First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. This paper extends the stochastic gradient langevin dynamics by using the Reimannian structure and applies it into probability simplex. The idea appears to be quite interesting. But there are several confusing parts that I don't quite get. Maybe the authors can elaborate those a bit.

A unique benefit to this approach is that each transition's TD error can be

We consider the problem of fitting variational posterior approximations using stochastic optimization methods.

We consider the problem of fitting variational posterior approximations using stochastic optimization methods.

First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. This paper presents a Bayesian approach to state and parameter estimation in nonlinear state-space models, while also learning the transition dynamics through the use of a Gaussian process (GP) prior. The inference mechanism is based on particle Markov chain Monte Carlo (PMCMC) with the recently-introduced idea of ancestor sampling. The paper also discusses computational efficiencies to be had with respect to sparsity and low-rank Cholesky updates. This is a technically sound and strong paper with clear and accessible presentation.

Ideally, we would prefer an estimator that learns any distribution.

An essential aspect of any machine learning system is understanding what the model does not know.

In the framework specified in section 2.2, we use a first-order Markov chain with two states as our Figure S1: For four different cells, the posterior distribution function is computed and depicted. Here, we concentrate only on trials within original environments, where we know the correct environment and hence can assess how well is the decoding. In this approach, instead of using a state-space structure, we use the likelihoods given by Eq. (1) of Each plot shows a histogram of the average probability (over time) of correctly decoding the trials within unambiguous environments. Fig. S3 shows the decoded environment for a few sample trials based on the neural activity of the whole population. In some trials (e. g. trials 65 & 25) we observe few fluctuations, while in other In Eq. 6, we use a history dependent, gamma-distributed generalized linear model with identity link.