Recursive Bayesian Networks: Generalising and Unifying Probabilistic Context-Free Grammars and Dynamic Bayesian Networks

Abstract: Probabilistic context-free grammars (PCFGs) and dynamic Bayesian networks (DBNs) are widely used sequence models with complementary strengths and limitations. While PCFGs allow for nested hierarchical dependencies (tree structures), their latent variables (non-terminal symbols) have to be discrete. In contrast, DBNs allow for continuous latent variables, but the dependencies are strictly sequential (chain structure). Therefore, neither can be applied if the latent variables are assumed to be continuous and also to have a nested hierarchical dependency structure. In this paper, we present Recursive Bayesian Networks (RBNs), which generalise and unify PCFGs and DBNs, combining their strengths and containing both as special cases. RBNs define a joint distribution over tree-structured Bayesian networks with discrete or continuous latent variables. The main challenge lies in performing joint inference over the exponential number of possible structures and the continuous variables. We provide two solutions: 1) For arbitrary RBNs, we generalise inside and outside probabilities from PCFGs to the mixed discrete-continuous case, which allows for maximum posterior estimates of the continuous latent variables via gradient descent, while marginalising over network structures. 2) For Gaussian RBNs, we additionally derive an analytic approximation of the marginal data likelihood (evidence) and marginal posterior distribution, allowing for robust parameter optimisation and Bayesian inference. The capacity and diverse applications of RBNs are illustrated on two examples: In a quantitative evaluation on synthetic data, we demonstrate and discuss the advantage of RBNs for segmentation and tree induction from noisy sequences, compared to change point detection and hierarchical clustering. In an application to musical data, we approach the unsolved problem of hierarchical music analysis from the raw note level and compare our results to expert annotations.


Reviews and author responses:

 Thanks for your comments. Some additions for completeness:

- We _do_ describe how MAP inference in general RBNs can be achieved via gradient descent (Section 2.2 Inference: Maximum Posterior Inference) but also point out that the problem is highly non-convex. Also note that full-posterior inference in general RBNs is usually intractable (due to the exponentially growing number of nested integrals), which is why we focussed on the Gaussian case in our solution.
- For a better understanding of hierarchical structures in music, please consider the papers we cited [2, 3, 8, 10, 38, 46], in particular, the seminal book by Lerdahl & Jackendoff (1983). Concrete harmonic progressions that go beyond your example but that are captured by our model would be [extended fifth progressions](https://en.wikipedia.org/wiki/Circle_of_fifths#Modulation_and_chord_progression) in Western classical music (also see our next point) or equivalently the [ii-V-I progression](https://youtu.be/oxpKr6T_ctE?t=799) in Jazz and Pop music. The fact that Bach's music is fundamentally fifth-based, as revealed by our model, therefore is a highly non-trivial, empirical confirmation of music theoretical insights.
- In the expert analysis of the C-major prelude, most non-identity transitions turn out to be (noisy) fifth transpositions when accounting for ambiguities in the labels. In left-most derivation order (ignoring identities and bass notes "/") we have for the first half of the piece: C --> G7 --> Dm7 [2xfifth]; C --> G --> D7 --> Am [3xfifth]; G --> D7 --> Cmaj7$\approx$Am [2xfifth]; C --> Bdim7$\approx$G7 --> Dm --> C#dim7$\approx$A7 [3xfifth]; C --> G7 --> Fmaj7$\approx$Dm7 [2xfifth]; G7 --> Dm7 [1xfifth]. Here X$\approx$Y denotes chords that have a very similar surface pattern (i.e. raw notes) and might easily be confused by an unexperienced human analyst (such as the notes G–C–E, which could amongst others be labeled as a C major chord in second inversion, a G major chord with 64-suspension "Gsus64", or an A minor seventh chord with omitted root). This means that in the first half, virtually _all_ transitions can be explained as hierarchical fifth transpositions. The second half mostly follows a similar pattern (with some exceptions that are somewhat more noisy when explained via fifth transpositions).

Lerdahl F, Jackendoff R (1983) A generative theory of tonal music. MIT press

 Thanks to the authors for their further comments.

The authors propose the general RBN which includes the influential PCFG model, and the authors propose inference/learning for the Gaussian RBN.  The model itself accomodates much more general models (e.g., the case where we have both discrete and continuous non-terminals), but it does not seem we know how to do inference/learning with them, which to me is a big gap in what is claimed (the general RBN), versus what can actually be used now (Gaussian RBN with a single variable).  Perhaps this is subjective, but this gap is too big for my taste --- I thank the authors again for their clarifications regarding these distinctions, but what is claimed in the beginning (general RBN) is still not very close to what we get at the end (in terms of what we can learn or reason with).

As for musical structure : I still do not see what conclusions or insights that we get about, e.g., Bach's Prelude No 1, based on the model that was learned.  The transposition based model used in the experiments doesn't seem to really encode any realistic structure about music, besides what I suggested earlier (if a C note appears first, than a G note should appear later).


 Thanks for author's reply, and I think the phrase is clearer now with the explanation.

 Thank you for the further clarification. RBNs do not impose any kind of restriction on the number or type (discrete/continuous) of variables in a cell (see Definition 1). The only restriction (which at the same time is the key to efficient inference) is that the structures are tree-shaped/recursive. This is analogous to how DBNs are defined for sequential structures. So we can draw the following equivalences:

- sequential/chain structure
  - HMMs (with discrete latent state) are equivalent to regular grammars
  - Kalman filters (single Gaussian latent variable) are a special case of a "continuous regular grammar"
  - DBNs (with an arbitrary number of discrete or continuous, also non-Gaussian, variables per time slice) fully generalise regular grammars and comprise arbitrary Bayesian networks of discrete/continuous variables with a sequential/chain structure
- hierarchical/tree-shaped structure
  - RBNs with only discrete variables are equivalent to PCFGs
  - Gaussian RBNs with a single continuous variable (the basic form described in our paper) are the hierarchical equivalent of Kalman filters and a special case of a "continuous PCFG"
  - Extended Gaussian RBNs with mixed discrete/continuous variables (as used in the application to music) are more general but still only a special case of RBNs
  - General RBNs (with an arbitrary number of discrete or continuous, also non-Gaussian variables per cell) fully generalise PCFGs and comprise arbitrary Bayesian networks of discrete/continuous variables with a hierarchical/tree-shaped structure

Therefore, calling DBNs "continuous regular grammars" is misleading because they also comprise mixed discrete/continuous Bayesian networks. Calling RBNs "continuous PCFGs" is misleading for the same reason. Kalman filters and basic Gaussian RBNs (i.e. special cases of DBNs and RBNs, respectively) are what comes closest to continuous versions of regular and context-free grammars, respectively.

Concerning musical structure, as you pointed out correctly, an HMM can only simulate _bounded_ hierarchical structure, but music (just as language) is assumed to exhibit _unbounded_ hierarchical dependencies. We hopes this clarifies the importance of aiming for a more general model class.

 Thanks for your response which properly addressed two points I made in my review.

 My issue is not whether the proposed RBN can do something more substantial than a "continuous PCFG", but is it really general enough to justify the name "recursive Bayesian network".  Just because it has discrete elements (such as structure learning, etc.) doesn't make it a "discrete/continuous Bayesian network" in a more general sense that I had meant.  Hybrid Bayesian networks are a sub-area of Bayesian networks which deal with general unrestricted Bayesian network structures that support both discrete and continuous variables.

So, for me, the paper still currently seems to make bigger claims than seems justified.

Is it fair to call DBNs (or Kalman filters, etc.) "continuous regular grammars"?  The authors raise this point but don't say either way.  If it is fair, than I think it is also fair to call RBNs "continuous context free grammars".  If not, then why?

On hierarchical structure of music: while again, I am not involved in music information retrieval etc., I still don't get the significance of the task, or any of the results provided.  A "dominant-tonic" relationship in the proposed model seems to suggest that if a C note appears as the first note, that a G note should appear later (perhaps I am oversimplifying).  An HMM with extra latent variables could also simulate some bounded hierarchical structure.


 As mentioned in our general comment, we agree that a simple and more concrete example will be beneficial and we will add one to illustrate the inference procedure.

**"Convexity" of Gaussian RBNs**

To be clear: Gaussian RBNs do not generally have a convex (log) posterior and gradient descent is therefore not guaranteed to find the global maximum.

By saying in line 173 that Gaussian RBNs are "otherwise" convex (i.e. if we did not have to marginalise over structures), we wanted to highlight the fact that marginalising over the unknown structure turns a convex model into a non-convex one. Specifically, for a fixed structure, a Gaussian RBN reduces to a normal Bayesian network with Gaussian distributions, which effectively corresponds to a single joint Gaussian that is (log) convex and can be solved exactly. However, marginalising over the (exponentially many) network structures in a Gaussian RBN means that the posterior effectively is a mixture of Gaussians (one for each possible structure), which may be highly non-convex. We will rephrase the sentence to make this clear.

 **Monte-Carlos approaches**

Monte-Carlos (MC) approaches are indeed an interesting option for inference in general (i.e. also non-Gaussian) RBNs. Evaluating the point-wise posterior (i.e. with fixed values for the continuous variables but marginalising over structures) is straight-forward and PyTorch readily provides a gradient. This means that in principle hybrid/Hamiltonian MC (HMC) methods can be used for sampling from the full-posterior distribution. In practice, however, we found standard HMC methods to be highly volatile and suffering from vanishing gradient problems due to the mixed discrete-continuous character of RBNs: A sub-optimal value of one high-level variable may "cut off" an entire set of low-level variables. For the future, we think that MC methods that exploit the hierarchical character of RBNs are very promising. In particular, the fact that generating with an RBN can be understood as a probabilistic program opens up a number of options for exploiting the (mainly MC-based) inference methods developed in that area.

**RBN "killer application"**

The main goal in our evaluations was to demonstrate the advantage of RBNs against the baseline method and showcase how more challenging applications may be tackled. In that sense, we agree that the presented results can only hint at the potential for future applications. We think that RBNs will be particularly strong in domains where symbolic categories exist (such as major/minor keys or different chord types in music), but where the boundaries between these categories are fuzzy and fluent (e.g. music theorists differentiate between different kinds of minor that cannot always be clearly separated). Technically, this may be for instance realised by taking a PCFG (for the clearly symbolic parts) and _expanding_ it to an RBN (see Section 2.1 and Appendix A.2.2), thereby endowing each symbol with fuzzy, continuous properties. We think, this could well become a "killer application" in many domains.

 As mentioned in our general comment above, we agree that a concrete accompanying example will be beneficial and we will add one to better the connection of Figures 2 and 3 to the inference procedure from Section 2.

**Sum-Product Networks**

The relation between RBNs and Sum-Product Networks (SPNs) is the same as for PCFGs (which is only briefly mentioned in Poon and Domingos, 2011): For a given RBN/PCFG, the process of parsing an input sequence of length $n$ can be rewritten as a SPN with $O(n^2)$ nodes. Evaluating the SPN is then equivalent to parsing with the RBN/PCFG. However, this also means that the exact same challenges arise in the continuous case. That is, point-wise evaluation with a fixed assignment of the continuous variables is tractable, but computing the full posterior distribution is not (due to the combinatorial explosion induced by the alternating sum and product nodes). For that reason, SPNs do not solve any of the problems addressed in our paper, but merely provide an equivalent reformulation. We still think that the connection to SPNs should be pointed out in the paper and will add a corresponding statement.

 **Mixed discrete-continuous case**

The case of mixed discrete-continuous variables is indeed interesting, but for RBNs it does not add much complexity to the inference process (this is different for normal Bayesian networks!) and is covered by our results. 

As mentioned in our general comment above, in RBNs we are _always_ facing the more complex mixed discrete-continuous case, because the discrete network structure (described by the discrete structural variables) is jointly distributed with the continuous variables. Marginalising over the latent structure produces mixtures over the continuous variables, even for the case of simple Gaussians.

In contrast, for a normal Bayesian network (BN), the fully Gaussian case is tractable because there are no discrete variables and simple products of Gaussians are again Gaussians. Introducing Gaussian mixtures in a normal BN then adds the kind of problems we are addressing in Section 2.3 on Gaussian RBNs: In each product of two variables, the number of mixture components is multiplied, which leads to a combinatorial explosion. Our adaptive approximation scheme is designed to solve exactly this problem. (Side note: All of this is not a problem in the purely discrete case, because a mixture of categorical distributions again produces a _single_ categorical, which is why inference in PCFGs is tractable.)

The advantage of having solved the more complex mixed case is that explicitly using mixed discrete-continuous non-terminal or terminal variables now comes for free. In fact, our Gaussian RBN for music has a discrete variable $\tau$ for the transposition, so it actually _is_ mixed discrete-continuous. In our adaptive approximation scheme, the additional mixture components arising from $\tau$ are dealt with in the same way as the mixture components arising from marginalising over the latent structure: in each step, they are approximated with a single Gaussian to avoid a combinatorial explosion.

A second way of introducing more complex discrete structures is by using multiple non-terminal variables (e.g. a Gaussian and a categorical to describe a generative process that may alternate between evolving in a continuous and a discrete space). While not featured in the examples employed in our evaluation (which are in extended Chomsky normal form with only a single non-terminal and terminal variable), this does not fundamentally change anything and is fully covered by our definition in Section 2.1 and the equations for the general case (Appendix A.3).

**Why not call it "continuous PCFG"?**

Whether a certain name conveys the right associations is a difficult question and we do not think there is a clear right or wrong here. But there are some reasons why we feel "continuous PCFG" might be somewhat misleading. First, RBNs do not just define "a new type of PCFG" that can eventually be rewritten as a normal PCFGs. Instead, they require fundamentally new approaches to make inference tractable, as described in the paper. Second, there is not a simple one-to-one correspondence between a PCFGs and "its RBN version": _Abstraction_ of a PCFG (Section 2.1, Appendix A.2.1) uniquely defines an equivalent RBN in Chomsky normal form, which might then be relaxed to make it continuous. But _expansion_ (Section 2.1, Appendix A.2.2) offers a second path to transform a PCFG into an RBN that results in a more powerful model with each symbol of the PCFG becoming a distinct (and possibly continuous) random variable. Finally, using the flexible factor graph notation from Figure 2 allows for easily incorporating a high degree of structure into the RBN definition (such as factorised variables etc as in dynamic Bayesian networks), which is typical for Bayesian networks in general. In the end, we think that RBNs are "continuous PCFGs" as much or as little as dynamic Bayesian networks are "continuous regular grammars".

**Hierarchical structure in music**

It is true that purely sequential models (HMMs, DBNs, recurrent neural networks etc) are able to capture some local structures in music (such as simple chord progressions). However, dependencies such as a dominant-tonic relation do not only occur locally at the chord level but also at much deeper levels (e.g. between sections of a piece; e.g. the "middle part" of a piece in A–B–A form is often in the dominant key). It is therefore generally understood that Western (but also e.g. Indian) music exhibits deep hierarchical structure (see our references [2, 8, 38] in the paper), which simple time-series models are unable to capture.

**Raw note input**

The challenge with raw note input is that the existing context-free grammars for music operate on chord symbols that have to be annotated by musical experts (or automatically inferred, which generally is unreliable and also an open research problem, precisely because it requires hierarchical structural information to resolve some ambiguities). While topic models, such as the one by Hu & Saul (2009), are able to process raw note input, they are unable to capture these hierarchical dependencies.

**Continuous latent variables for discrete data**

Using continuous latent variables has proven highly beneficial for modelling discrete data. The probably best known example are vector embeddings for words (e.g. word2vec; Mikolov et al, 2013) and other structured data types, such as graphs (Grohe M, 2020). The approach taken in our evaluations (categorical distributions over the surface elements parameterised by the continuous latent variables) is just one (rather simple) way to map raw notes to a continuous space and any other method may be use instead. 

- Grohe M (2020) word2vec, node2vec, graph2vec, X2vec: Towards a Theory of Vector Embeddings of Structured Data. In: Proceedings of the 39th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems. Association for Computing Machinery, New York, NY, USA
- Mikolov T, Sutskever I, Chen K, et al (2013) Distributed representations of words and phrases and their compositionality. In: Advances in neural information processing systems. pp 3111–3119

 We would like to thank all reviewers for their feedback, which is very helpful for further improving the paper. Some points that relate to more than one review or that are of general interest are addressed in this comment. For a more detailed point-by-point response, please see our comments below the respective reviews.

**Main contribution:** We would like to stress that the primary contribution of our paper lies in the theoretical advancement of unifying and generalising PCFGs and DBNs and deriving tractable solutions for the arising inference problems. The empirical evaluations are crucial to demonstrate feasibility and the method's advantage on characteristic problems, but they can only give a first idea of the method's potential on the diverse application domains. Considering the widespread use of PCFGs and the way that continuous vector embeddings have revolutionised many areas of machine learning, we think that the continuous generalisation of PCFGs (which are RBNs) has a huge potential for many applications that can hardly be gauged at the current stage. (Also see our comment to reviewer K4dB.)

**The mixed discrete-continuous character of RBNs:** Even in the fully continuous case (e.g. only Gaussian distributions), RBNs always have a mixed discrete-continuous character, because the discrete network structure is jointly distributed with the continuous variables. This is the reason why full-posterior inference is hard, even for the most benign case of only Gaussian distributions. (Also see our comments to reviewers K4dB and VzMU.)

**Simple example:** We agree that a simple and concrete example may greatly benefit understanding and intuition. We will illustrate the inference procedure with a minimal but complete example of a Gaussian RBN applied to a short sequence (probably in the Appendix because of space limitations), drawing the connection to Figures 2 and 3, and showing how the marginal probability of a latent variable and the probability of a specific structure are computed for a given sequence. (Also see our comments to reviewers VzMU and gxvj.)

This paper proposes the "recursive Bayesian network", which appears to be an extension of probabilstic context-free grammars (PCFGs) with non-terminals that correspond to continuous latent variables.
  I have some reservations, partially in how the paper is pitched and then presented.  The concept of a "recursive Bayesian network" is appealing and ambitious, but perhaps too aggressive considering the presented results.  Mainly, I am not sure if the experiments are sufficient for such a bold claim.

The fully Gaussian (prior,non-terminal,terminal) case is the main focus of the paper.  The fully discrete case corresponds to PCFGs.  It doesn't seem like hte mixed-case is considered in this paper, which to my knowledge would be challenging (as is the case with reasoning and learning with mixed discrete/continuous Bayesian networks).  So I think it is a bit of a stretch to say that RBNs generalize both PCFGs and continuous DBNs.  Figure 1, suggests a continuous PCFG would be a more apt label.

The synthetic experiment in Section 3.1 seems to simulate data from an RBN and then tries to learn the model back using an RBN compared to hierarchical clustring, which is ifine.  I would be more straightforward with saying the synthetic data was simulated by an RBN, as it was only mentioned in the appendix.

I believe the significance of the music model needs to be expanded.  For example, as a non-expert in machine learning for music, with some elementary background in music and music theory, I could not see the significance of extracting the hierarchical tree structure of a song.  I would expect a simple time-series model (like an HMM or a DBN) would also capture a dominant-tonic progression, and perhaps even more complex chord sequences, e.g., the common I-IV-V-I.  Another latent variable model was previously proposed that also takes raw notes as input:

  "A probabilistic topic model for music analysis"
  by Hu & Saul, NIPS 2009

and I imagine a dynamic/temporal extension would also be sufficient (perhaps even using just a dynamic topic model for text).  Further, it seemed like the data was discrete, so again I am not sure why we would want to use a Gaussian RBN in this case.

To summarize, I believe the paper oversells its claims unneccessarily.  Perhaps a continuous PCFG would be interesting enough, although even then the experiments do not seem well motivated for learning hierarchical models with latent continuous variables.
 ok

The paper presents Recursive Bayesian Networks (RBN) which combine the strengths of probabilistic context-free grammars and dynamic Bayesian networks into a new class of probabilistic models. Specially, RBNs allow nested structures over the latent variables as well both discrete and continuous variables. Inference in these models is cast as a generalisation of inside and outside probabilities from probabilistic context-free grammars. Experimental results on a real-world music application demonstrate the effectiveness of the proposed approach.
  The paper is fairly well written and organised. The quality of the presentation is overall fairly good. I have two main comments which I think need to be addressed.

1. While I appreciate the effort the authors put in to carefully describe the technical details of the proposed approach, the paper is quite dense and without a more concrete accompanying running example it is not easy to follow all the details especially by readers that are not familiar with PCFGs. For instance, I found Figures 2 and 3 a little confusing as it was hard to connect them to the inference procedure from section 2.

2. Sum-Product Networks (SPNs) and their more recent extensions (Mixed SPNs - MSPNs, Conditional SPNs - CSPNs) are probabilistic models that can in principle capture hierarchical (or nested) structures over both discrete and continuous variables. Therefore, I was wondering if there isn't a closer connections (relation) between these models and the proposed RBNs.

H. Poon and P. Domingos. Sum-Product Networks: a new deep architecture. UAI 2011.

A. Molina, A. Vergari, N. Di Mauro, S. Natarajan, F. Esposito, K. Kersting. Mixed Sum-Product networks: a deep architecture for hybrid domains. AAAI 2018

X. Shao, A. Molina, A. Vergari, K. Stelzner, R. Peharz, T. Liebig, K. Kersting. Conditional Sum-Product Networks: imposing structure on deep probabilistic architectures. PGM 2020.
 The limitations are discussed in the paper and there is no potential negative societal impact of this work.

As the title indicates a new formalism: recursive Bayesian networks is
presented which is basically the union of PCFGs and DBNs. RBNs are
properly defined and their relation to PCFGs and DBNs clearly
explained (with examples). Since RBNs have both (hierarchical)
discrete structure and continuous variables and are not "finite"
(since PCFGs alone are not), inference is quite tricky. An extension
of inside-outside computation (for PCFGs) is extended to RBNs. A
subclass of RBNs - Gaussian RBNs -  are focused on, and
GaussianRBN-specific inference methods are presented. Two sorts of
experiments are presented: tree induction on synthetic data and music
analysis.
  The paper is very well-written. The goals are ambitious since RBNs are
indeed highly expressive. So, as ever, the question is whether they
are too expressive. Although the (inference) methods are well-chosen
(I certainly can't think of any better options!) and show some
mathematical sophistication, I did not go away from the paper with a
real feel for how big a problem RBNs can be applied to.

In their approach to inference the authors have
opted for analytical approximation (e.g. approximating a mixture of
Gaussians by a single Gaussian (by moment matching)). An alternative
option might have been some sort of Monte Carlo method. I don't know
whether a good Monte Carlo (MC) method exists for RBNs but I think it
worth exploring: one often sees MC methods in the various formalisms
invented in the SRL literature (perhaps because it is relatively
easy!).

The authors promise to show us "the capacity and diverse applications
of RBNs". The music analysis presented was certainly interesting, and
the trained GRBN captured some aspects of harmony, but, for all I
know, some much simpler method could also have done this. I think it
is fair to say we do not yet have a "killer application" for RBNs.

In summary, I have yet to be convinced of important real-world
applications of RBNs, but perhaps it is unreasonably early to expect
that. The authors have designed sensible inference methods for a
highly-expressive probabilistic formalism and moreover explained what
they are doing very well.
 These have not been addressed. Doing so would be difficult given that RBNs are only introduced here. As mentioned in the main review it is hard to predict their impact.

Author proposed a new probabilistic model, Recursive Bayesian networks, that describes the generative process of sequence of random variables. After the model definition, author further demonstrated multiple inference procedures (exact and approximate) (marginal and structural) for the model. Lastly, author demonstrated the application of the model in areas of change point detection and hierarchical music analysis.  Originality:
The paper propose a novel probabilistic model that describes a the joint distribution of a sequence. The model not only describes the likelihood of each symbols in the sequence being generated, it also describes the likelihood of the underlying hierarchical structure, e.g. a chain or a balanced tree. As a result, this new model generalizes two previous sequence model, Probabilistic Context-Free Grammars (PCFGs) and Dynamic Bayesian Networks(DBNs), by supporting continuous latent variables (which was not supported in PCFG) and uncertainties in tree structures (which is not supported in DBNs) simultaneously. 

Quality:
The formulation of RBN and its inference algorithm seems correct to me.

Clarity:
There are abundant of results that authors are trying to present, and it somewhat makes the paper dense to read. For example, the demonstration of the inference algorithm currently consists of equations in its most general form. It would be nice that inference procedure can be demonstrated with some simple examples. For example, how the marginal probability of a non-terminal latent variable, and the probably of a structure given the sequence is computed.

Significance:
The work is significant. First, RBN generalizes two existing powerful sequence models, PCFGs and DBNs, and it can model problems that none of its processor can handle solely. Secondly, author demonstrates the applicability of the RBNs in multiple domains, i.e. change point detection and  hierarchical music analysis. In particular, it shows quantitative improvements on the task of change point detection compared to the existing dedicated solution.

Minor Edits:
ln:92 "The left-hand side l_i of a transition p_i" It is unclear what the left-hand side means here.
ln:144 "non-terminal variable alpha_i:k" -> "non-terminal variable a_i:k"
ln:173  "convex models such as Gaussian RBNs" -> what does it mean that Gaussian RBNs are convex? Does that mean gradient descent can find the exact Maximum Posterior result for Gaussian RBN? Yes.

Meta-review: There have been mixed opinions about the manuscript, even after the short discussion. The paper describes an interesting model, some properties and algorithms. I share some concerns of a reviewer who is unsatisfied about how the work is presented, as it can give the impression to do more than it actually does. Indeed there are many direction here that could be worth investigating, and the presentation and choices could be better. It is unclear to me how tractable and how general things really are: they do not seem as efficient/tractable/general as one gets the impression in the beginning. On the other hand, the remaining reviewers are positive, in particular because there is potential for RBNs to be an important model, and the steps described here may be enough for justifying publication at neurips 2021.

Decision: accept