be a section that focuses on describing the assumptions they use in their method.

2. Given that the authors are about to clarify and shorten their assumptions, it is still of great interest whether the proposed method can work if the assumptions are slightly violated. Now the experiments are conducted using fully synthetic/simulated data that strictly follows the assumptions. I suggest authors prepare an example where it is not abundantly clear whether the assumptions still hold (a real data example), or at least generate synthetic data that intentionally violates the assumption. Some really strict assumptions in my opinion are:

(i) the bias is independent of the in-situ observations

(ii) the bias is time independent

For example, if the authors can conduct still a synthetic experiment where the generated bias is slightly dependent on the time and climate model output, and run their algorithm against this case, this would make the proposed algorithm stronger. Another option for the authors is to refer to existing literature and argue that some of the assumptions hold for most cases in real applications.

3. In terms of the structure of the manuscript, I personally have to understand the contribution of the manuscript until I start reading line 290 in page 12. It would help if the authors can make this clear very early in the manuscript.

Minor comments:

1. Is "1 process" just doing a GP for Y and "2 process" doing both GP for Y and Z? It is not very clear when I read through Figure 7 and its associated experiment section.

2. Line 115: "the their" is a typo

3. I personally find that Appendix A is easier to follow than some of the text in the main body. Authors may want to consider restructuring the manuscript