Transcript Speech, Perception, & AI

Uncertain Reasoning over Time
Artificial Intelligence
CMSC 25000
February 22, 2007
Noisy-Channel Model
• Original message not directly observable
– Passed through some channel b/t sender, receiver + noise
– From telephone (Shannon), Word sequence vs acoustics
(Jelinek), genome sequence vs CATG, object vs image
• Derive most likely original input based on observed
Bayesian Inference
• P(W|O) difficult to compute
– W – input, O – observations
W *  arg max P(W | O)
W
 arg max
W
P(O | W ) P(W )
P(O)
 arg max P(O | W ) P(W )
W
–
Generative and Sequence
Applications
• AI: Speech recognition!, POS tagging,
sense tagging, dialogue, image
understanding, information retrieval
• Non-AI:
– Bioinformatics: gene sequencing
– Security: intrusion detection
– Cryptography
Hidden Markov Models:
Probabilistic Reasoning Over Time
Natural Language Processing
CMSC 25000
February 22, 2007
Agenda
• Hidden Markov Models
–
–
–
–
Uncertain observation
Temporal Context
Recognition: Viterbi
Training the model: Baum-Welch
• Speech Recognition
– Framing the problem: Sounds to Sense
– Speech Recognition as Modern AI
Modelling Processes over Time
• Infer underlying state sequence from observed
• Issue: New state depends on preceding states
– Analyzing sequences
• Problem 1: Possibly unbounded # prob tables
– Observation+State+Time
• Solution 1: Assume stationary process
– Rules governing process same at all time
• Problem 2: Possibly unbounded # parents
– Markov assumption: Only consider finite history
– Common: 1 or 2 Markov: depend on last couple
Hidden Markov Models (HMMs)
• An HMM is:
– 1) A set of states: Q  qo , q1,...,qk
– 2) A set of transition probabilities: A  a01 ,...,amn
• Where aij is the probability of transition qi -> qj
– 3)Observation probabilities:
B  bi (ot )
• The probability of observing ot in state i
– 4) An initial probability dist over states:
• The probability of starting in state i
– 5) A set of accepting states
i
Three Problems for HMMs
• Find the probability of an observation sequence given a
model
– Forward algorithm
• Find the most likely path through a model given an
observed sequence
– Viterbi algorithm (decoding)
• Find the most likely model (parameters) given an observed
sequence
– Baum-Welch (EM) algorithm
Bins and Balls Example
• Assume there are two bins filled with red
and blue balls. Behind a curtain, someone
selects a bin and then draws a ball from it
(and replaces it). They then select either the
same bin or the other one and then select
another ball…
– (Example due to J. Martin)
Bins and Balls Example
.6
.7
.4
Bin 1
Bin 2
.3
Bins and Balls
• Π Bin 1: 0.9; Bin 2: 0.1
• A
Bin1 Bin2
Bin1 0.6 0.4
Bin2
• B
Red
Blue
0.3
0.7
Bin 1 Bin 2
0.7
0.4
0.3
0.6
Bins and Balls
• Assume the observation sequence:
– Blue Blue Red (BBR)
• Both bins have Red and Blue
– Any state sequence could produce observations
• However, NOT equally likely
– Big difference in start probabilities
– Observation depends on state
– State depends on prior state
Bins and Balls
Blue Blue Red
111
112
121
122
211
212
221
222
(0.9*0.3)*(0.6*0.3)*(0.6*0.7)=0.0204
(0.9*0.3)*(0.6*0.3)*(0.4*0.4)=0.0077
(0.9*0.3)*(0.4*0.6)*(0.3*0.7)=0.0136
(0.9*0.3)*(0.4*0.6)*(0.7*0.4)=0.0181
(0.1*0.6)*(0.3*0.7)*(0.6*0.7)=0.0052
(0.1*0.6)*(0.3*0.7)*(0.4*0.4)=0.0020
(0.1*0.6)*(0.7*0.6)*(0.3*0.7)=0.0052
(0.1*0.6)*(0.7*0.6)*(0.7*0.4)=0.0070
Answers and Issues
• Here, to compute probability of observed
– Just add up all the state sequence probabilities
• To find most likely state sequence
– Just pick the sequence with the highest value
• Problem: Computing all paths expensive
– 2T*N^T
• Solution: Dynamic Programming
– Sweep across all states at each time step
• Summing (Problem 1) or Maximizing (Problem 2)
Forward Probability
 j (t )  P(o1 , o2 ,..,ot , qt  j |  )
 j (1)   j b j (o1 ),1  j  N
N

 j (t  1)    i (t )aij b j (ot 1 )
 i 1

N
P(O |  )    i (T )
i 1
Where α is the forward probability, t is the time in utterance,
i,j are states in the HMM, aij is the transition probability,
bj(ot) is the probability of observing ot in state bj
N is the max state, T is the last time
Forward Algorithm
• Idea: matrix where each cell forward[t,j] represents probability
of being in state j after seeing first t observations.
• Each cell expresses the probability:
forward[t,j] = P(o1,o2,...,ot,qt=j|w)
• qt = j means "the probability that the tth state in the sequence of
states is state j.
• Compute probability by summing over extensions of all paths
leading to current cell.
• An extension of a path from a state i at time t-1 to state j at t is
computed by multiplying together:
i. previous path probability from the previous cell forward[t1,i],
ii. transition probability aij from previous state i to current
state j
iii. observation likelihood bjt that current state j matches
observation symbol t.
Forward Algorithm
Function Forward(observations length T, state-graph) returns
best-path
Num-states<-num-of-states(state-graph)
Create path prob matrix forwardi[num-states+2,T+2]
Forward[0,0]<- 1.0
For each time step t from 0 to T do
for each state s from 0 to num-states do
for each transition s’ from s in state-graph
new-score<-Forward[s,t]*at[s,s’]*bs’(ot)
Forward[s’,t+1] <- Forward[s’,t+1]+new-score
Viterbi Algorithm
• Find BEST sequence given signal
– Best P(sequence|signal)
– Take HMM & observation sequence
• => seq (prob)
• Dynamic programming solution
– Record most probable path ending at a state i
• Then most probable path from i to end
• O(bMn)
Viterbi Code
Function Viterbi(observations length T, state-graph) returns best-path
Num-states<-num-of-states(state-graph)
Create path prob matrix viterbi[num-states+2,T+2]
Viterbi[0,0]<- 1.0
For each time step t from 0 to T do
for each state s from 0 to num-states do
for each transition s’ from s in state-graph
new-score<-viterbi[s,t]*at[s,s’]*bs’(ot)
if ((viterbi[s’,t+1]==0) || (viterbi[s’,t+1]<new-score))
then
viterbi[s’,t+1] <- new-score
back-pointer[s’,t+1]<-s
Backtrace from highest prob state in final column of viterbi[] & return
Learning HMMs
• Issue: Where do the probabilities come from?
• Supervised/manual construction
• Solution: Learn from data
– Trains transition (aij), emission (bj), and initial (πi)
probabilities
• Typically assume state structure is given
– Unsupervised
– Baum-Welch aka forward-backward algorithm
• Iteratively estimate counts of transitions/emitted
• Get estimated probabilities by forward comput’n
– Divide probability mass over contributing paths
Manual Construction
• Manually labeled data
– Observation sequences, aligned to
– Ground truth state sequences
•
•
•
•
•
Compute (relative) frequencies of state transitions
Compute frequencies of observations/state
Compute frequencies of initial states
Bootstrapping: iterate tag, correct, reestimate, tag.
Problem:
– Labeled data is expensive, hard/impossible to obtain, may be
inadequate to fully estimate
• Sparseness problems
Unsupervised Learning
• Re-estimation from unlabeled data
– Baum-Welch aka forward-backward algorithm
– Assume “representative” collection of data
• E.g. recorded speech, gene sequences, etc
– Assign initial probabilities
• Or estimate from very small labeled sample
– Compute state sequences given the data
• I.e. use forward algorithm
– Update transition, emission, initial probabilities
Updating Probabilities
• Intuition:
– Observations identify state sequences
– Adjust probability of transitions/emissions
– Make closer to those consistent with observed
–
Increase P(Observations|Model)
• Functionally
– For each state i, what proportion of transitions from
state i go to state j
– For each state i, what proportion of observations match
O?
– How often is state i the initial state?
Estimating Transitions
• Consider updating transition aij
– Compute probability of all paths using aij
– Compute probability of all paths through i (w/ and w/o i->j)
i
j
Forward Probability
 j (t )  P(o1 , o2 ,..,ot , qt  j |  )
 j (1)   j b j (o1 ),1  j  N


 j (t  1)    i (t )aij b j (ot 1 )
 i 1

N
N
P(O |  )    i (T )
i 1
Where α is the forward probability, t is the time in utterance,
i,j are states in the HMM, aij is the transition probability,
bj(ot) is the probability of observing ot in state bj
N is the max state, T is the last time
Backward Probability
i (T )  1
N
i (t )   aijb j (ot 1 ) j (t  1)
i 1
N
P (O |  )    j b j (o1 ) j (1)
j 1
Where β is the backward probability, t is the time in sequence,
i,j are states in the HMM, aij is the transition probability,
bj(ot) is the probability of observing ot in state bj
N is the final state, and T is the last time
Re-estimating
• Estimate transitions
from i->j
 t (i, j ) 
 i (t )aijb j (ot 1 )  j (t  1)
P(O |  )
 (i, j )


   (i, j )
T 1
aˆij
T 1
t 1
N
t 1
• Estimate
observations in j
t
j 1
P(qt  j , O |  )  j (t )  j (t )
 j (t ) 

P(O |  )
P(O |  )
t 1s.t.ot vk
bˆ j (vk ) 
T
T

t 1
ˆ i   i (1)
• Estimate initial i
t
 j (t )
 j (t )
Speech Recognition
• Goal:
– Given an acoustic signal, identify the
sequence of words that produced it
– Speech understanding goal:
• Given an acoustic signal, identify the meaning intended by
the speaker
• Issues:
– Ambiguity: many possible pronunciations,
– Uncertainty: what signal, what word/sense
produced this sound sequence
Decomposing Speech Recognition
• Q1: What speech sounds were uttered?
– Human languages: 40-50 phones
• Basic sound units: b, m, k, ax, ey, …(arpabet)
• Distinctions categorical to speakers
– Acoustically continuous
• Part of knowledge of language
– Build per-language inventory
– Could we learn these?
Decomposing Speech Recognition
• Q2: What words produced these sounds?
– Look up sound sequences in dictionary
– Problem 1: Homophones
• Two words, same sounds: too, two
– Problem 2: Segmentation
• No “space” between words in continuous speech
• “I scream”/”ice cream”, “Wreck a nice beach”/”Recognize
speech”
• Q3: What meaning produced these words?
– NLP (But that’s not all!)
Signal Processing
• Goal: Convert impulses from microphone
into a representation that
– is compact
– encodes features relevant for speech recognition
• Compactness: Step 1
– Sampling rate: how often look at data
• 8KHz, 16KHz,(44.1KHz= CD quality)
– Quantization factor: how much precision
• 8-bit, 16-bit (encoding: u-law, linear…)
(A Little More) Signal Processing
• Compactness & Feature identification
– Capture mid-length speech phenomena
• Typically “frames” of 10ms (80 samples)
– Overlapping
– Vector of features: e.g. energy at some frequency
– Vector quantization:
• n-feature vectors: n-dimension space
– Divide into m regions (e.g. 256)
– All vectors in region get same label - e.g. C256
Speech Recognition Model
• Question: Given signal, what words?
• Problem: uncertainty
– Capture of sound by microphone, how phones
produce sounds, which words make phones, etc
• Solution: Probabilistic model
– P(words|signal) =
–
P(signal|words)P(words)/P(signal)
– Idea: Maximize P(signal|words)*P(words)
• P(signal|words): acoustic model; P(words): lang model
Language Model
• Idea: some utterances more probable
• Standard solution: “n-gram” model
– Typically tri-gram: P(wi|wi-1,wi-2)
• Collect training data
– Smooth with bi- & uni-grams to handle sparseness
– Product over words in utterance
Acoustic Model
• P(signal|words)
– words -> phones + phones -> vector quantiz’n
• Words -> phones
– Pronunciation dictionary lookup
• Multiple pronunciations?
– Probability distribution
» Dialect Variation: tomato
t
ow
» +Coarticulation
– Product along path
0.2 ow
t
0.8 ax
0.5 aa
m
0.5 ey
t
ow
0.5 aa
m
0.5 ey
t
ow
Pronunciation Example
• Observations: 0/1
Acoustic Model
• P(signal| phones):
– Problem: Phones can be pronounced differently
• Speaker differences, speaking rate, microphone
• Phones may not even appear, different contexts
– Observation sequence is uncertain
• Solution: Hidden Markov Models
– 1) Hidden => Observations uncertain
– 2) Probability of word sequences =>
• State transition probabilities
– 3) 1st order Markov => use 1 prior state
Acoustic Model
• 3-state phone model for [m]
– Use Hidden Markov Model (HMM)
0.3
0.9
0.4
Transition probabilities
Onset 0.7 Mid 0.1 End 0.6 Final
C3:
C1: C2:
0.3
0.5 0.2
C5:
C3: C4:
0.1
0.2 0.7
C6:
C4: C6:
0.4
0.1 0.5
Observation probabilities
– Probability of sequence: sum of prob of paths
ASR Training
• Models to train:
–
–
–
–
Language model: typically tri-gram
Observation likelihoods: B
Transition probabilities: A
Pronunciation lexicon: sub-phone, word
• Training materials:
– Speech files – word transcription
– Large text corpus
– Small phonetically transcribed speech corpus
Training
• Language model:
– Uses large text corpus to train n-grams
• 500 M words
• Pronunciation model:
– HMM state graph
– Manual coding from dictionary
• Expand to triphone context and sub-phone models
HMM Training
• Training the observations:
– E.g. Gaussian: set uniform initial mean/variance
• Train based on contents of small (e.g. 4hr) phonetically
labeled speech set (e.g. Switchboard)
• Training A&B:
– Forward-Backward algorithm training
Does it work?
• Yes:
– 99% on isolated single digits
– 95% on restricted short utterances (air travel)
– 89+% professional news broadcast
• No:
–
–
–
–
77% Conversational English
67% Conversational Mandarin (CER)
55% Meetings
?? Noisy cocktail parties
N-grams
• Perspective:
– Some sequences (words/chars) are more likely
than others
– Given sequence, can guess most likely next
• Used in
–
–
–
–
Speech recognition
Spelling correction,
Augmentative communication
Other NL applications
Probabilistic Language Generation
• Coin-flipping models
– A sentence is generated by a randomized
algorithm
• The generator can be in one of several “states”
• Flip coins to choose the next state.
• Flip other coins to decide which letter or word to
output
Shannon’s Generated Language
• 1. Zero-order approximation:
– XFOML RXKXRJFFUJ ZLPWCFWKCYJ
FFJEYVKCQSGHYD QPAAMKBZAACIBZLHJQD
• 2. First-order approximation:
– OCRO HLI RGWR NWIELWIS EU LL NBNESEBYA
TH EEI ALHENHTTPA OOBTTVA NAH RBL
• 3. Second-order approximation:
– ON IE ANTSOUTINYS ARE T INCTORE ST BE S
DEAMY ACHIND ILONASIVE TUCOOWE AT
TEASONARE FUSO TIZIN ANDY TOBE SEACE
CTISBE
Shannon’s Word Models
• 1. First-order approximation:
– REPRESENTING AND SPEEDILY IS AN GOOD
APT OR COME CAN DIFFERENT NATURAL HERE
HE THE A IN CAME THE TO OF TO EXPERT
GRAY COME TO FURNISHES THE LINE
MESSAGE HAD BE THESE
• 2. Second-order approximation:
– THE HEAD AND IN FRONTAL ATTACK ON AN
ENGLISH WRITER THAT THE CHARACTER OF
THIS POINT IS THEREFORE ANOTHER METHOD
FOR THE LETTERS THAT THE TIME OF WHO
EVER TOLD THE PROBLEM FOR AN
UNEXPECTED
Corpus Counts
• Estimate probabilities by counts in large
collections of text/speech
• Issues:
– Wordforms (surface) vs lemma (root)
– Case? Punctuation? Disfluency?
– Type (distinct words) vs Token (total)
Basic N-grams
• Most trivial: 1/#tokens: too simple!
• Standard unigram: frequency
– # word occurrences/total corpus size
• E.g. the=0.07; rabbit = 0.00001
– Too simple: no context!
2
n
• Conditional
of
word
P(w1n )  Pprobabilities
(
w
)
P
(
w
|
w
)
P
(
w
|
w
)...
P
(
w
|
w
1
2
1
3
1
n
1)
n
sequences  P(wk | w1k 1 )
k 1
Markov Assumptions
• Exact computation requires too much data
• Approximate probability given all prior
wds
– Assume finite history
– Bigram: Probability of word given 1 previous
• First-order Markov
n1
n1
P
(
w
|
w
)

P
(
w
|
w
n
1
n
n N 1 )2 previous
– Trigram: Probability
of word
given
n
• N-gram approximation
n
Bigram sequence
P( w1 )   P( wk | wk 1 )
k 1
Issues
• Relative frequency
– Typically compute count of sequence
• Divide by prefix
C ( wn wn 1 )
P( wn | wn 1 ) 
C ( wn 1 )
• Corpus sensitivity
– Shakespeare vs Wall Street Journal
• Very unnatural
• Ngrams
Evaluating n-gram models
• Entropy & Perplexity
– Information theoretic measures
– Measures information in grammar or fit to data
– Conceptually, lower bound on # bits to encode
H ( XH(X):
)  
( x)alog
• Entropy:
Xpis
random
2 p( x) var, p: prob fn
xX
– E.g. 8 things:
H number as code => 3 bits/trans
– Alt. short 2
code if high prob; longer if lower
• Can reduce
• Perplexity:
Computing Entropy
• Picking horses (Cover and Thomas)
• Send message: identify horse - 1 of 8
– If all horses 8equally likely, p(i) = 1/8
H ( X )  1 / 8 log1 / 8   log1 / 8  3bits
i 1
– Some horses
8 more likely:
) log4:
p(1/16;
i)  2bits
 3:p(i1/8;
• H1:( X½;) 2:¼;
5,6,7,8: 1/64
i 1
Entropy of a Sequence
• Basic sequence
1
n
H (W1n )  
1
p(W1n ) log2 p(W1n )

n W1n L
• Entropy of
language: infinite
lengths
–
1
H
(
L
)

lim

p ( w1 ,...,wn ) log p ( w1 ,...,wn )

Assume stationary
n 
n W L
& ergodic
1
H ( L)  lim  log p ( w1 ,...,wn )
n 
n
Cross-Entropy
• Comparing models
– Actual distribution unknown
– Use simplified model to estimate
• Closer match will have lower cross-entropy
1
H ( p, m)  lim   p( w1 ,...,wn ) log m( w1 ,...,wn )
n 
n W L
1
H ( p, m)  lim  log m( w1 ,...,wn )
n 
n
Perplexity Model Comparison
• Compare models with different history
• Train models
– 38 million words – Wall Street Journal
• Compute perplexity on held-out test set
– 1.5 million words (~20K unique, smoothed)
• N-gram Order |
– Unigram
– Bigram
– Trigram
|
|
|
Perplexity
962
170
109
Entropy of English
• Shannon’s experiment
– Subjects guess strings of letters, count guesses
– Entropy of guess seq = Entropy of letter seq
– 1.3 bits; Restricted text
• Build stochastic model on text & compute
– Brown computed trigram model on varied corpus
– Compute (per-char) entropy of model
– 1.75 bits
Speech Recognition as
Modern AI
• Draws on wide range of AI techniques
– Knowledge representation & manipulation
• Optimal search: Viterbi decoding
– Machine Learning
• Baum-Welch for HMMs
• Nearest neighbor & k-means clustering for
signal id
– Probabilistic reasoning/Bayes rule
• Manage uncertainty in signal, phone, word
mapping