structured inference networks for nonlinear state space models (wip)

13 March 2017

notes on (Krishnan et al., 2017).

• understanding: 7/10
• code: github.com/clinicalml/structuredinference

basically vae on state space models (SSMs): learn model parameters of SSMs and at the same time learn an inference network. the SSM under consideration is the standard SSM. but the transition is a neural net. emission is also a neural net. everything is gaussian.

the novelty lies in

1. form of \(q\)
2. the reformulation of the ELBO

form of \(q\)

\(q\) takes in the form \begin{align} q_{\phi}(x_{1:T} \given y_{1:T}) = q_{\phi}(x_1 \given x_1, \dotsc, x_T) \prod_{t = 2}^T q_{\phi}(x_t \given x_{t - 1}, y_t, \dotsc, y_T), \end{align} i.e. condition only on the last \(x\) and the future \(y_t\)s. this comes from considering the conditional independence structure of the posterior…

other forms of \(q_{\phi}\) such as

• \(q_{\phi}(x_1 \given x_1, \dotsc, x_T) \prod_{t = 2}^T q_{\phi}(x_t \given y_1, \dotsc, y_T)\),
• \(q_{\phi}(x_1 \given x_1) \prod_{t = 2}^T q_{\phi}(x_t \given y_1, \dotsc, y_t)\),
• \(q_{\phi}(x_1 \given x_1, \dotsc, x_T) \prod_{t = 2}^T q_{\phi}(x_t \given x_{t - 1}, y_1, \dotsc, y_T)\),
• \(q_{\phi}(x_1 \given x_1) \prod_{t = 2}^T q_{\phi}(x_t \given x_{t - 1}, y_1, \dotsc, y_t)\),

but \(q_{\phi}(x_1 \given x_1, \dotsc, x_T) \prod_{t = 2}^T q_{\phi}(x_t \given x_{t - 1}, y_t, \dotsc, y_T)\) performs best.

reformulation of the ELBO

the elbo has some sort of weird form because everything is gaussian. reparametrization trick is thus not needed… check eq. 6.

experiments

experiments on

• 2d linear gaussian state space model:
  • \(x\) is in \(\mathbb R^2\)
  • \(y\) is in \(\mathbb R^2\)
  • \(T\) is 25
  • \(N\) (number of data sets) is 5000
• polyphonic music
  • \(x\) is in ??
  • \(y\) is in \(\{0, 1\}^{88}\)
  • \(T\) is ??
  • \(N\) is ??
• ehr patient data
  • \(x\) is in \(\mathbb R^{200}\)
  • \(y\) is in \(\{0, 1\}^{48}\)
  • \(T\) is 18
  • \(N\) is 15000

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references

1. Krishnan, R. G., Shalit, U., & Sontag, D. (2017). Structured Inference Networks for Nonlinear State Space Models. AAAI.
   • Bib

   @inproceedings{krishnan2016structured,
     title = {Structured Inference Networks for Nonlinear State Space Models},
     author = {Krishnan, Rahul G and Shalit, Uri and Sontag, David},
     booktitle = {AAAI},
     year = {2017}
   }
   

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