# structured inference networks for nonlinear state space models (wip)

13 March 2017

notes on (Krishnan et al., 2017).

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

## references

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