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Tea – Time - Talks. Every Friday 3.30 pm ICS 432. We Need Speakers (you)! Please volunteer. Philosophy: a TTT (tea-time-talk) should approximately take 15 mins. (extract the essence only). Email: welling@ics. Embedded HMM’s Radford Neal Matt Beal Sam Roweis University of Toronto.

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Tea time talks l.jpg

Tea – Time - Talks

Every Friday 3.30 pm

ICS 432


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We Need Speakers (you)!Please volunteer.Philosophy: a TTT (tea-time-talk)should approximately take 15 mins.(extract the essence only).Email: welling@ics


Embedded hmm s radford neal matt beal sam roweis university of toronto l.jpg
Embedded HMM’sRadford NealMatt BealSam RoweisUniversity of Toronto


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


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Graphical Modelcontinuous domain

hidden

observed


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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 l.jpg
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 hmm8 l.jpg
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”.


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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


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Note: The probabilities of the HMM are not normalized,

so it should it should be treated as an undirected graphical model