Van Gael et al ICML 2008 Presented by Daniel Johnson Introduction Infinite Hidden Markov Model iHMM is n onparametric approach to the HMM New inference algorithm for iHMM Comparison with Gibbs sampling algorithm ID: 246264
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Slide1
Beam Sampling for the Infinite Hidden Markov Model
Van Gael, et al. ICML 2008
Presented by Daniel JohnsonSlide2
Introduction
Infinite Hidden Markov Model (
iHMM
) is
n
onparametric approach to the HMM
New inference algorithm for
iHMM
Comparison with Gibbs sampling algorithm
ExamplesSlide3
Hidden Markov Model (HMM)
Markov Chain with finite state space 1,…,K
Hidden state sequence:
s
= (s
1
, s
2
, … ,
s
T
)
π
ij
= p(
s
t
= j|s
t-1
=
i
)
Observation sequence:
y
= (y
1
, y
2
, … ,
y
T
)
Parameters
ϕ
s
t
such that
p(
y
t
|s
t
) = F(
ϕ
s
t
)
Known:
y
,
π
,
ϕ
,
F
Unknown:
sSlide4
Infinite Hidden Markov Model (
iHMM
)
Known:
y,
F
Unknown:
s, π, ϕ, KStrategy: use BNP priors to deal with additional unknowns: Slide5
Gibbs Methods
Teh
et al., 2006: marginalize out
π
,
ϕ
Update prediction for each s
t individually Computation of O(TK)Non-conjugacy handled in standard Neal wayDrawback: potential slow mixingSlide6
Beam Sampler
Introduce auxiliary variable
u
Conditioned on
u
, # possible trajectories finite
Use dynamic programming filtering algorithm Avoid marginalizing out π, ϕIteratively sample u, s, π, ϕ
, β
,
α
,
γSlide7
Auxiliary Variable u
Sample each u
t
~ Uniform(0,
π
s
t-1
st)u acts as a threshold on πOnly trajectories with πst-1st
≥ u
t
are possibleSlide8
Forward-Backward Algorithm
Forwards: compute p(s
t
|y
1:t
,u
1:t)
from t = 1..TBackward: compute p(st|st+1,y1:T,u1:T) and sample st from t = T..1Slide9
Non-Sticky ExampleSlide10
Sticky ExampleSlide11
Example: Well DataSlide12
Issues/Conclusions
Beam sampler is elegant and fairly straight forward
Beam sampler allows for bigger steps in the MCMC state space than the Gibbs method
Computational cost similar to Gibbs method
Potential for poor mixing
Bookkeeping can be complicated