Tea – Time - Talks Every Friday 3.30 pm ICS 432

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Embedded HMM’s Radford Neal Matt Beal Sam Roweis University of Toronto

Question: Can we efficiently sample in non-linear state space models with hidden variables (e.g. non- linear Kalman filter).

Graphical Model continuous domain hidden observed

Inference One option is Gibbs sampling. However: if random variables are tightly coupled, the Markov chain mixes very slowly. This is because we need to change a large number of variables simultaneously.

Idea: Embed an HMM! 1. Choose a distribution at every time slice t. i) Define a forward kernel and a backward kernel: such that: (note: not necessarily detailed balance) The kernels will be used to sample K states embedded the continuous domain of

Idea: Embed an HMM! 1. Choose a distribution at every time slice t. 2. Sample M states from distribution as follows: i) Define a forward kernel and a backward kernel: and the current state sequence ii) Pick a number uniformly at random between iii) Apply the forward kernel times, starting at and apply backward kernel times, starting at 3. Sample from the `embedded HMM’ using “forward-backward”.

Sampling from the eHMM REPEAT: 1. Starting at the current state sequence, sample K states by applying the forward kernel J times (J chosen uniform at random) and the backward kernel K-J-1 times. This defines the embedded state space. 2. Sample states using the forward-backward algorithm from the following distribribution: x & y discrete! proof of detailed balance: see paper

Note: The probabilities of the HMM are not normalized, so it should it should be treated as an undirected graphical model