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CS 59000 Statistical Machine learning
Lecture 24
Yuan (Alan) Qi
Purdue CS
Nov. 20 2008
Outline
• Review of K-medoids, Mixture of Gaussians,
Expectation Maximization (EM), Alternative
view of EM
• Hidden Markvo Models, forward-backward
algorithm, EM for learning HMM
parameters, Viterbi Algorithm, Linear state
space models, Kalman filtering and
smoothing
K-medoids Algorithm
Mixture of Gaussians
Mixture of Gaussians:
Introduce latent variables:
Marginal distribution:
Conditional Probability
Responsibility that component k takes for
explaining the observation.
Maximum Likelihood
Maximize the log likelihood function
Severe Overfitting by Maximum Likelihood
When a cluster has only data point, its variance
goes to 0.
Maximum Likelihood Conditions (1)
Setting the derivatives of
to zero:
Maximum Likelihood Conditions (2)
Setting the derivative of
to zero:
Maximum Likelihood Conditions (3)
Lagrange function:
Setting its derivative to zero and use the
normalization constraint, we obtain:
Expectation Maximization for Mixture Gaussians
Although the previous conditions do not
provide closed-form conditions, we can use
them to construct iterative updates:
E step: Compute responsibilities
.
M step: Compute new mean , variance ,
and mixing coefficients .
Loop over E and M steps until the log
likelihood
stops to increase.
General EM Algorithm
EM and Jensen Inequality
Goal: maximize
Define:
We have
From Jesen’s Inequality, we see
bound of
.
is a lower
Lower Bound
is a functional of the distribution
Since
.
and
,
is a lower bound of the log likelihood
function
. (Another way to see the
lower bound without using Jensen’s
inequality)
Lower Bound Perspective of EM
• Expectation Step:
Maximizing the functional lower bound
the distribution
.
over
• Maximization Step:
Maximizing the lower bound over the parameters .
Illustration of EM Updates
Sequential Data
There are temporal dependence between data points
Markov Models
By chain rule, a joint distribution can be re-written as:
Assume conditional independence, we have
It is known as first-order Markov chain
High Order Markov Chains
Second order Markov assumption
Can be generalized to higher order Markov
Chains. But the number of the parameters
explores exponentially with the order.
State Space Models
Important graphical models for many dynamic
models, includes Hidden Markov Models (HMMs)
and linear dynamic systems
Questions: order for the Markov assumption
Hidden Markov Models
Many applications, e.g., speech recognition,
natural language processing, handwriting
recognition, bio-sequence analysis
From Mixture Models to HMMs
By turning a mixture Model into a dynamic model,
we obtain the HMM.
Let model the dependence between two consecutive
latent variables by a transition probability:
HMMs
Prior on initial latent variable:
Emission probabilities:
Joint distribution:
Samples from HMM
(a) Contours of constant probability density for the emission
distributions corresponding to each of the three states of the latent
variable. (b) A sample of 50 points drawn from the hidden Markov
model, with lines connecting the successive observations.
Inference: Forward-backward Algorithm
Goal: compute marginals for latent variables.
Forward-backward Algorithm: exact inference
as special case of sum-product algorithm on
the HMM.
Factor graph representation (grouping
emission density and transition probability
in one factor at a time):
Forward-backward Algorithm as Message Passing Method (1)
Forward messages:
Forward-backward Algorithm as Message Passing Method (2)
Backward messages (Q: how to compute it?):
The messages actually involves X
Similarly, we can compute the following (Q: why)
Rescaling to Avoid Overflowing
When a sequence is long, the forward message will
become to small to be represented by the dynamic
range of the computer. We redefine the forward
message
as
Similarly, we re-define the backward message
as
Then, we can compute
See detailed derivation in textbook
Viterbi Algorithm
Viterbi Algorithm:
• Finding the most probable sequence of
states
• Special case of sum-product algorithm on
HMM.
What if we want to find the most probable
individual states?
Maximum Likelihood Estimation for HMM
Goal: maximize
Looks familiar? Remember EM for mixture of
Gaussians… Indeed the updates are similar.
EM for HMM
E step:
Computed from forward-backward/sum-product
algorithm
M step:
Linear Dynamical Systems
Equivalently, we have
where
Kalman Filtering and Smoothing
Inference on linear Gaussian systems.
Kalman filtering: sequentially update scaled
forward message:
Kalman smoothing: sequentially update state
beliefs based on scaled forward and
backward messages:
Learning in LDS
EM again…
Extension of HMM and LDS
Discrete latent variables: Factorized HMMs
Continuous latent variables: switching Kalman filtering models