Hidden Markovian Model
Some Definitions
• Finite automation is defined by a set of states,
and a set of transitions between states that
are taken based on the input observations
• A weighted finite-state automation is a simple
augmentation of the finite automaton in
which each arc is associated with a probability,
indicating how likely that path is to be taken
– Sum of all outgoing arcs from a particular state
should equal zero
Markov Chain
• A Markov chain is a special case of a weighted
automaton in which the input sequence
uniquely determines which states the
automation will go through
• Markov chain is only useful for assigning
probabilities to unambiguous sequences
Hidden Markovian Model (HMM)
• Hidden State
– The states are not directly observable in the world
instead they have to be inferred through other
means
HMM
• HMM
– A set of N states
– A set of O observations
– A special start and end state
– Transition Probability
– Emission Probability
HMM
• Transition Probability
– At each time instant the system may change its
state from the current state to another state, or
remain in the same state, according to a certain
probability distribution
• Emission Probability
– A sequence of observation likelihoods, each
expressing the probability of a particular
observation being emitted by a particular state
Example
• Imagine you are a climatologist in the year
2799 studying the history of global warming
– No records of weather in Baltimore for Summer
2007
– We have Jason Eisner’s Diary which has how many
ice creams he had each day
– Our goal is to estimate the climate based on the
observations we have. For simplicity we are going
to assume only two states, hot and cold.
Markov Assumptions
• A first order HMM instantiates two simplifying
assumptions
– The probability of a particular state is dependent
only on the previous state
– The probability of an output observation is
dependent only on the state that produced the
observation and not any other states or any other
observations
HMM usage
• There are 3 important ways in which HMM is
used,
– Computing Likelihood
• Given an HMM l = (A,B) and an observation sequence
O, determine the likelihood P(O|l)
– Decoding
• Given an observation sequence O and an HMM l =
(A,B), discover the best hidden state sequence Q.
– Learning
• Given an observation sequence O and the set of states
in the HMM, learn the HMM parameters A and B.
Computing Likelihood
• Lets assume 3 1 3 is our observation sequence
• The real problem here is that we are not
aware of the hidden state sequence
corresponding to the observation sequence
• This is to compute the total probability of the
observations just by summing over all possible
hidden state sequences
Forward Algorithm
• For N number of states and T sequences there
can be upto N^T possible hidden sequences
and when N and T are considerably large there
can be an issue here…
• Forward Algorithm
– It is a kind of dynamic programming algorithm, i.e,
algorithm that uses a table to store intermediate
values as it builds up the probability of the
observation sequence