The hierarchical hidden Markov model (HHMM) is a generalization of the hidden Markov model (HMM) that models sequences with structure at many length/time scales [FST98]. Unfortunately, the original infer- is ence algorithm is rather complicated, and takes the length of the sequence, making it impractical for many domains. In this paper, we show how HHMMs are a special kind of dynamic Bayesian network (DBN), and thereby derive a much simpler inference algorithm, which only takes time. Furthermore, by drawing the connection between HHMMs and DBNs, we enable the application of many stan- dard approximation techniques to further speed up inference.

Gaussian Processes (GPs) provide a powerful probabilistic framework for interpolation, forecasting, and smoothing, but have been hampered by computational scaling issues. Here we prove that for data sampled on one dimension (e.g., a time series sampled at arbitrarily-spaced intervals), approximate GP inference at any desired level of accuracy requires computational effort that scales linearly with the number of observations; this new theorem enables inference on much larger datasets than was previously feasible. To achieve this improved scaling we propose a new family of stationary covariance kernels: the Latent Exponentially Generated (LEG) family, which admits a convenient stable state-space representation that allows linear-time inference. We prove that any continuous integrable stationary kernel can be approximated arbitrarily well by some member of the LEG family. The proof draws connections to Spectral Mixture Kernels, providing new insight about the flexibility of this popular family of kernels. We propose parallelized algorithms for performing inference and learning in the LEG model, test the algorithm on real and synthetic data, and demonstrate scaling to datasets with billions of samples.

The hierarchical hidden Markov model (HHMM) is a generalization of the hidden Markov model (HMM) that models sequences with structure at many length/time scales [FST98].

The hierarchical hidden Markov model (HHMM) is a generalization of the hidden Markov model (HMM) that models sequences with structure at many length/time scales [FST98].

The hierarchical hidden Markov model (HHMM) is a generalization of the hidden Markov model (HMM) that models sequences with structure at many length/time scales [FST98].