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Sequence Models

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Sequence Models

With slides by me, Joshua Goodman, Fei Xia

- Language Modeling
- Ngram Models
- Hidden Markov Models
- Supervised Parameter Estimation
- Probability of a sequence
- Viterbi (or decoding)
- Baum-Welch

Language Model: A distribution that assigns a probability to language utterances.

e.g., PLM(“zxcv ./,mweaafsido”) is zero;

PLM(“mat cat on the sat”) is tiny;

PLM(“Colorless green ideas sleep furiously”) is bigger;

PLM(“A cat sat on the mat”) is bigger still.

- Information Retrieval
- Handwriting recognition
- Speech Recognition
- Spelling correction
- Optical character recognition
- Machine translation
- …

Speech Recognition: convert an acoustic signal (sound wave recorded by a microphone) to a sequence of words (text file).

Straightforward model:

But this can be hard to train effectively (although see CRFs later).

Speech Recognition: convert an acoustic signal (sound wave recorded by a microphone) to a sequence of words (text file).

Traditional solution: Bayes’ Rule

Acoustic Model

(easier to train)

Language Model

Ignore: doesn’t matter for picking a good text

So far, we’ve been making the exchangeability, or bag-of-words, assumption:

The order of words is not important.

It turns out, that’s actually not true (duh!).

“cat mat on the the sat” ≠ “the cat sat on the mat”

“Mary loves John” ≠ “John loves Mary”

Problem: How can we define a model that

- assigns probability to sequences of words (a language model)
- the probability depends on the order of the words
- the model can be trained and computed tractably?

- Language Modeling
- Ngram Models
- Hidden Markov Models
- Supervised parameter estimation
- Probability of a sequence (decoding)
- Viterbi (Best hidden layer sequence)
- Baum-Welch

P(Francisco | eggplant) vs P(stew | eggplant)

- “Francisco” is common, so backoff, interpolated methods say it is likely
- But it only occurs in context of “San”
- “Stew” is common, and in many contexts
- Weight backoff by number of contexts word occurs in

Backoff:

Interpolation:

- Language Modeling
- Ngram Models
- Hidden Markov Models
- Supervised parameter estimation
- Probability of an observation sequence
- Viterbi (Best hidden layer sequence)
- Baum-Welch

A dynamicBayes Net (dynamic because the size can change).

The Oi nodes are called observed nodes.

The Xi nodes are called hidden nodes.

X1

X2

…

XT

O1

O2

…

OT

NLP

- HMMs have been used in a variety of applications, but especially:
- Speech recognition
(hidden nodes are text words, observations are spoken words)

- Part of Speech Tagging
(hidden nodes are parts of speech, observations are words)

- Speech recognition

X1

X2

…

XT

O1

O2

…

OT

NLP

HMMs assume that:

- Xi is independent of X1 through Xi-2, given Xi-1 (Markov assumption)
- Oi is independent of all other nodes, given Xi
- P(Xi | Xi-1) and P(Oi | Xi) do not depend on i (stationary assumption)
Not very realistic assumptions about language – but HMMs are often good enough, and very convenient.

X1

X2

…

XT

O1

O2

…

OT

NLP

An HMM predicts that the probability of observing a sequence o = <o1, o2, …, oT> with a particular set of hidden states x = <x1, … xT> is:

To calculate, we need:

- Prior: P(x1) for all values of x1

- Observation: P(oi|xi) for all values of oi and xi

- Transition: P(xi|xi-1) for all values of xi and xi-1

- A set of hidden states H = {h1, …, hN} that are the values which hidden nodes may take.
- A vocabulary, or set of states V = {v1, …, vM} that are the values which an observed node may take.
- Initial probabilities P(x1=hj) for all j
- Written as a vector of N initial probabilities, called π
- πj = P(x1=hj)

- Transition probabilities P(xt=hj | xt-1=hk) for all j, k
- Written as an NxN ‘transition matrix’ A
- Aj,k = P(xt=hj | xt-1=hk)

- Observation probabilities P(ot=vk|st=hj) for all k, j
- written as an MxN ‘observation matrix’ B
- Bj,k = P(ot=vk|st=hj)

- H = {DT, NN, VB, IN, …}, the set of all POS tags.
- V = the set of all words in English.
- Initial probabilities πj are the probability that POS tag hj can start a sentence.
- Transition probabilities Ajk represent the probability that one tag (e.g., hk=VB) can follow another (e.g., hj=NN)
- Observation probabilities Bjk represent the probability that a tag (e.g., hj=NN) will generate a particular word (e.g., vk=cat).

- Language Models
- Ngram modeling
- Hidden Markov Models
- Supervised parameter estimation
- Probability of a sequence
- Viterbi: what’s the best hidden state sequence?
- Baum-Welch: unsupervised parameter estimation

o1

ot-1

ot

ot+1

oT

A

A

A

A

X1

Xt-1

Xt

Xt+1

XT

B

B

B

B

B

- Given an observation sequence and states, find the HMM parameters (π, A, and B) that are most likely to produce the sequence.
- For example, POS-tagged data from the Penn Treebank

o1

ot-1

ot

ot+1

oT

A

A

A

A

x1

xt-1

xt

xt+1

xT

B

B

B

B

B

- Language Modeling
- Ngram models
- Hidden Markov Models
- Supervised parameter estimation
- Probability of a sequence
- Viterbi
- Baum-Welch

Suppose I asked you, ‘What’s the probability of seeing a sentence v1, …, vT on the web?’

If we have an HMM model of English, we can use it to estimate the probability.

(In other words, HMMs can be used as language models.)

- If we knew the hidden states that generated each word in the sentence, it would be easy:

Via marginalization, we have:

Unfortunately, if there are N values for each xi (h1 through hN),

Then there are NT values for x1,…,xT.

Brute-force computation of this sum is intractable.

x1

xt-1

xt

xt+1

xT

o1

ot-1

ot

ot+1

oT

- Special structure gives us an efficient solution using dynamic programming.
- Intuition: Probability of the first t observations is the same for all possible t+1 length state sequences.
- Define:

x1

xt-1

xt

xt+1

xT

o1

ot-1

ot

ot+1

oT

x1

xt-1

xt

xt+1

xT

o1

ot-1

ot

ot+1

oT

x1

xt-1

xt

xt+1

xT

o1

ot-1

ot

ot+1

oT

x1

xt-1

xt

xt+1

xT

o1

ot-1

ot

ot+1

oT

x1

xt-1

xt

xt+1

xT

o1

ot-1

ot

ot+1

oT

x1

xt-1

xt

xt+1

xT

o1

ot-1

ot

ot+1

oT

x1

xt-1

xt

xt+1

xT

o1

ot-1

ot

ot+1

oT

x1

xt-1

xt

xt+1

xT

o1

ot-1

ot

ot+1

oT

x1

xt-1

xt

xt+1

xT

o1

ot-1

ot

ot+1

oT

Probability of the rest of the states given the first state

x1

xt-1

xt

xt+1

xT

o1

ot-1

ot

ot+1

oT

Forward Procedure

Backward Procedure

Combination

- Language modeling
- Ngram models
- Hidden Markov Models
- Supervised parameter estimation
- Probability of a sequence
- Viterbi: what’s the best hidden state sequence?
- Baum-Welch

Find the hidden state sequence that best explains the observations

Viterbi algorithm

o1

ot-1

ot

ot+1

oT

x1

xt-1

hj

o1

ot-1

ot

ot+1

oT

The state sequence which maximizes the probability of seeing the observations to time t-1, landing in state hj, and seeing the observation at time t

o1

ot-1

ot

ot+1

oT

x1

xt-1

xt

xt+1

Recursive Computation

o1

ot-1

ot

ot+1

oT

x1

xt-1

xt

xt+1

xT

Compute the most likely state sequence by working backwards

- Language modeling
- Ngram models
- Hidden Markov Models
- Supervised parameter estimation
- Probability of a sequence
- Viterbi
- Baum-Welch: Unsupervised parameter estimation

o1

ot-1

ot

ot+1

oT

A

A

A

A

B

B

B

B

B

- Given an observation sequence, find the model that is most likely to produce that sequence.
- No analytic method
- Given a model and observation sequence, update the model parameters to better fit the observations.

o1

ot-1

ot

ot+1

oT

A

A

A

A

B

B

B

B

B

Probability of traversing an arc

Probability of being in state i

o1

ot-1

ot

ot+1

oT

A

A

A

A

B

B

B

B

B

Now we can compute the new estimates of the model parameters.

o1

ot-1

ot

ot+1

oT

A

A

A

A

B

B

B

B

B

- Guarantee: P(o1:T|A,B,π) <= P(o1:T|Â,B̂,π̂)
- In other words, by repeating this procedure, we can gradually improve how well the HMM fits the unlabeled data.
- There is no guarantee that this will converge to the best possible HMM, however (only guaranteed to find a local maximum).

o1

ot-1

ot

ot+1

oT

A

A

A

A

B

B

B

B

B

We can use the special structure of this model to do a lot of neat math and solve problems that are otherwise not tractable.