Particle Filtering Sometimes |X| is too big to use exact inference - - PowerPoint PPT Presentation

Particle Filtering

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ØSometimes |X| is too big to use exact inference

• |X| may be too big to even store

B(X)

• E.g. X is continuous
• |X|2 may be too big to do updates

ØSolution: approximate inference

• Track samples of X, not all values
• Samples are called particles
• Time per step is linear in the

number of samples

• But: number needed may be large
• In memory: list of particles

ØThis is how robot localization works in practice

ØElapse of time

B’(Xt)=Σxt-1p(Xt|xt-1)B(xt-1) ØObserve B(Xt) ∝p(et|Xt)B’(Xt)

ØRenormalize

B(xt) sum up to 1

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Forward algorithm vs. particle filtering

Forward algorithm Particle filtering

• Elapse of time

x--->x’

• Observe

w(x’)=p(et|x)

• Resample

resample N particles

Today

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ØSpeech recognition

• A massive HMM!

ØIntroduction to machine learning

Speech and Language

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ØSpeech technologies

• Automatic speech recognition (ASR)
• Text-to-speech synthesis (TTS)
• Dialog systems

ØLanguage processing technologies

• Machine translation
• Information extraction
• Web search, question answering
• Text classification, spam filtering, etc…

Digitizing Speech

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The Input

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ØSpeech input is an acoustic wave form

Graphs from Simon Arnfield’s web tutorial on speech, sheffield: http://www.psyc.leeds.ac.uk/research/cogn/speech/tutorial/

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The Input

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ØFrequency gives pitch; amplitude gives volume

• Sampling at ~8 kHz phone, ~16 kHz mic

ØFourier transform of wave displayed as a spectrogram

• Darkness indicates energy at each frequency

Acoustic Feature Sequence

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ØTime slices are translated into acoustic feature vectors (~39 real numbers per slice) ØThese are the observations, now we need the hidden states X

State Space

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Øp(E|X) encodes which acoustic vectors are appropriate for each phoneme (each kind of sound) Øp(X|X’) encodes how sounds can be strung together ØWe will have one state for each sound in each word ØFrom some state x, can only:

• Stay in the same state (e.g. speaking slowly)
• Move to the next position in the word
• At the end of the word, move to the start of the next word

ØWe build a little state graph for each word and chain them together to form our state space X

HMMs for Speech

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Transitions with Bigrams

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Decoding

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ØWhile there are some practical issues, finding the words given the acoustics is an HMM inference problem ØWe want to know which state sequence x1:T is most likely given the evidence e1:T: ØFrom the sequence x, we can simply read off the words

( ) ( )

1: 1:

* 1: 1: 1: 1: 1:

argmax | argmax ,

T T

T T T x T T x

x p x e p x e = =

Machine Learning

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ØUp until now: how to reason in a model and how to make optimal decisions ØMachine learning: how to acquire a model on the basis of data / experience

• Learning parameters (e.g. probabilities)
• Learning structure (e.g. BN graphs)
• Learning hidden concepts (e.g. clustering)

Parameter Estimation

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ØEstimating the distribution of a random variable ØElicitation: ask a human ØEmpirically: use training data (learning!)

• E.g.: for each outcome x, look at the empirical rate of

that value:

• This is the estimate that maximizes the likelihood of the

data

( ) ( )

count total samples

ML

x p x =

( ) ( )

,

i i

L x p x

θ

θ =∏

( )

1 3

ML

p r =

Estimation: Smoothing

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ØRelative frequencies are the maximum likelihood estimates (MLEs) ØIn Bayesian statistics, we think of the parameters as just another random variable, with its own distribution

( ) ( )

count total samples

ML

x p x =

( ) ( )

argmax | argmax

ML i i

p X p X

θ θ θ

θ θ = =

∏

( ) ( ) ( ) ( ) ( ) ( )

argmax | argmax | argmax |

MAP

p X p X p p X p X p

θ θ θ

θ θ θ θ θ θ = = =

????

Estimation: Laplace Smoothing

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ØLaplace’s estimate:

• Pretend you saw every outcome
• nce more than you actually did

( ) ( )

ML LAP

p X p X = =

pLAP x

( ) =

c x

( )+1

c x

( )+1

! " # $

x

∑

= c x

( )+1

N + X

Estimation: Laplace Smoothing

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ØLaplace’s estimate (extended):

• Pretend you saw every outcome

k extra times

• What’s Laplace with k=0?
• k is the strength of the prior

ØLaplace for conditionals:

• Smooth each condition

independently:

( ) ( ) ( )

,0 ,1 ,100 LAP LAP LAP

p X p X p X = = =

( ) ( )

,

=

LAP k

c x k p x N k X + +

( ) ( ) ( )

,

, | =

LAP k

c x y k p x y c y k X + +

Example: Spam Filter

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Ø Input: email Ø Output: spam/ham Ø Setup:

• Get a large collection of

example emails, each labeled “spam” or “ham”

• Note: someone has to

hand label all this data!

• Want to learn to predict

labels of new, future emails

Ø Features: the attributes used to make the ham / spam decision

• Words: FREE!
• Text patterns: $dd, CAPS
• Non-text:

senderInContacts

• ……

Example: Digit Recognition

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Ø Input: images / pixel grids Ø Output: a digit 0-9 Ø Setup:

• Get a large collection of example

images, each labeled with a digit

• Note: someone has to hand label

all this data!

• Want to learn to predict labels of

new, future digit images

Ø Features: the attributes used to make the digit decision

• Pixels: (6,8) = ON
• Shape patterns:

NumComponents, AspectRation, NumLoops

• ……

A Digit Recognizer

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ØInput: pixel grids ØOutput: a digit 0-9

Naive Bayes for Digits

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ØSimple version:

• One feature Fij for each grid position <i,j>
• Possible feature values are on / off, based on whether

intensity is more or less than 0.5 in underlying image

• Each input maps to a feature vector, e.g.
• Here: lots of features, each is binary valued

ØNaive Bayes model: ØWhat do we need to learn?

→ F

0,0 = 0 F 0,1 = 0 F 0,2 =1 F 0,3 =1 F 0,4 = 0 F 15,15 = 0

p Y | F

0,0F 15,15

( )∝ p Y ( )

p F

i, j |Y

( )

i, j

∏

General Naive Bayes

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ØA general naive Bayes model: ØWe only specify how each feature depends on the class ØTotal number of parameters is linear in n

Y × F

n

parameters

p Y,F

1Fn

( ) =

p Y

( )

p F

i |Y

( )

i

∏

Y parameters n × Y × F parameters

Inference for Naive Bayes

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ØGoal: compute posterior over causes

• Step 1: get joint probability of causes and evidence
• Step 2: get probability of evidence
• Step 3: renormalize

p Y, f1 fn

( ) =

p y1, f1 fn

( )

p y2, f1 fn

( )

 p yk, f1 fn

( )

! " # # # # # # $ % & & & & & &

p y1

( )

p fi | c1

( )

i

∏

p y2

( )

p fi | c2

( )

i

∏

 p yk

( )

p fi | ck

( )

i

∏

" # $ $ $ $ $ $ $ $ % & ' ' ' ' ' ' ' '

p f

1 f n

( )

p Y | f1 fn

( )

General Naive Bayes

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ØWhat do we need in order to use naive Bayes?

• Inference (you know this part)
• Start with a bunch of conditionals, p(Y) and the p(Fi|Y) tables
• Use standard inference to compute p(Y|F1…Fn)
• Nothing new here
• Estimates of local conditional probability tables
• p(Y), the prior over labels
• p(Fi|Y) for each feature (evidence variable)
• These probabilities are collectively called the parameters of

the model and denoted by θ

• Up until now, we assumed these appeared by magic, but…
• … they typically come from training data: we’ll look at this now

Examples: CPTs

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1 0.1 2 0.1 3 0.1 4 0.1 5 0.1 6 0.1 7 0.1 8 0.1 9 0.1 0.1

( )

p Y

( )

3,1

| p F

• n Y

=

1 0.05 2 0.01 3 0.90 4 0.80 5 0.90 6 0.90 7 0.25 8 0.85 9 0.60 0.80

( )

5,5

| p F

• n Y

=

1 0.01 2 0.05 3 0.05 4 0.30 5 0.80 6 0.90 7 0.05 8 0.60 9 0.50 0.80

Important Concepts

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Ø Data: labeled instances, e.g. emails marked spam/ham

• Training set
• Held out set
• Test set

Ø Features: attribute-value pairs which characterize each x Ø Experimentation cycle

• Learn parameters (e.g. model probabilities) on training set
• (Tune hyperparameters on held-out set)
• Compute accuracy of test set
• Very important: never “peek” at the test set!

Ø Evaluation

• Accuracy: fraction of instances predicted correctly

Ø Overfitting and generalization

• Want a classifier which does well on test data
• Overfitting: fitting the training data very closely, but not

generalizing well