Week7-3HMM1

Markov model

• sequence of random variables that are not independent
  
  • weather report
  • text

Properties

• limited horizon
  
  • $P(X_{t+1} = s_k \mid X_1, ..., X_t) = P(X_{t+1} = s\mid X_t)$(first order)
• time invariant(stationary)
  
  • $P(X_2 = s_k) = P(X_1)$

Visible MM

$$\begin{align} P(X_1, ..., X_T) &= P(X_1)P(X_2 \mid X_1)P(X_3 \mid X_1, X_2) ... P(X_T \mid X_1, X_2, ..., X_{T-1})\\ &= P(X_1)P(X_2 \mid X_1) P(X_3 \mid X_2) ... P(X_t \mid X_{t-1}) \end{align}$$

Hidden MM

• Motivation
  
  • observing a sequence of symbols
  • the sequence of state that led to the generation of the symbols is hidden
• Definition
  
  • Q = sequence of states
  • O = sequence of observations, drawn from a vocabulary
  • $q_0, q_f$ = special (start, final) states
  • A = state transition probabilities
  • B = symbol emission probabilities
  • $\Pi$ = initial state probabilities
  • $\mu(A,B,Pi)$ = complete probabilistic model

used to model state sequences and observation sequences

Generative algorithm

• pick start state from $Pi$
• For t = 1…T
  
  • move to another state based on A
  • emit an observation based on B

Example

State probability

Emission probability

• Initial
  

  $$P(A\mid start) = 1.0 \\ P(B\mid start) = 0.0$$

• Transition
  

  $$P(A \mid A) = 0.8\\ P(A \mid B) = 0.6\\ P(B \mid A) = 0.2\\ P(B \mid B) = 0.4$$

• Emission: see previous table

observation of the sequence “yz”

• Possible sequences of states:
  
  • AA
  • AB
  • BA
  • BB

$$\begin{align} p(yz) &= p(yz \mid AA) + p(yz \mid AB) + p(yz \mid BA) + p(yz \mid BB)\\ &=1.0 \times 0.2 \times 0.8 \times 0.1 \\ &=1.0 \times 0.2 \times 0.2 \times 0.2 \\ &=1.0 \times 0.5 \times 0.6 \times 0.2 \\ &=1.0 \times 0.5 \times 0.4 \times 0.1 \end{align}$$

In this way we could compute all the possibilities of any sequence.

State and transitions

• states:

  • the states encode the most recent history
  • the transitions encode likely sequences of states
  • use MLE to estimate the transition probabilities

• transitions:

  • estimating the emission probabilities
  • possible to use standard smoothing and heuristic methods

Sequence of observation

• observers can only see the emitted symbols
• observation likelihood
  
  • given the observation sequence S and the model $\mu$, what is the probability $P(S \mid \mu)$ that the sequence was generated by that model
• HMM turn to language model

Tasks with HMM

• tasks
  
  • Given a $\mu(A, B, \Pi)$, find the distribution $P(O\mid \mu)$
  • Given $O, \mu$, what is $(X_1. X_2, ..., X_t)$
  • Given O and a space of all possible $\mu$, find the best $\mu$
• decoding
  
  • tag each token with a label

Inference

• find the most likely tag, given the word
  
  • $t^\ast = \arg\max_t p(t \mid w)$
• given the model $\mu$, we could find the best sequence of tags ${\{ t_i \}_1^n}^\ast$ given the sequence of words $\{w_i\}_1^n$
• too many ways for combinations

Viterbi algorithm

• Find the best path up to observation i and state s(partial best path), and if we condition on the whole string of i, we will get the best path for the whole sentence.

  • dynamic programming
  • memoization
  • backpointers

• initial state
  

• we could calculate the first state $P(t = 1)$
  

• say if we want to calculate $P(B , t = 2)$ and calculating $P(A, t = 2)$ is similar
  

• and we could find the best path and best sequence of states
  

• finally we could find the best sequence for all observations

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