Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: This paper considers inference in Hidden Continuous-Time semi-Markov Chains. The authors derive a tractable algorithm for inference in these models. Notably, the authors consider the challenging case of irregularly spaced data. The introduced algorithm is of a forward-backward type; the output of this algorithm can be used for smoothing. This brings inference in the model class in line with other similar models. 

strengths_and_weaknesses: Overall, this paper makes a good contribution. I have enjoyed reading the paper and it is well-written and (together with the supplementary material) provides sufficient detail to understand the proposed methodology. The derivation of the forward-backward equations for the considered model is non-trivial and at the same time a very natural question to ask. A more principled inference method as an alternative to existing ad hoc ones is certainly welcome. 

Why not provide a description of the (main) forward-backward algorithm step by step in the main paper instead of leaving in to supplementary section C, where it is nicely described? 




questions: Could you give one example of an application of such a model? Where would you find this kind of data? The method is presented in entirety without any reference to applications. It would be good to have an idea of the practical scale of applicability of this method. 

From a numerics point of view, I don't really understand the discussion in the first paragraph of section 4.2, which seems somewhat crucial to the practical implementation of the method. You say that using (13) is numerically more costly but at the same time more stable. This is in comparison to the method presented in appendix C.

Minor comments:

typo on line 333 "where" should be "were" 
typo on line 347 "testet" should be "tested"

limitations: Yes.
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: The paper describes a class of models which the authors characterize as being "continuous time hidden semi-Markov chains" and provides an inferential framework for this class of models generalising the classical algorithms for HMM inference to this context.
strengths_and_weaknesses: This is a timely piece of work in the sense that there have been several innovations relating to inference for HSMMs in recent years and the development herein seems somewhat more principled than many of them. Working directly in continuous time seems to allow for a relatively clean derivation of forward and backward equations that facilitate the inferential tasks of interest. My impression is that there is currently interest in performing inference for this type of model and hence the work is likely to find an audience.

The main weaknesses, to my mind are:
1. The bulk of the paper is dedicated to fairly routine computations and the extent of the novelty is perhaps a little limited.
2. A lack of connection with the wider literature on hidden continuous time processes.
3. The empirical study lacks any compelling application or use case to showcase the model or demonstrate its broad importance.
4. The manuscript seems to have been a little hastily prepared and would benefit from careful proofreading (see the minor details in the next section for a few example typos)
questions: 1. My first thought on seeing this paper was that in continuous time something like a hidden semi-Markov model is actually very much more natural than it is in discrete time where one has to jump through artificial hoops to induce the non-geometric holding time distribution. To what extent are hidden semi-Markov models different to the setting in which one has a latent piecewise deterministic Markov process in the sense of [Davis, M. H. A. (1984). "Piecewise-Deterministic Markov Processes: A General Class of Non-Diffusion Stochastic Models". Journal of the Royal Statistical Society. Series B (Methodological). 46 (3): 353–388. doi:10.1111/j.2517-6161.1984.tb01308.x. JSTOR 2345677.]? 

2. Following up on 1, how does what is presented here relate to what is known for filtering in the setting in which one has a latent PDMP? As the PDMP class is much broader than that considered here it's very probable that substantial benefits can be obtained by considering this particular context but it would be nice to see this discussed a little.

3. Are there settings in which you envisage the method having real world application? The numerical example in this paper is underwhelming in that regard.

Minor details:
line 80: This seems to describe an overly general observation space Y given it's assumed to be real and to have a known conditional distribution function.
line 85: this is the intersection isn't what's intended, is it? It's the intersection of possibly several points and hence typically empty. I think what's wanted is actually the intesection of the events so the intersection hould come outside the { } and it might be clearer if the implicit \omega were made explicit.
(2) could be made clearer: what precisely does "Markov at jump instances" mean. Can you give a reference?
l189 (??)
l317 time-differemces,
l331 "detailled"
l342 10^-4 and 10^-3
l344 "adative"
l347 testet"

limitations: The authors have indicated that societal implications are N/A for this work and given the nature of the paper I think that is appropriate. I didn't note any particular limitations that are not discussed.
review_title: null

final_score: rating: 5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.
review_content: summary: This paper propose inference algortihms for hidden continuous time semi-Markov
chains (latent CTSMCs). Specifically it focuses on homogeneous and direction-time 
independent CTSMCs. Both inference of posterior probabilities of latent states 
given observations, as well as maximum-a posteriori state path are addressed.
The foundation of the algorithm is the observation that given a transition
event at a specific time, past and future dynamics become independant. Therefore,
the core element is to update the intensity of such event (termed current) as a 
function of time and state. These updates involve integral equations that are solved 
using numerical methods.

strengths_and_weaknesses: Strengths:
This work tackles a very fundamental and important problem. The authors seem
exploit the structured representation of direction-time Independence in a elegant way,
namely by focusing on the representation of currents. 
 
Weaknesses:
Although the setup is well written, starting from Section 2.2 the derivations become
hard to follow and it is hard to judge all the elements. The initial conditions are not 
clear. Additionally, the numerical algorithm as well as the overall complexity is not 
well described. See suggestions below regarding missing descriptions and flow.

questions: **Suggestions**
1. The role of current is introduced in Section 2.1. At this point it will be
good to state what are the challenges in computing them and to outline the approach
for doing so.
2. Give detailed explanations about the initial conditions introduced after
Equation 5 and in the algorithm summary.
3. Give a short intuitive explanation about how integral equations are solved,
what is the complexity of the solvers and the overall method. This is important
in addition to the references that appear.
4. Provide a summary of all the steps involved in the algorithm.

**Questions and minor comments** 
1. Lines 112, 114: Should 'leaving' be replaced by 'entering' in line 112 and vice 
	versa in line 114?
2. Line 126: the current seem to be a density of a joint probability not a probability;
is likelihood is a better term?
3. Figure 1: Do the Gamma distributions refer to sojourn time?
4. Line 134: What is the meaning of 'assume a terminal condition to be fixed at $T$?
Does it mean the we observed the end state? Do we observe also how the entry time
to this state?
5. Line 141: the opening sentence is not clear. 
6. **Equation 5:** what is $g_p$? Why is there an addition here? What are the properties
of this function? Also in line 153: What properties should $g_\phi$ satisfy? 
7. Line 154: the motivation to demanding a transition shortly after $t$ is not 
clear.
8. Line 189: fix the reference. 
9. 201: Can you give a time-domain expression for $\kappa$ and explain what is gained
10. Equation 13: can you give an explicit reminder-reference there to how $\phi_\alpha$ 
and $\psi_\beta$ are computed as by-products? 
11. Line 317: fix 'differemces'

limitations: The authors adequately addressed the limitations.
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: The paper generalizes latent state inference from HSMMs to continuous-time chains in latent space. In this case, the posterior is not simply proportional to the usual forward and backward probabilities. Instead, the transition random variables ("currents") are Markov. The authors take the limit of step size approaching 0 for these currents. The paper shows that the marginal $p(x, t)$, in this case, is found by a convolution over the input currents. Similarly, the output currents are a convolution over the input currents. Including observations, the authors derive Kolmogorov forward and backward equations, leading to algorithms akin to the classical sum-product (belief propagation) and max-sum (Viterbi) message-passing algorithms, but in continuous time. An additional advantage of the continuous formulation is the availability of adaptive step-sizing in the continuous case. The algorithms, as well as adaptive step-sizing, are evaluated in 1D experiments. 
strengths_and_weaknesses: Strengths:
- Strong theoretical foundation.
- Important generalization of well-known important algorithms to the continuous-time case.
- Adaptive step-sizing.

Weaknesses:
- The paper was a bit hard to follow from time to time. This could either be due to my lack of background knowledge but also perhaps due to the quite liberal suppression of function arguments and sometimes rather cluttered mathematics mid-sentence.


Minor:
- Some typos: l271: "therefor", l317: "differemces", l189: equation reference failed, l274: what does "(x)" refer to?

questions: - What is the black line in figures 2b,c? Is it the underlying ground truth state? If so, why does the blue HSMM with adaptive step size line in 2c seem to perform so poorly?
limitations: The authors properly discuss the limitations in the conclusion.
review_title: null

Meta-review: metareview: All of the authors agree that the work meets the NeurIPS standards, with the two lowest-scoring reviewers upping their recommendation from 4 to 5 on rebuttal. The work is described as "a fundamental and important problem" and "timely", "a good contribution". 

Reviewer 6Ntb summarises the technical contribution: 
"The paper generalizes latent state inference from HSMMs to continuous-time chains in latent space. In this case, the posterior is not simply proportional to the usual forward and backward probabilities. Instead, the transition random variables ("currents") are Markov. The authors take the limit of step size approaching 0 for these currents"
It's clear to me that the the work has been communicated really well, since all of the reviewers were able to grasp the paper and there was very little misunderstanding in the discussions. 

There were some recommendations from the reviewers - please ensure these are fixed in the camera-ready version. 

confidence: Certain
Decision: Accept