Recently proposed quasi-Bayesian (QB) methods initiated a new era in Bayesian computation by directly constructing the Bayesian predictive distribution through recursion, removing the need for expensive computations involved in sampling the Bayesian posterior distribution. This has proved to be data-efficient for univariate predictions, but extensions to multiple dimensions rely on a conditional decomposition resulting from predefined assumptions on the kernel of the Dirichlet Process Mixture Model, which is the implicit nonparametric model used. Here, we propose a different way to extend Quasi-Bayesian prediction to high dimensions through the use of Sklar's theorem by decomposing the predictive distribution into one-dimensional predictive marginals and a high-dimensional copula. Thus, we use the efficient recursive QB construction for the one-dimensional marginals and model the dependence using highly expressive vine copulas. Further, we tune hyperparameters using robust divergences (eg. energy score) and show that our proposed Quasi-Bayesian Vine (QB-Vine) is a fully non-parametric density estimator with \emph{an analytical form} and convergence rate independent of the dimension of data in some situations. Our experiments illustrate that the QB-Vine is appropriate for high dimensional distributions ($\sim$64), needs very few samples to train ($\sim$200) and outperforms state-of-the-art methods with analytical forms for density estimation and supervised tasks by a considerable margin.
Paper
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Summary
Quasi-Bayes, in the sense of building a model as a consistent sequence of predictive distributions, has recently received a lot of interest. The initial construction in [25] used products of bivariate copulas to build their predictives; the authors propose here to extend the construction to vine copulas, a flexible model for multivariate distributions.
Strengths
* Quasi-Bayes in the wake of [25] is indeed a hot topic in Bayesian statistics, and the paper is potentially impactful. * The initial copula constructions in [25] felt a bit constrained, and more flexibility would intuitively help. * I overall enjoyed reading the paper, though I have many (hopefully constructive) comments below.
Weaknesses
## Major 1. p1 L1 I would avoid flattering formulations like "[25] initiated a new era of Bayesian computation" and L33 "[25] heralded a revolution in Bayesian methods", "liberating Bayes". While you are free to think it and predict a revolution, that kind of statements about a 2023 paper are necessarily statements of opinion or personal predictions at best, and we should let time tell whether a new era or a revolution happened. You can say, e.g., that [25] proposed a stimulating change of paradigm, that is a fact. More pragmatically, starting a paper with opinion statements puts the reader in a suspicious mood. 2. Abstract: the sentence "But extensions [...] used" is unclear. What assumptions? What kernel? 3. Figure 1 is not particularly useful, I'd say, and the fonts in the decomposition are a bit strange. I know ICLR papers, for example, tend to have a small summary figure at the beginning, but I believe that it doesn't especially help here. The space gained could be used to give e.g. a more formal introduction to vine copulas in the main text. 4. p2 L42 "DPMM kernel structure". Maybe remind the reader on p1, when you first mention the DPMM, of what it is and what you call the kernel. It will not be obvious to many. 5. The introduction should contain a sentence defining vine copulas, or "vines". It is likely not a standard notion in the NeurIPS community, and the introduction should make it clear what the title of the paper means. Actually, the definition is somehow informally given in the caption of Figure 1, but I had missed it at first reading. I would avoid putting anything else in a caption than the description of what the figure shows. Important details should go to the main text. 6. p2 L56 "under the assumption of well-identified simplified vine copula model". It is unclear at this stage what you mean here. 7. Eqn 2: as far as I understand, going from the first to the second line requires Sklar's formula, at least if I follow the derivation of [25]. If you also use it, this should be mentioned. There is a reference to Appendix A, where copula tools are introduced, such as Sklar's theorem, but I'm not sure the argument is explained. 8. p3 L 82-82, in what sense does $c^{(n)}(\cdot,\cdot)$ converge to $1$? How does is guarantee a convergence for $p^{(n)}$? Why is the copula symmetric? 9. The paragraph L102--L113 is a bit dense and can gain in clarity. In what sense and under what assumptions does the "univariate R-BP model converge to a limiting distribution ... with density..."? What do you mean by "Newton's algorithm"? Also, there are steps to explain if you want to talk about how quasi-Bayes "approximates a posterior density"; so far, we have only talked about prediction. What parameter do you consider? How do you approximate the posterior density? You could refer to the corresponding passages in [25] and explain them in a few words. Or avoid discussing parameter inference if it is not central to your contribution. 10. p4 L128 what do you mean by "the predictives are unconditional marginal densities"? The sentence is puzzling. 11. Importantly, the authors of [25] restrict their designs of predictive distributions so that they satisfy their martingale condition (Eqn (4.1) in the Arxiv v2 version). They use their Corollary 1 for univariate copulas, and note in Section 4.3 that a multivariate extension of their Corollary 1 is not easy, which prompts them to take instpiration from factorized kernels in DPMMs, for which they can guarantee (4.1). So the obvious question is: how do you guarantee that your vine predictives satisfy Fong el al.'s martingale condition? Or, alternatively, how do you guarantee the existence of the limiting martingale posterior $P^{(\infty)}$? It is possible that I am missing an obvious argument. 12. Lemma 3.2 uses $P^{(\infty)}$, which should be formally defined. Or at least, a reference should be given to the formal definition in [25]. But I think that giving a formal definition would help, since it would entail checking the martingale assumption of [25]; see bullet 11 above. 13. Overall, guarantees like Lemma 3.2 or Theorem 3.3 are guarantees in probability, under the distribution formed by the product of your predictives, am I right? In that case, these guarantees are meaningful only if I believe that the product of predictives is actually a good model for the data generating process. Is this realistic? Even under the classical simplifying assumption on conditional copulas that you are making? 14. p5 L146-173 the introduction to vine copulas is a bit confusing and informal. This section should be more formal, as understanding vine copulas is key to the paper, and I think not standard to many NeurIPS readers. I suggest having a subsection of Section 2 that formally introduces vine copulas, independently of their application to quasi-Bayes, and gives your example (23). Then Section 3 can focus on how you use vine copulas to define your joint predictive distributions. Also, I would use Pseudocode at the end of Section 3, to summarize all the steps in your procedure: hyperparameter tuning, estimation of the pair copulas, etc. I am not 100% confident that I can write this pseudocode from the text only. 15. Relatedly, having pseudocode would allow you commenting on the computational cost of the different steps. p9 L335 you mention "a search over a vast model space during estimation", but this has not been commented on before. 16. Theorem 3.3: what do you mean by "Assuming a correctly identified simplified vine strucgture for $c^{(\infty)}(\mathbf{u})$?". If I understand well, a more standard phrasing would be to formally introduce your model, say as $P$, and then state that under the distribution $P$, such and such is true. 17. The caption of Table 1 mentions error bars of two standard deviations averaged over 5 runs. I am unconfortable with the number of runs, which is intuitively too low for a $2\\hat{\sigma}$ interval to make sense, be it through a CLT or a Chebyshev argument. ## Minor * p1 L29 the notation $\mathbf{p}^{n}(\mathbf{x})$ has not been defined yet, so I would simply avoid using the notation here. * p2 L52 while you are not the only ones to do so, I don't see the need to boldface "main contributions". * Theorem A.1: I would define what a copula distribution is. * p3 L87 "intractable": following [25], I would even say it is not particularly desireable, as they try to bypass the need to specify a prior-likelihood pair. * In Eqn (8), the densities should appear with their arguments. * Section 5: don't you miss a log in your expressions of the log score?
Questions
Q1. Can you give pseudocode for your method applied to a particular task, including all hyperparameter tuning steps like pair-copula estimation? Q2. Can you then discuss the computational cost of each step, and identify the bottlenecks? Q3. A convincing answer to items 11-12-13 in my list of major comments would also make me reconsider my mark.
Rating
5
Confidence
4
Soundness
3
Presentation
2
Contribution
3
Limitations
This is fundamental work, and no immediate negative societal consequence is foreseen.
Rebuttal by Authors part 2 - proof
# Reply 11: We do not claim to converge to a multivariate $P^\infty(\mathbf{x})$. The $P^\infty(\mathbf{x})$ is only relevant when one wants to do predictive resampling to impute the unobserved data, as done in the martingale posterior for parameter inference. We do not investigate predictive resampling and do not advertise our work as such, nor focus on parameter inference. We need the marginals to converge (lemma 3.2). Then, for our theorem 3.3, this is purely a statement on the density as done in the vine copula literature, not on the limiting $P^\infty(\mathbf{x})$ martingale posterior in multiple dimensions. Similarly, theorem 3.1 adapts Sklar's Theorem to multivariate densities, holding for any $n$. For experiments, we show that our model approximates well all the datasets, doing density estimation, for which a $P^\infty(\mathbf{x})$ is not required. Our paper focuses on modelling predictive densities in an efficient way, which is a worthwhile pursuit as classical density estimators such as KDE are known to scale poorly with dimension. However, we were able to prove the martingale condition, see point 4 in the main reply, and the proof below. (We could not get it to format properly, we express our deepest apologies and hope it is still readable.) We write $\mathbf{c}^{(i)}= \mathbf{v}^{(i)}$ for clarity and denote inputs to copulas as vectors $(P_1(x_1),\ldots,P_d(x_d))=[P_i(x_i)]$: $$ \begin{aligned} &\int \mathbf{p}^{(n)}(\mathbf{x})\cdot\mathbf{p}^{(n-1)}(\mathbf{x}^n)d\mathbf{x}^n\ =& \int \prod_{i=1}^d \left( p^{(n-1)}(x_i)\cdot c^{n}(P^{(n-1)}(x_i),P^{(n-1)}(x_i^n)) \right) \cdot \mathbf{v}^{(n)}([P_i^{(n)}(x_i)]) \cdot \mathbf{p}^{(n-1)}(\mathbf{x}^n) d\mathbf{x}^n\ =& \prod_{i=1}^d\left(p^{(n-1)}(x_i)\right)\cdot\mathbf{v}^{(n-1)}([P_i^{(n-1)}(x_i)])\ \cdot \int\prod_{i=1}^d\left( p^{(n-1)}(x_i^n)\cdot c^{n}(P^{(n-1)}(x_i),P^{(n-1)}(x_i^n))\right)\cdot\frac{\mathbf{v}^{(n)}([P_i^{(n)}(x_i)])}{\mathbf{v}^{(n-1)}([P_i^{(n-1)}(x_i)])}\cdot\mathbf{v}^{(n-1)}([P_i^{(n-1)}(x_i^n)])d\mathbf{x}^n\=&\mathbf{p}^{(n-1)}(\mathbf{x})\frac{1}{\mathbf{v}^{(n-1)}([P_i^{(n-1)}(x_i)])}\int\prod_{i=1}^d\left(c^{n}(P^{(n-1)}(x_i),u_i^n)\right)\cdot\mathbf{v}^{(n)}([P_i^{(n)}(x_i)])\cdot\mathbf{v}^{(n-1)}([u_i^n]) d\mathbf{u}^n\=&\mathbf{p}^{(n-1)}(\mathbf{x})\frac{1}{\mathbf{v}^{(n-1)}([P_i^{(n-1)}(x_i)])}\int\prod_{i=1}^d\left(c^{n}(P^{(n-1)}(x_i),u_i^n)\right)\cdot\mathbf{v}^{(n-1)}([u_i^n])\cdot\mathbf{v}^{(n)}([(1-\alpha_n)\cdot P_i^{(n-1)}(x_i)+\alpha\cdot H(P_i^{(n)}(x_i)|u^n)])d\mathbf{u}^n\=&\mathbf{p}^{(n-1)}(\mathbf{x})\frac{\mathbf{v}^{(n)}([P_i^{(n-1)}(x_i)])}{\mathbf{v}^{(n-1)}([P_i^{(n-1)}(x_i)])}=\mathbf{p}^{(n-1)}(\mathbf{x}). \end{aligned}$$ The first equality is applying Sklar on $ \mathbf{p}^{(n)}(\mathbf{x})$ and using recursion (4) on the ensuing marginal densities $p_i^{(n)}(x_i)$. The second equality is Sklar on $\mathbf{p}^{(n-1)}(\mathbf{x}^n)$ and writing out the recursive ratio of copulas for $\mathbf{v}^{(n)}$ (Equation (8) of the main text). The third step is obtained by the substitution $du_i^n=p_i^{(n)}(x_i^n)dx$. Lastly, we use equation (4) for the cdf inside the copula. Then, the result follows by noticing that the bivariate copulas and the copula $ \mathbf{v}^{(n-1)}([u_i^n])$ integrate to 1 by copula properties (see e.g. the proof of theorem 6 in [9]), and $[(1-\alpha_n)\cdot P_i^{(n-1)}(x_i) + \alpha \cdot H(P_i^{(n)}(x_i)|u^n)]$ integrates into $[P_i^{(n-1)}(x_i)]$ due to it being a martingale marginally for each $i$. Consequently, predictive resampling is also possible with our approach and is as simple as sampling from the fitted copula, and marginally updating each univariate R-BP. This can be done in parallel across dimensions, instead of sequentially as in [25,31], which is computationally much more appealing and opens an interesting avenue for future work. Thank you for your perceptive remark, which has guided us to new insights.
Rebuttal by Authors - part 3
# Reply 12: We note that $P^\infty(x)$ used in lemma 3.2 is univariate, with the martingale condition being established for the univariate R-BP in [25] already. Following your advice, in the camera ready, we will dedicate a paragraph to the introduction of the martingale condition and the proof of the QB-Vine satisfying it. # Reply 16: We thank you for this comment, and have rewritten the statement with that structure. # Reply 17: Please see our reply to Q2 of reviewer iVqh, explaining our limitation due to [31]. With their code unavailable, we assume the best case scenario for other methods, believing those intervals are adequate there, but study the intervals of the QB-Vine in more detail, running 15 more runs for each point Figure 2 (see pdf). # Reply minor points: We incorporated the changes in our manuscript. # Reply Q1: Please find the pseudo-code in Figure 2 of the pdf, including hyperparameter estimation. Some brief clarification: the training data can be permuted in full or partially depending on how much data is available, amd is captured by the variable $M$ giving the number of permutations. We keep $M=10$. Other hyperparameters are the number of points $B_1,\ldots,B_l$ to select the bandwidth of the vine. Our experiments were run with a grid of size $50$ between $2-4$ for UCI dataset and $0.5-3$ for the rest. $V=0.8$ to have $5-$fold cross-validation, and $J=100$. $\rho$ can be optimised with gradients and converged very quickly in our experiments, requiring less than five evaluations on average. # Reply Q2: The computational cost of each step comes from the existing method, R-BP or vines [65,], with the exception that we half the time of the R-BP [25,39] due to the optimisation of the Energy score. Due to space constraints, please see our reply to weakness 3 of reviewer iVqh. Finally, we thank the reviewer many times for their great suggestions and critical comments. We truly believe the paper is stronger as a result, and express our greatest thanks for dedicating your time to our work. We hope we addressed your concerns, and remain available to discuss any further points you would wish to raise.
Thanks a lot for the detailed clarifications. Trusting the authors to include the proposed changes, I will increase my score to a 5. I am reluctant to increase more, because if I draw a parallel with a journal submission, for the latter I would have liked to review and carefully proofread a revised version of the paper.
Summary
This paper develops the recently proposed quasi-Bayesian methods by applying vine copula (hence named QB-Vine) to the recursive Bayesian predicative distributions and bypassing the need for expensive posterior integration. The proposed method consists of two parts: independent recursion of marginals by bivariate Gaussian copulas; and estimating the simplified vine copula to capture data dependence hence relaxing the kernel assumption for DPMM. Error bounds for both the distribution functions and the copulas are provided to justify the proposed QB-Vine. Numerical examples including density estimation and regression/classification are used to showcase the advantage compared with the state-of-the-art alternatives.
Strengths
The paper proposes a novel Bayesian method to compute the predictive distribution. The proposed method, QB-Vine demonstrates numerical advantage over alternatives. Theoretic characterization on the errors is provided.
Weaknesses
The dimension of the problems is relatively low (up to 64).
Questions
Should there be $|\mathcal S_{ij}$ after the second conditional distribution in equation (10)? Did you have dimension specific bandwidth $b$ and correlation $\rho$ for all the numerical experiments? Did it increase the overall computation time significantly by having them different for each dimension? Line 223: "convergence" should be "converges" Figure 2: why LPS increases from 500 samples to 900 samples for QB-Vine?
Rating
6
Confidence
4
Soundness
3
Presentation
3
Contribution
2
Limitations
See weakness
Summary
The authors propose a novel method for modeling high-dimensional distributions (for density estimation and supervised learning), where they break the estimation task into estimation of univariate marginals and estimation of a multivariate copula. To expedite the univariate estimation tasks they utilize the novel quasi-Bayesian (QB) estimation method, and they use a simplified vine copula to approximate the multivariate copula. They present empirical results that compare their method to others in density estimation and supervised learning.
Strengths
- The authors present a well-defined problem that is of potential interest to conference readership, place it within the existing literature, and clearly demarcate their innovations. - They break down the estimation problem into two sub tasks (estimation of the univariate marginals and the copula), and offer tangible contributions for both tasks, and provide convergence results for their estimator. - Empirical results are convincing regarding authors' method's practical usefulness.
Weaknesses
Although mostly outside of my expertise, this is a well-written paper overall. However, there are a number of potential changes that can improve the readability and accessibility of the paper. I list these in the section below.
Questions
- L29: The paper makes a confusing start regarding its notation. $n$ is used here without being introduced, on L69 it's referred to as $K$, which leads to further confusion. This is likely the most important variable in the paper, so a proper introduction is required - especially because the recursive nature of the approach in question may not be as obvious for a reader from a different subfield. - L163: Potential implications of using simplified vine copula can be discussed here, with a reference to forthcoming Thm 3.3. - L174: Any particular reasons for the choice of Cauchy as the initial distribution? - L188: Please use the additional space afforded after reviews to make this discussion more explicit.
Rating
7
Confidence
2
Soundness
3
Presentation
3
Contribution
3
Limitations
The authors provide a satisfactory discussion of the limitations of the paper.
Summary
Inspired by previous quasi-Bayesian (QB) methods, the recursive decomposition of Bayesian predictive posterior distributions, and vine copulas, the work introduces a new adaptation of QB to higher dimensions. The driving idea, introduced in Section 3 consists of making an adaptation of the previous decomposition for univariate densities to a higher number of dimensions, which ends up being a product of marginals times a vine copula with a certain design. Such vine copula models the conditioning among dimensions since the marginals do not. This one works via a decomposition into $d(d-1)/2$ elements that capture the dependency structure. Experiments on UCI datasets with a log-predictive metric show improvement wrt the selected SOTA methods with datasets of small N.
Strengths
I would like to add some points of strength that I think are worth to be mentioned: - The work introduces well enough the problem of higher dimension d in certain Bayesian inference problems and how some SOTA methods and frameworks might struggle when this is larger than a few dozens. Additionally, the reference to previous works and advances is important and well pointed out in my opinion, giving the right credit for the technical parts to each one of the previously published works. - The paper is in general concise, and I could follow the technical details -- so I don't think there are details missing in the math part, despite some lack of analysis in certain directions that I'd had loved to see. - In general, the idea of breaking the conditioning in this way, exploiting both recursive predictives and an additional element that captures correlations in a moderately-scalable way (i.e. vine copulas) is interesting to me and for the paper.
Weaknesses
Some comments that I would like to add in terms of weak parts or ideas that I consider kind of a problem (or at least I'm concerned about): - I think the technical derivations, and in general the notation could be improved. From the usual Bayesian perspective and for readers familiar with probabilistic methods, it is not really orthodox and the subscript-superscript system could be confusing at times. The work introduces too many things sometimes instead of focusing on clarity and highlighting the actual contribution and strong ideas proposed in the manuscript. At least, that's what I get from reading it. - The contribution and novelty are a big concern to me. Honestly, I feel that if Eq. (7) is the big contribution or the point of novelty of the paper, it would not be enough, as it is just an adaptation of the previous derivations in Section 2 / Background to a vine copula. Additionally, the ratio in Eq. (8) is trivial, right? nothing really to derive there, or am I wrong here? - The experiments are very limited, on small UCI datasets and only show one empirical result which is the improvement with the dimensions, however, no discussion about complexity/computational cost, run times, sensitivity to the choice of the vine copula system, etc is made. In that regard, I fear a bit how scalable really is the vine copula with such $d/(d-1)/2$ mechanism to capture conditioning.
Questions
**Q1** -- What is the scalability of the simplified vine copula in L146-L152? Could the authors add some extra info about the computational cost per iteration n? **Q2** -- Why so much focus on the log-predictive metric? Wasn't there an additional informative metric that could give the reader more insights about the performance compared with other inference methods? Is there a log() operator missing in the equation on L271? **After rebuttal comment:** Dear authors, thanks for your rebuttal and the clarifications made regarding my points of weakness and concerns. Some things are clearer now to me, so I am happy to increase my score to 5 now. However, I still feel that some issues raised in my review have not or at difficult to fix atm, and I feel that there are not yet strong reasons to clearly accept the paper (i.e. clarity/readability and empirical results are yet limited, and I still have doubts on the novelty side). I don't object to acceptance, but won't be the supporter of that either. Best of luck.
Rating
5
Confidence
3
Soundness
2
Presentation
3
Contribution
2
Limitations
Yes
I have raised my score
Thanks to the authors for addressing my concerns. I appreciate the added results which make the numerics more convincing.
I thank the authors for their response. I believe the modifications they propose will improve the paper.
Closing comments
We thank reviewers QJH9, 2HTp and qqFA for their positive review of our work and for increasing their scores. We believe this process has been fruitful in improving our manuscript and has led to a more cohesive paper. We kindly note the following improvements in response to the reviews: For weakness 1 of reviewer iVqh, we have improved the clarity of our paper in-line with reviewers iVqh's comments and those in Q1/Q2 of QJH9 and in Major 3,5,6,14 of qqFA. Regarding contribution, reviewers iVqh's comment is echoed by Major 2 of qqFA. We thank reviewers for their favourable view of the contribution of our work, citing "..., and clearly demarcate their innovations."-QJH9, "...a novel Bayesian method to compute the predictive distribution."-2HTp, "..., and the paper is potentially impactful."-qqFA. Lastly, we address comments on computational scalability, found in weakness 3/Q1 of iVqh, the comment on weakness 1 of 2HTp, and Major14/Q1/Q2 of reviewer qqFA.
Decision
Accept (poster)