Bayesian Hierarchical Copula Models with a Dirichlet–Laplace Prior

Round 1

Reviewer 1 Report

The authors use multiple chains of MCMC to generate posterior samples. Is there any convergence checks for the posterior samples? It would be nice to confirm that all results are converged.

In the simulation study, equal number of observations were sampled. However, this might not be feasible in the real life e.g. patients with rare diseases. How do the authors deal with different group sizes?

When validating the approach, the authors computed MSE by comparing observation and the value from the posterior samples, with the full dataset. Have the authors consider leaving out 10 or 20 percent of the data and do some additional validation exercise to assess the model calibration?

Some minor comments on the notations used in the paper. First, "Logis" is rarely used for logistic transformation. Usually, the term "Logit" is used. Second, the authors use the expression such as "i = 1, ..., m". Strictly speaking, this is wrong because i cannot be 1, ..., m simultaneously. Instead, it is better to use "i \in {1, ..., m}", where \in is the latex expression.

Author Response

1. The authors use multiple chains of MCMC to generate posterior samples. Is there any convergence checks for the posterior samples?

It would be nice to confirm that all results are converged.

Thanks for the question. In the new version we report diagnostic checks for the chains

In the simulation study, equal number of observations were sampled. However, this might not be feasible in the real life e.g. patients with rare diseases. How do the authors deal with different group sizes?

This extension can be easily introduced in the model as documented by the first real data application

When validating the approach, the authors computed MSE by comparing observation and the value from the posterior samples, with the full dataset. Have the authors consider leaving out 10 or 20 percent of the data and do some additional validation exercise to assess the model calibration?

In order to be more exhaustive on this issue, we have introduced a posterior predictive check in the form of WAIC as in Gelman et al. 1996 (Statistica Sinica, POSTERIOR PREDICTIVE ASSESSMENT OF MODEL FITNESS VIA REALIZED DISCREPANCIES). We have done it in the two real data examples.

Time constraints for presenting the new version of the paper did not allow us to perform it in the simulation study

Some minor comments on the notations used in the paper. First, "Logis" is rarely used for logistic transformation. Usually, the term "Logit" is used. Second, the authors use the expression such as "i = 1, ..., m". Strictly speaking, this is wrong because i cannot be 1, ..., m simultaneously. Instead, it is better to use "i \in {1, ..., m}", where \in is the latex expression.

Thanks for pointing them! We have made changes accordingly.

Reviewer 2 Report

This paper presents a method with a Bayesian hierarchical copula model using a new dirichlet-laplace prior. The work is based upon a previous Bayesian hierarchical copula model but with a modified prior distribution. The method is evaluated through simulations and financial time series data. 

I have the following suggestions:

1. The paper includes a proof that the previous work led to improper posterior. I suggest the authors talk more about what improper posterior implies. My understanding is that if the distribution is improper, the MCMC sampling results would be meaningless. I think the paper can add more discussions around there. 

2. From the appendix, my reaction is that it is not straightforward to see if a proper prior leads to proper or improper posterior in Bayesian inference. So my natural question is, would the Dirichlet-laplace prior chosen by the authors lead to a prior posterior? Is there any proof about it?

3. line 25 and line 106. “The use of small parameters of the inverse Gamma priors simply hides the problem without actually solving it Berger [2].” It would be better to illustrate the “problem” mentioned here. What problem is hidden by the previous approach and why can the proposed approach solve it?

4. section 3.2.1 Column Vertebral Data application shows a direct comparison between the proposed approach and the referenced previous work. Are there any conclusions that can be drawn from the comparison? Which method is better or are they similar? In terms of both computation and inference results.

minor language issues: 

line 25 and 106: “the use of small parameters of the inverse Gamma priors simply hides the problem without actually solving it Berger [2].” – as shown in Berger[2]

Author Response

This paper presents a method with a Bayesian hierarchical copula model using a new Dirichlet-Laplace prior. The work is based upon a previous Bayesian hierarchical copula model but with a modified prior distribution. The method is evaluated through simulations and financial time series data. 

I have the following suggestions:

1. The paper includes a proof that the previous work led to improper posterior. I suggest the authors talk more about what improper posterior implies. My understanding is that if the distribution is improper, the MCMC sampling results would be meaningless. I think the paper can add more discussions around there. 

Thanks for the suggestion, we have included a sentence on the danger of using improper posteriors.

2. From the appendix, my reaction is that it is not straightforward to see if a proper prior leads to proper or improper posterior in Bayesian inference. So my natural question is, would the Dirichlet-Laplace prior chosen by the authors lead to a prior posterior? Is there any proof about it?

In a parametric setting, It can be proved that when a proper prior distribution is adopted and the likelihood is bounded (as in our case) the resulting posterior distribution is proper no matter what is the sample size and no matter what is the (finite) number of parameters.

3. line 25 and line 106. “The use of small parameters of the inverse Gamma priors simply hides the problem without actually solving it Berger [2].” It would be better to illustrate the “problem” mentioned here. What problem is hidden by the previous approach and why can the proposed approach solve it?

We have included a sentence which explains our statement.

4. section 3.2.1 Column Vertebral Data application shows a direct comparison between the proposed approach and the referenced previous work. Are there any conclusions that can be drawn from the comparison? Which method is better or are they similar? In terms of both computation and inference results.

We have included a sentence about comparing methods

and evaluated the WAIC for the model

minor language issues: 

line 25 and 106: “the use of small parameters of the inverse Gamma priors simply hides the problem without actually solving it Berger [2].” – as shown in Berger[2]

Thanks for pointing it!