Paper: Improved Variational Bayesian Phylogenetic Inference using Mixtures (ICLR-2024)

Reviewer comment (Novelty):
The mixture approximation for variational posterior has been already used in previous works on VAE. Therefore, the novelty is restricted to the application of Phylogenetic Inference.

Author response:
- **On the Novelty and Application of VBPI-Mixtures:** We acknowledge that mixtures of variational approximations and normalizing flows have recently been studied in VI and machine learning. However, applying these techniques to Bayesian phylogenetic inference is non-trivial, and the efforts have been previously celebrated at NeurIPS and ICLR (NFs: Zhang, (2020); GNNs: Zhang, (2023)). Hence, our novel application of mixtures in BBVI represents a meaningful advancement in phylogenetic modeling, which will be appreciated by the ICLR community.

Reviewer comment (Novelty):
Although the usage of Normalizing Flow could lead to an improvement in performance, they are not new in the VI context. Similar to the mixture approximation, they could be new in the application of Phylogenetic Inference.

Author response:
- **On the Novelty and Application of VBPI-Mixtures:** We acknowledge that mixtures of variational approximations and normalizing flows have recently been studied in VI and machine learning. However, applying these techniques to Bayesian phylogenetic inference is non-trivial, and the efforts have been previously celebrated at NeurIPS and ICLR (NFs: Zhang, (2020); GNNs: Zhang, (2023)). Hence, our novel application of mixtures in BBVI represents a meaningful advancement in phylogenetic modeling, which will be appreciated by the ICLR community.

Reviewer comment (Evaluation):
The improvement in NLL compared to other baselines in Table 2 is not significant.

Author response:
- **Regarding Improvements in NLL and Other Metrics:** We understand your concerns about the incremental improvements in NLL as shown in Table 2. However, we emphasize that in the realm of VI, the sole focus should not be on achieving the lowest NLL. Instead, the consistency and robustness of the model in approximating the target distribution are equally crucial. Our findings show significant improvements in the variance of estimates for the marginal likelihood, especially with an increase in the 'S' parameter. This reduction in variance is a vital indicator of the model's improved capability in capturing the uncertainty of phylogenetic structures, which is a key contribution of our study.

Reviewer comment (Writing):
In section 3.1.1, "we conclude that extending the VIMCO estimator to S > 1 cannot be trivially achieved without our derivation provided above," what exactly do you mean by this?

Author response:
- **Clarification on VIMCO Extension for Discrete Mixtures:**
    We are happy to improve the clarity of this statement by the following alternative formulation: "we emphasize that naively replacing all occurrences of variational approximations in the VIMCO estimator with mixtures of variational approximations will not result in the correct estimator (to see this, please see Appedix C). To this end, we provide the necessary derivations in the next section and in Appendix C."

    Appendix C will then be extended with the following text, highlighting the importance of our derivations in our proposed work: "We demonstrate how naively introducing mixtures without carefully performing the derivation gives the incorrect expression. We do this by inspecting Eq. (5) in (Mnih and Rezende, 2016) and employing their notation. In fact, it is sufficient to consider the first term in Eq. (5) in (Mnih and Rezende ,2016), i.e. 
    
    $E_{Q(h^{1:K}|x)}\left[\sum_j \hat{L}(h^{1:K})\nabla_{\theta}\log Q(h^j|x)\right],$
    
    to see that the naive introduction of a mixture $Q(h|x) = \frac{1}{S}\sum_{s=1}^S Q_s(h|x)$, where each component has parameters $\theta_s,$ gives 
    
    $\frac{1}{S}\sum_{s=1}^SE_{Q_s(h_s^{1:K}|x)}\left[\sum_j \hat{L}(h_s^{1:K})\nabla_{\theta_s}\log \frac{1}{S}\sum_{s'=1}^S Q_{s'}(h_s^j|x)\right],$
    
    which is an incorrect expression for the first term of the VIMCO estimator for mixtures. Instead, the correct expression, as we have showed in our submission, is 
    
    $\frac{1}{S}E_{Q_s(h_s^{1:K}|x)}\left[\sum_j \hat{L}(h_s^{1:K})\nabla_{\theta_s}\log Q_{s}(h_s^j|x)\right].$ 
    
    As such, carefully performing the derivation is necessary to arrive at the correct expression for the objective."
    
    
    As a final remark on the utility of our derivation of the VIMCO estimator for mixtures: In the original VBPI paper (Zhang and Matsen IV, 2019), the VIMCO estimator is derived for their specific choices of model and variational distributions. The derivation (given in their Appendix A) closely follows the original VIMCO paper (Mnih and Rezende, 2016). In a similar vein, we extend this VIMCO estimator for a new class of variational distributions, mixture distributions, which proved to be a less straightforward derivation than the one provided in (Zhang and Matsen IV, 2019). To our knowledge, this derivation is not available in the literature, and so our extension is an important contribution to the field of mixture learning in BBVI for discrete latent variable models.