Paper:
This paper presents a novel variant of hierarchical hidden Markov models (HMMs), the multiscale hidden Markov model (MSHMM), and an associated spectral estimation and prediction scheme that is consistent, finds global optima, and is computationally efficient. Our MSHMM is a generative model of multiple HMMs evolving at different rates where the observation is a result of the additive emissions of the HMMs. While estimation is relatively straightforward, prediction for the MSHMM poses a unique challenge, which we address in this paper. Further, we show that spectral estimation of the MSHMM outperforms standard methods of predicting the asset covariance of stock prices, a widely addressed problem that is multiscale, non-stationary, and requires processing huge amounts of data.

Review:
The paper focuses on a very particular HMM structure which involves multiple, independent HMMs. Each HMM emits an unobserved output with an explicit duration period. This explicit duration modelling captures multiple scale of temporal resolution. The actual observations are a weighted linear combination of the emissions from each latent HMM. The structure allows for fast inference using a spectral approach.
I found the paper unclear and lacking in detail in several key aspects:
1. It is unclear to me from Algorithm 2 how the weight vectors w are estimated. This is not adequately explained in the section on estimation.
2. The authors make the assumption that each HMM injects noise into the unobserved output which then gets propagated into the overall observation. What are reasons for his choice of model over a simpler model where the output of each HMM is uncorrupted?
3. The simulation example does not really demonstrate the ability of the MSHMM to do anything other than recover structure from data simulated under an MSHMM. It would be more interesting to apply to data simulated under non-Markovian or other setups that would enable richer frequency structures to be included and the ability of MSHMM to capture these.
4. The real data experiments shows some improvements in predictive accuracy with fast inference. However, the authors do not give a sufficiently broad exploration of the representations learnt by the model which allows us to understand the regimes in which the model would be advantageous.
Overall, the paper presents an interesting approach but the work lacks maturity. Furthermore, simulation and real data examples to explore the properties and utility of the method are required.