Variational Learning for Switching State-Space Models

Motivation(s)

Time series probabilistic models commonly fall under HMM or SSM (state-space model a.k.a. stochastic linear dynamic systems). The former has discrete states while the latter’s state is continuous. Most real-world processes cannot be characterized purely using HMM or SSM. Existing hybrid models assume a single real-valued observed state vector (see Figure 2).

Proposed Solution(s)

This paper generalized the hybrid models to multiple real-valued hidden state vectors. The derived inference algorithm consists of applying structured variational approximation to switching state-space models. The derived learning algorithm consists of forward-backward recursions on a HMM and Kalman smoothing recursions on each state-space model. The HMM’s states determine the soft assignment of each observation to a state-space model. The SSM’s prediction errors determine the observation probabilities for the HMM. See Figure 3 for a graphical model representation.

Evaluation(s)

The proposed hybrid model performed much better than simple dynamical models on sleep apnea data. However, the method is easily influenced by model selection and initialization.

Future Direction(s)

  • How to apply belief propagation to state switching?

Question(s)

  • What is the purpose of the hamiltonian and Boltzmann distribution?

Analysis

A very intuitive way to apply HMM and SSM on real-world problems. Thinking in terms of graphical models and then deriving the factorizations can shed light into why exact EM would not work.

Although the idea is very elegant and quite intuitive, the experiment is not very convincing w.r.t real world applications. The sleep apnea is simple enough to use alternative proposals instead of implementing this complicated hybrid.

Notes

  • Variational Approximation

    • The general strategy of using a parameterized approximating distribution.

    • The free parameters of the distribution are called variational parameters.

  • Generative Model

    • See Equation (7) and (8)

  • Learning

    • The exact E-step for switching state-space models is intractable.

      • The posterior probability of the real-valued states is a Gaussian mixture with \(M^T\) terms.

      • Hidden state variables are marginally independent, but become conditionally dependent given the observation sequence.

    • The solution is to use a variational approximation; see Figure 4 and (14)

References

GH00

Zoubin Ghahramani and Geoffrey E Hinton. Variational learning for switching state-space models. Neural computation, 12(4):831–864, 2000.