An HDP-HMM for systems with state persistence

Abstract

The hierarchical Dirichlet process hidden Markov model (HDP-HMM) is a flexible, nonparametric model which allows state spaces of unknown size to be learned from data. We demonstrate some limitations of the original HDP-HMM formulation (Teh et al., 2006), and propose a sticky extension which allows more robust learning of smoothly varying dynamics. Using DP mixtures, this formulation also allows learning of more complex, multimodal emission distributions. We further develop a sampling algorithm that employs a truncated approximation of the DP to jointly resample the full state sequence, greatly improving mixing rates. Via extensive experiments with synthetic data and the NIST speaker diarization database, we demonstrate the advantages of our sticky extension, and the utility of the HDP-HMM in real-world applications. Disciplines Statistics and Probability AbstractThe hierarchical Dirichlet process hidden Markov model (HDP-HMM) is a flexible, nonparametric model which allows state spaces of unknown size to be learned from data. We demonstrate some limitations of the original HDP-HMM formulation (Teh et al., 2006), and propose a sticky extension which allows more robust learning of smoothly varying dynamics. Using DP mixtures, this formulation also allows learning of more complex, multimodal emission distributions. We further develop a sampling algorithm that employs a truncated approximation of the DP to jointly resample the full state sequence, greatly improving mixing rates. Via extensive experiments with synthetic data and the NIST speaker diarization database, we demonstrate the advantages of our sticky extension, and the utility of the HDP-HMM in real-world applications.

Nonparametric Bayesian Identification of Jump Systems with Sparse Dependencies

“…When modeling dynamical processes with mode persistence, the flexible nature of the HDP-HMM prior allows for mode sequences with unrealistically fast dynamics to have large posterior probability. Recently, it has been shown (Fox et al [2008a]) that one may mitigate this problem by instead considering a sticky HDP-HMM where π j is distributed as follows: π j ∼ DP(αβ + κδ j ) (5) Here, (αβ + κδ j ) indicates that an amount κ > 0 is added to the j th component of αβ, thus increasing the expected probability of self-transition. When κ = 0 the original HDP-HMM is recovered.…”

“…See Fox et al [2008a] for details of sampling ({π k }, β, α, κ, γ) given z 1:T . The sampler for the HDP-SLDS is identical with the additional step of sampling the state sequence, x 1:T , and conditioning on the state sequence when resampling dynamic parameters.…”

“…Sampling z 1:T As shown in Fox et al [2008a], the mixing rate of the Gibbs sampler for the HDP-HMM can be dramatically improved by using a truncated approximation to the HDP, such as the weak limit approximation, and jointly sampling the mode sequence using a variant of the forward-backward algorithm. Specifically, we compute backward messages m t+1,t (z t ) ∝ p(ψ t+1:T |z t ,ψ t , π, θ) and then recursively sample each z t conditioned on z t−1 from (24) Joint sampling of the mode sequence is especially important when the observations are directly correlated via a dynamical process since this correlation further slows the mixing rate of the sampler of Teh et al [2006].…”

“…In this paper we use a variant of the HDP-HMM-the sticky HDP-HMM of Fox et al [2008a]-that provides improved control over the number of modes inferred; such control is crucial for the problems we examine. An extension of the sticky HDP-HMM formulation to learning SLDS and S-VAR with an unknown number of modes was presented in Fox et al [2008b], however, relying on knowledge of the model order.…”

Variational Hidden Conditional Random Fields with Coupled Dirichlet Process Mixtures

“…Over the past decade, nonparametric methods have been successfully applied to many existing graphical models, allowing them to grow the number of latent states as necessary to fit the data [1][2][3][4][5][6]. Infinite HCRFs were first presented in [7] and since exact inference for such models with an infinite number of parameters is intractable, inference was based on a Markov chain Monte Carlo (MCMC) sampling algorithm.…”