Camera camera = Camera.open(); - - PowerPoint PPT Presentation

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Camera camera = Camera.open(); - - PowerPoint PPT Presentation

Feb 06, 2023 163 likes •2.12k views

Camera camera = Camera.open(); camera.setDisplayOrientation(90); Camera camera = Camera.open(); Camera camera = Camera.open(); camera.setDisplayOrientation(90);

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∈

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Camera camera = Camera.open(); camera.setDisplayOrientation(90);

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Camera camera = Camera.open(); camera.setDisplayOrientation(90); Camera camera = Camera.open(); camera.setDisplayOrientation(90); SurfaceHolder holder = getHolder(); holder.addCallback(this); holder.setType(SurfaceHolder.STP); MediaRecorder r = new MediaRecorder(); r.setAudioSource(MediaRecorder.AS); r.setVideoSource(MediaRecorder.VS); r.setOutFormat(MediaRecorder.MPEG4); camera.unlock(); r.setCamera(camera);

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Console.WriteLine("Hi"); … System.out.println("Hi"); …

C# Java

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function FZ(e, t) { var n = []; var r = e.length; var i = 0; for (; i < r; i += t) if (i + t < r) n.push(e.substring(i, i + t)); else n.push(e.substring(i, r)); return n; } function chunkData(str, step) { var colNames = []; var len = str.length; var i = 0; for (; i < len; i += step) if (i + step < len) colNames.push(str.substring(i, i + step)); else colNames.push(str.substring(i, len)); return colNames; }

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   

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Probability Refresher

  • :  P(x) = 1
  • P(x,y),

:  P(x,y) = 1

  • : P(y | x) = P(y, x) / P(x)
  • P(y | x) = P(x | y) )  P(y) / P (x)

x y

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Probability Refresher

  • P(x, y) = P(x) )  P(y)

– : x  y

  • P(X, Y | Z) = P(X | Z)  P(Y | Z)

– X Y Z. –

X  Y | Z

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∈

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Camera camera = Camera.open(); camera.setDisplayOrientation(90); camera.unlock(); MediaRecorder rec = new MediaRecorder(); rec.setCamera(camera); rec.setAudioSource(MediaRecorder.AudioSource.MIC); rec.setVideoSource(MediaRecorder.VideoSource.DEFAULT); rec.setOutputFormat(MediaRecorder.OutputFormat.MPEG_4); rec.setAudioEncoder(1); rec.setVideoEncoder(3); rec.setOutputFile("file.mp4"); ...

? ? ?

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Camera camera = Camera.open(); camera.setDisplayOrientation(90); camera.unlock(); MediaRecorder rec = new MediaRecorder(); rec.setCamera(camera); rec.setAudioSource(MediaRecorder.AudioSource.MIC); rec.setVideoSource(MediaRecorder.VideoSource.DEFAULT); rec.setOutputFormat(MediaRecorder.OutputFormat.MPEG_4); rec.setAudioEncoder(1); rec.setVideoEncoder(3); rec.setOutputFile("file.mp4"); ...

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Camera camera = Camera.open(); camera.setDisplayOrientation(90); camera.unlock(); MediaRecorder rec = new MediaRecorder(); rec.setCamera(camera); rec.setAudioSource(MediaRecorder.AudioSource.MIC); rec.setVideoSource(MediaRecorder.VideoSource.DEFAULT); rec.setOutputFormat(MediaRecorder.OutputFormat.MPEG_4); rec.setAudioEncoder(1); rec.setVideoEncoder(3); rec.setOutputFile("file.mp4"); ...

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Camera camera = Camera.open(); camera.setDisplayOrientation(90); camera.unlock(); MediaRecorder rec = new MediaRecorder(); rec.setCamera(camera); rec.setAudioSource(MediaRecorder.AudioSource.MIC); rec.setVideoSource(MediaRecorder.VideoSource.DEFAULT); rec.setOutputFormat(MediaRecorder.OutputFormat.MPEG_4); rec.setAudioEncoder(1); rec.setVideoEncoder(3); rec.setOutputFile("file.mp4"); ...

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Camera camera = Camera.open(); camera.setDisplayOrientation(90); camera.unlock(); MediaRecorder rec = new MediaRecorder(); rec.setCamera(camera); rec.setAudioSource(MediaRecorder.AudioSource.MIC); rec.setVideoSource(MediaRecorder.VideoSource.DEFAULT); rec.setOutputFormat(MediaRecorder.OutputFormat.MPEG_4); rec.setAudioEncoder(1); rec.setVideoEncoder(3); rec.setOutputFile("file.mp4"); ...

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MediaRecorder rec = new MediaRecorder(); rec.setCamera(camera);

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MediaRecorder rec = new MediaRecorder(); rec.setCamera(camera);

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From Programs to Sentences

she = new X(); me = new Y(); me.sleep(); if (random()) { me.eat(); } she.enter(); me.talk(she);

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From Programs to Sentences

she = new X(); me = new Y(); me.sleep(); if (random()) { me.eat(); } she.enter(); me.talk(she);

Typestate analysis

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From Programs to Sentences

she = new X(); me = new Y(); me.sleep(); if (random()) { me.eat(); } she.enter(); me.talk(she);

Typestate analysis Alias analysis

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From Programs to Sentences

she = new X(); me = new Y(); me.sleep(); if (random()) { me.eat(); } she.enter(); me.talk(she);

Typestate analysis Alias analysis

me:

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From Programs to Sentences

she = new X(); me = new Y(); me.sleep(); if (random()) { me.eat(); } she.enter(); me.talk(she);

Typestate analysis Alias analysis

me:

Yinit sleep talk

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From Programs to Sentences

she = new X(); me = new Y(); me.sleep(); if (random()) { me.eat(); } she.enter(); me.talk(she);

Typestate analysis Alias analysis

me: Yinit sleep talk

Yinit sleep eat talk

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From Programs to Sentences

she = new X(); me = new Y(); me.sleep(); if (random()) { me.eat(); } she.enter(); me.talk(she);

Typestate analysis Alias analysis

me: Yinit sleep talk

Yinit sleep eat talk

she:

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From Programs to Sentences

she = new X(); me = new Y(); me.sleep(); if (random()) { me.eat(); } she.enter(); me.talk(she);

Typestate analysis Alias analysis

me: Yinit sleep talk

Yinit sleep eat talk

she: Xinit enter talkparam1

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Learn Regularities

⇔

Xinit enter talkparam1 talkparam1 enter

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Statistical Language Models

s = w1 w2 w3 … wn

P( w1 w2 w3 … wn ) P( w1 w2 w3 … wn ) = ∏i=1..n P( wi | w1 … wi-1 )

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Statistical Language Models

s = w1 w2 w3 … wn

P( w1 w2 w3 … wn ) P( w1 w2 w3 … wn ) = ∏i=1..n P( wi | w1 … wi-1 ) P( The quick brown fox jumped ) = P( The ) P( quick | The ) P( brown | The quick ) P( fox | The quick brown ) P( jumped | The quick brown fox )

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N-gram language model

n-1 P( wi | w1 … wi-1 ) ≈ P( wi | wi-n+1 … wi-1 )

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N-gram language model

n-1 P( wi | w1 … wi-1 ) ≈ P( wi | wi-n+1 … wi-1 )

n-1 words

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N-gram language model

n-1 P( wi | w1 … wi-1 ) ≈ P( wi | wi-n+1 … wi-1 )

n-1 words P( jumped | The quick brown fox ) ≈ P( jumped | brown fox ) ≈ #( brown fox jumped ) #( brown fox ) #( n-gram )

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Tri-gram language model

3-grams # of occurrences

brown fox jumped

125

brown fox walked

45

brown fox snapped

30

P (jumped | brown fox) ~= 125/200 ~= 0.625 P (brown fox jumped) ~= P (brown)  P (fox | brown)  P (jumped | brown fox) 200 / 600  200 / 200  125 / 200 ~= 0.208

P (w1  w2  w3  … wn) ~= P (w1)  P (w2  w1)  P (w3  w1  w2)  …  P (wn  wn-2  wn-1)

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Key Problem: Sparsity of Data

#( brown fox jumped ) #( brown fox ) P( jumped | brown fox ) ≈

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Smoothing: Intuitively

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P( wi | wi-N+1) =

i-1

P( wi | wi-N+1)

i-1

B(wi-N+1)  P( wi | wi-N+2)

i-1

wi-N+1

i

i-1

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Training: Witten-Bell Interpolated

Pwbi (wi | wi-N+1) =

i-1

Fwbi (wi-N+1)

i

Bwbi (wi-N+1)  Pwbi( wi | wi-N+2)

i-1 i-1

+

Fwbi (wi-N+1) =

i

 PML( wi | wi-N+1)

i-1

N(ws) = | {w : # (

s

w ) > 0} |

i-N+1

i-1

w

N( )

i-N+1

i-1

w

N( )

+

i-N+1

i-1

#(w ) w

s

w

(1 - Bwbi (wi-N+1))

i-1

Bwbi (wi-N+1) =

i-1

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wi-K+1

i

Pwbi (wi | wi-N+1)

i-1

Bwbi (wi-K+1)

i-1

Pwbi (wi | wi-K+1)

i-1

wi-N+1

i

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N-gram language model

n-1 P( wi | w1 … wi-1 ) ≈ P( wi | wi-n+1 … wi-1 )

n-1 words

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Recurrent Neural Networks (RNN)

P( )

U V W

hidden layer input word

  • utput words
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Recurrent Neural Networks (RNN)

U V W

P( )

hidden layer input word

  • utput words

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Recurrent Neural Networks (RNN)

U V W

P( )

hidden layer input word

  • utput words

 

http://rnnlm.org/

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Code Completion

smsMgr = SmsManager.getDefault(); int length = message.length(); if (length > MAX_SMS_MESSAGE_LENGTH) { list = smsMgr.divideMessage(message); ? {smsMgr, list} // (Hole H1) } else { ? {smsMgr, message} // (Hole H2) }

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Code Completion

smsMgr = SmsManager.getDefault(); int length = message.length(); if (length > MAX_SMS_MESSAGE_LENGTH) { list = smsMgr.divideMessage(message); ? {smsMgr, list} // (Hole H1) } else { ? {smsMgr, message} // (Hole H2) } smsMgr: getDefaultresult divideMessage H1

getDefaultresult H2

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Code Completion

smsMgr = SmsManager.getDefault(); int length = message.length(); if (length > MAX_SMS_MESSAGE_LENGTH) { list = smsMgr.divideMessage(message); ? {smsMgr, list} // (Hole H1) } else { ? {smsMgr, message} // (Hole H2) } smsMgr: getDefaultresult divideMessage H1

getDefaultresult H2

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Code Completion

smsMgr = SmsManager.getDefault(); int length = message.length(); if (length > MAX_SMS_MESSAGE_LENGTH) { list = smsMgr.divideMessage(message); ? {smsMgr, list} // (Hole H1) } else { ? {smsMgr, message} // (Hole H2) } list: divideMessageresult H1 smsMgr: getDefaultresult divideMessage H1

getDefaultresult H2

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Code Completion

smsMgr = SmsManager.getDefault(); int length = message.length(); if (length > MAX_SMS_MESSAGE_LENGTH) { list = smsMgr.divideMessage(message); ? {smsMgr, list} // (Hole H1) } else { ? {smsMgr, message} // (Hole H2) } list: divideMessageresult H1 smsMgr: getDefaultresult divideMessage H1

getDefaultresult H2

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Code Completion

smsMgr = SmsManager.getDefault(); int length = message.length(); if (length > MAX_SMS_MESSAGE_LENGTH) { list = smsMgr.divideMessage(message); ? {smsMgr, list} // (Hole H1) } else { ? {smsMgr, message} // (Hole H2) } list: divideMessageresult H1 smsMgr: getDefaultresult divideMessage H1

getDefaultresult H2

message: length divideMessageparam1 length H2

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Code Completion

smsMgr = SmsManager.getDefault(); int length = message.length(); if (length > MAX_SMS_MESSAGE_LENGTH) { list = smsMgr.divideMessage(message); ? {smsMgr, list} // (Hole H1) } else { ? {smsMgr, message} // (Hole H2) } list: divideMessageresult H1 smsMgr: getDefaultresult divideMessage H1

getDefaultresult H2

message: length divideMessageparam1 length H2

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getDefaultresult divideMessage H1 getDefaultresult H2 divideMessageresult H1 length H2 13

Code Completion

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getDefaultresult divideMessage sendMultipartTextMessage 0.0033 getDefaultresult divideMessage sendTextMessage 0.0016 getDefaultresult sendTextMessage 0.0073 getDefaultresult sendMultipartTextMessage 0.0010 divideMessageresult sendMultipartTextMessageparam3 0.0821 length length 0.0132 length split 0.0080 length sendTextMessageparam3 0.0017

Code Completion

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getDefaultresult divideMessage sendMultipartTextMessage 0.0033 getDefaultresult divideMessage sendTextMessage 0.0016 getDefaultresult sendTextMessage 0.0073 getDefaultresult sendMultipartTextMessage 0.0010 divideMessageresult sendMultipartTextMessageparam3 0.0821 length length 0.0132 length split 0.0080 length sendTextMessageparam3 0.0017

Code Completion

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getDefaultresult divideMessage sendMultipartTextMessage 0.0033 getDefaultresult divideMessage sendTextMessage 0.0016 getDefaultresult sendTextMessage 0.0073 getDefaultresult sendMultipartTextMessage 0.0010 divideMessageresult sendMultipartTextMessageparam3 0.0821 length length 0.0132 length split 0.0080 length sendTextMessageparam3 0.0017

Code Completion

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getDefaultresult divideMessage sendMultipartTextMessage 0.0033 getDefaultresult divideMessage sendTextMessage 0.0016 getDefaultresult sendTextMessage 0.0073 getDefaultresult sendMultipartTextMessage 0.0010 divideMessageresult sendMultipartTextMessageparam3 0.0821 length length 0.0132 length split 0.0080 length sendTextMessageparam3 0.0017

Code Completion

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getDefaultresult divideMessage sendMultipartTextMessage 0.0033 getDefaultresult divideMessage sendTextMessage 0.0016 getDefaultresult sendTextMessage 0.0073 getDefaultresult sendMultipartTextMessage 0.0010 divideMessageresult sendMultipartTextMessageparam3 0.0821 length length 0.0132 length split 0.0080 length sendTextMessageparam3 0.0017

Code Completion

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smsMgr = SmsManager.getDefault(); int length = message.length(); if (length > MAX_SMS_MESSAGE_LENGTH) { list = smsMgr.divideMessage(message); smsMgr.sendMultipartTextMessage(...list...); } else { smsMgr.sendTextMessage(...message...); }

Code Completion

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∈

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∈

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∈

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function f(a) { var b = document.getElementById(a); return b; }

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function f(a) { var b = document.getElementById(a); return b; } getElementById f document a b

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function f(a) { var b = document.getElementById(a); return b; } getElementById f document a b

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Facts to be predicted are dependent Millions of candidate choices Must quickly learn from huge codebases Prediction should be fast (real time)

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Factors

  • A factor (or potential)  is a function from a set of

random variables D to a real number R  : D  R

  • The set of variables D is the scope of the factor 

– we are typically concerned with non-negative factors

  • Intuition: captures affinity, agreement, compatibility
  • f the variables in D
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Factors: Example

A B value 30 1 5 1 1 1 1 10

  • –
  • –

1 (0,0) = 30 says we believe Alice and Bob agree on 0 with belief 30 – 3 (C,D) says that Ceco and Dobri argue all the time 

1 (A,B)

B C value 100 1 1 1 1 1 1 100

2 (B,C)

C D value 1 1 100 1 100 1 1 1

3 (C,D)

D A value 100 1 1 1 1 1 1 100

4 (D,A)

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Factors: Example

A B value 30 1 5 1 1 1 1 10

  • –
  • –

1 (0,0) = 30 says we believe Alice and Bob agree on 0 with belief 30 – 3 (C,D) says that Ceco and Dobri argue all the time 

1 (A,B)

B C value 100 1 1 1 1 1 1 100

2 (B,C)

C D value 1 1 100 1 100 1 1 1

3 (C,D)

D A value 100 1 1 1 1 1 1 100

4 (D,A)

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Defining a Global Probabilistic Model

P(A, B, C, D) = F(A,B,C,D) / Z(A,B,C,D)

F(A, B, C, D) = 1 (A,B)  2 (B,C)  3 (C,D)  4 (D,A) Z(A,B,C,D) =  F(a, b, c, d)

a  A, b  B c  C, d  D

Partition function Joint probability distribution

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P(A,B,C,D)

A B C D Unnormalized (F) Normalized (P = F/Z) 300,000 0.04 1 300,000 0.04 1 300,000 0.04 1 1 30 4.1  10-6 … … … … … …

Z(A,B,C,D) = 7,201,840

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SLIDE 87

P(A,B,C,D)

A B C D Unnormalized (F) Normalized (P = F/Z) 300,000 0.04 1 300,000 0.04 1 300,000 0.04 1 1 30 4.1  10-6 … … … … … …

Z(A,B,C,D) = 7,201,840 P(A, B, C, D): P(B = 1) =  P(A, 1, C, D) ~= 0.732

a  A c  C d  D

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( ) 1 (

)  2 ( )  …  m ( ) ( ) ( )  1 ( )  2 ( )  …  m ( )

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( ) 1 (

)  2 ( )  …  m ( ) ( ) ( )  1 ( )  2 ( )  …  m ( )

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Graphs vs. Probability Distributions

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1 ( ) , … , m ( )

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1 ( ) , … , m ( )

( )

1 ( )  2 ( )  3 ( )  4 ( ) / Z

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1 ( ) , … , m ( )

( )

1 ( )  2 ( )  3 ( )  4 ( ) / Z

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1 ( ) , … , m ( )

( )

1 ( )  2 ( )  3 ( )  4 ( ) / Z

( )

1 ( )  2 ( ) / Z

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1 ( ) , … , m ( )

( )

1 ( )  2 ( )  3 ( )  4 ( ) / Z

( )

1 ( )  2 ( ) / Z

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1 ( ) , … , m ( )

( ) 1 (

)  2 ( ) / Z

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1 ( ) , … , m ( )

( ) 1 (

)  2 ( ) / Z

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1 ( ) , … , m ( )

( ) 1 (

)  2 ( ) / Z

( ) 1 (

) / Z

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1 ( ) , … , m ( )

( ) 1 (

)  2 ( ) / Z

( ) 1 (

) / Z

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1 ( ) , … , m ( )

( ) 1 (

)  2 ( )  3 ( )  4 ( ) / Z

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SLIDE 101

1 ( ) , … , m ( )

( ) 1 (

)  2 ( )  3 ( )  4 ( ) / Z

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Graphs vs. Probabilities

There is an important connection between a probability distribution that factorizes over a graph and the properties of the graph. In particular, we can discover various independence properties of the probabilistic distribution directly from the graph

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P(A, B | S) = P(A | S)  P(B | S)

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P(A, B | S) = P(A | S)  P(B | S)

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 

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I( )

I( ) I( )  I( )

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I( )

I( ) I( )  I( ) I( )  I( )

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(0, 0, 0, 0) = 1/8 (1, 0, 0, 0) = 1/8 (1, 1, 0, 0) = 1/8 (1, 1, 1, 0) = 1/8 (0, 0, 0, 1) = 1/8 (0, 0, 1, 1) = 1/8 (0, 1, 1, 1) = 1/8 (1, 1, 1, 1) = 1/8

I( ):

 

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SLIDE 110

(0, 0, 0, 0) = 1/8 (1, 0, 0, 0) = 1/8 (1, 1, 0, 0) = 1/8 (1, 1, 1, 0) = 1/8 (0, 0, 0, 1) = 1/8 (0, 0, 1, 1) = 1/8 (0, 1, 1, 1) = 1/8 (1, 1, 1, 1) = 1/8

I( )  I( ),

I( ):

 

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SLIDE 111

( ) (

) : ( )  ( )

( ) ( ) ( )  ( )  ( )  ( )  ( )  ( )  ( )  ( )

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SLIDE 112

( ):

( ):

( ):

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SLIDE 113

Graphical Models: So far

So far we learned what a Markov Network is, what it means for a probability distribution to factor over a graph and how to extract information about independence of variables from the graph. We next look at two equivalent representation of Markov Networks, one of which is amenable to inference queries (factor graphs) and another which is amenable to learning from data (e.g., log-linear form).

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SLIDE 114

𝑔1(D1) 𝑔𝑙(D𝑙) D𝑗 𝑔𝑗(D𝑗)

( )  𝑓𝑦𝑞−

𝑥𝑗  𝑔𝑗(D𝑗)

𝑙 𝑗 =1

( ) = 1 ( ) 𝑓𝑦𝑞−

𝑥𝑗  𝑔𝑗(D𝑗)

𝑙 𝑗 =1

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SLIDE 115

i (D𝑗) 𝑓𝑦𝑞 −𝑥𝑗  𝑔𝑗(D𝑗) 𝑓𝑦𝑞−

𝑥𝑗  𝑔𝑗(D𝑗)

𝑙 𝑗 =1

= 𝑓𝑦𝑞−𝑥1  𝑔1 D1  𝑓𝑦𝑞−𝑥2  𝑔2 D2 …  𝑓𝑦𝑞(−𝑥𝑙  𝑔𝑙 D𝑙 = 𝑓𝑦𝑞(−𝑥1  𝑔1 D1 − 𝑥2  𝑔2 D2 … − 𝑥𝑙  𝑔𝑙(D𝑙)) = 1 (D1)  2 D2 …  k (D𝑙)

( ) 1 (D1)  2 (D2)  …  k (Dk)

( )

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SLIDE 116

( )  𝑓𝑦𝑞−

𝑥𝑗  𝑔𝑗(D𝑗)

𝑙 𝑗 =1

( ) = 1 ( ) 𝑓𝑦𝑞−

𝑥𝑗  𝑔𝑗(D𝑗)

𝑙 𝑗 =1

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SLIDE 117

Feature Function: Example

1 : 1 ( , ) = 148.41 1  𝑔A = B A, B 𝑔1 A, B = 1 A = B and 𝑥1 =

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SLIDE 118

Factors vs. Features + Weights

A B value 30 1 5 1 1 1 1 10

1 (A,B)

B C value 100 1 1 1 1 1 1 100

2 (B,C)

C D value 1 1 100 1 100 1 1 1

3 (C,D)

D A value 100 1 1 1 1 1 1 100

4 (D,A)

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SLIDE 119

Factors vs. Features + Weights

A B value 30 1 5 1 1 1 1 10

1 (A,B)

i (D𝑗) 𝑓𝑦𝑞 −𝑥𝑗  𝑔𝑗(D𝑗)

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SLIDE 120

Factors vs. Features + Weights

A B value 30 1 5 1 1 1 1 10

1 (A,B)

A B value

  • 3.4

1

  • 1.61

1 1 1

  • 2.3

𝑥1  𝑔1(A, B)

𝑥1 = 1 𝑔1 A, B = value

i (D𝑗) 𝑓𝑦𝑞 −𝑥𝑗  𝑔𝑗(D𝑗)

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SLIDE 121

Factors vs. Features + Weights

A B value 30 1 5 1 1 1 1 10

1 (A,B)

B C value 100 1 1 1 1 1 1 100

2 (B,C)

A B value

  • 3.4

1

  • 1.61

1 1 1

  • 2.3

𝑥1  𝑔1(A, B)

𝑥1 = 1 𝑔1 A, B = value

i (D𝑗) 𝑓𝑦𝑞 −𝑥𝑗  𝑔𝑗(D𝑗)

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SLIDE 122

Factors vs. Features + Weights

A B value 30 1 5 1 1 1 1 10

1 (A,B)

B C value 100 1 1 1 1 1 1 100

2 (B,C)

A B value

  • 3.4

1

  • 1.61

1 1 1

  • 2.3

𝑥1  𝑔1(A, B)

B C value

  • 4.61

1 1 1 1

  • 4.61

𝑥2  𝑔2(B, C)

𝑥1 = 1 𝑔1 A, B = value 𝑥2 = −4.61 𝑔2 B, C = 1 if B = C 0 else

i (D𝑗) 𝑓𝑦𝑞 −𝑥𝑗  𝑔𝑗(D𝑗)

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SLIDE 123

Factors vs. Features + Weights

A B value 30 1 5 1 1 1 1 10

1 (A,B)

B C value 100 1 1 1 1 1 1 100

2 (B,C)

C D value 1 1 100 1 100 1 1 1

3 (C,D)

A B value

  • 3.4

1

  • 1.61

1 1 1

  • 2.3

𝑥1  𝑔1(A, B)

B C value

  • 4.61

1 1 1 1

  • 4.61

𝑥2  𝑔2(B, C)

𝑥1 = 1 𝑔1 A, B = value 𝑥2 = −4.61 𝑔2 B, C = 1 if B = C 0 else

i (D𝑗) 𝑓𝑦𝑞 −𝑥𝑗  𝑔𝑗(D𝑗)

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SLIDE 124

Factors vs. Features + Weights

A B value 30 1 5 1 1 1 1 10

1 (A,B)

B C value 100 1 1 1 1 1 1 100

2 (B,C)

C D value 1 1 100 1 100 1 1 1

3 (C,D)

A B value

  • 3.4

1

  • 1.61

1 1 1

  • 2.3

𝑥1  𝑔1(A, B)

B C value

  • 4.61

1 1 1 1

  • 4.61

C D value 1

  • 4.61

1

  • 4.61

1 1

𝑥2  𝑔2(B, C) 𝑥3  𝑔3(C, D)

𝑥1 = 1 𝑔1 A, B = value 𝑥2 = −4.61 𝑔2 B, C = 1 if B = C 0 else 𝑥3 = −4.61 𝑔3 C, D = 1 if C  D 0 else

i (D𝑗) 𝑓𝑦𝑞 −𝑥𝑗  𝑔𝑗(D𝑗)

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SLIDE 125

Factors vs. Features + Weights

A B value 30 1 5 1 1 1 1 10

1 (A,B)

B C value 100 1 1 1 1 1 1 100

2 (B,C)

C D value 1 1 100 1 100 1 1 1

3 (C,D)

D A value 100 1 1 1 1 1 1 100

4 (D,A)

A B value

  • 3.4

1

  • 1.61

1 1 1

  • 2.3

𝑥1  𝑔1(A, B)

B C value

  • 4.61

1 1 1 1

  • 4.61

C D value 1

  • 4.61

1

  • 4.61

1 1

𝑥2  𝑔2(B, C) 𝑥3  𝑔3(C, D)

𝑥1 = 1 𝑔1 A, B = value 𝑥2 = −4.61 𝑔2 B, C = 1 if B = C 0 else 𝑥3 = −4.61 𝑔3 C, D = 1 if C  D 0 else

i (D𝑗) 𝑓𝑦𝑞 −𝑥𝑗  𝑔𝑗(D𝑗)

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SLIDE 126

Factors vs. Features + Weights

A B value 30 1 5 1 1 1 1 10

1 (A,B)

B C value 100 1 1 1 1 1 1 100

2 (B,C)

C D value 1 1 100 1 100 1 1 1

3 (C,D)

D A value 100 1 1 1 1 1 1 100

4 (D,A)

A B value

  • 3.4

1

  • 1.61

1 1 1

  • 2.3

𝑥1  𝑔1(A, B)

B C value

  • 4.61

1 1 1 1

  • 4.61

C D value 1

  • 4.61

1

  • 4.61

1 1 D A value

  • 4.61

1 1 1 1

  • 4.61

𝑥2  𝑔2(B, C) 𝑥3  𝑔3(C, D) 𝑥4  𝑔4(D, A)

𝑥1 = 1 𝑔1 A, B = value 𝑥2 = −4.61 𝑔2 B, C = 1 if B = C 0 else 𝑥3 = −4.61 𝑔3 C, D = 1 if C  D 0 else 𝑥4 = −4.61 𝑔4 D, A = 1 if D = A 0 else

i (D𝑗) 𝑓𝑦𝑞 −𝑥𝑗  𝑔𝑗(D𝑗)

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SLIDE 127

Indicator Feature Function

D𝑗

𝑔ab A, B

𝑔 A, B

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SLIDE 128

Example: Using Indicator Function

A B value

  • 3.4

1

  • 1.61

1 1 1

  • 2.3

𝑥1  𝑔1(A, B)

B C value

  • 4.61

1 1 1 1

  • 4.61

C D value 1

  • 4.61

1

  • 4.61

1 1 D A value

  • 4.61

1 1 1 1

  • 4.61

𝑥2  𝑔2(B, C) 𝑥3  𝑔3(C, D) 𝑥4  𝑔4(D, A)

𝑥1 = 1 𝑔1 A, B = value 𝑥2 = −4.61 𝑔2 B, C = 1 if B = C 0 else 𝑥3 = −4.61 𝑔3 C, D = 1 if C  D 0 else 𝑥4 = −4.61 𝑔4 D, A = 1 if D = A 0 else

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SLIDE 129

Example: Using Indicator Function

A B value

  • 3.4

1

  • 1.61

1 1 1

  • 2.3

𝑥1  𝑔1(A, B)

𝑥1 = 1 𝑔1 A, B = value

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SLIDE 130

Example: Using Indicator Function

A B value

  • 3.4

1

  • 1.61

1 1 1

  • 2.3

𝑥1  𝑔1(A, B)

𝑥1 = 1 𝑔1 A, B = value 𝑔1

00 A, B = 1 if

A = 0, B = 0 𝑔1

01 A, B = 1 if

A = 0, B = 1 𝑔1

10 A, B = 1 if

A = 1, B = 0 𝑔1

11 A, B = 1 if

A = 1, B = 1

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SLIDE 131

Example: Using Indicator Function

A B value

  • 3.4

1

  • 1.61

1 1 1

  • 2.3

𝑥1  𝑔1(A, B)

𝑥1 = 1 𝑔1 A, B = value 𝑔1

00 A, B = 1 if

A = 0, B = 0 𝑔1

01 A, B = 1 if

A = 0, B = 1 𝑔1

10 A, B = 1 if

A = 1, B = 0 𝑔1

11 A, B = 1 if

A = 1, B = 1 𝑔1 A, B =

  • 3.4  𝑔1

00 A, B +

  • 1.61 𝑔1

01 A, B +

1 𝑔1

10 A, B +

  • 2.3  𝑔1

11 A, B

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SLIDE 132

Example: Using Indicator Function

A B value

  • 3.4

1

  • 1.61

1 1 1

  • 2.3

𝑥1  𝑔1(A, B)

𝑥1 = 1 𝑔1 A, B = value 𝑔1 A, B =

  • 3.4  𝑔1

00 A, B +

  • 1.61 𝑔1

01 A, B +

1 𝑔1

10 A, B +

  • 2.3  𝑔1

11 A, B

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SLIDE 133

Example: Using Indicator Function

A B value

  • 3.4

1

  • 1.61

1 1 1

  • 2.3

𝑥1  𝑔1(A, B)

B C value

  • 4.61

1 1 1 1

  • 4.61

𝑥2  𝑔2(B, C)

𝑥1 = 1 𝑔1 A, B = value 𝑥2 = −4.61 𝑔2 B, C = 1 if B = C 0 else 𝑔1 A, B =

  • 3.4  𝑔1

00 A, B +

  • 1.61 𝑔1

01 A, B +

1 𝑔1

10 A, B +

  • 2.3  𝑔1

11 A, B

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SLIDE 134

Example: Using Indicator Function

A B value

  • 3.4

1

  • 1.61

1 1 1

  • 2.3

𝑥1  𝑔1(A, B)

B C value

  • 4.61

1 1 1 1

  • 4.61

𝑥2  𝑔2(B, C)

𝑥1 = 1 𝑔1 A, B = value 𝑥2 = −4.61 𝑔2 B, C = 1 if B = C 0 else 𝑔2

00 B, C = 1 if

B = 0, C = 0 𝑔2

01 B, C = 1 if

B = 0, C = 1 𝑔2

10 B, C = 1 if

B = 1, C = 0 𝑔2

11 B, C = 1 if

B = 1, C = 1 𝑔1 A, B =

  • 3.4  𝑔1

00 A, B +

  • 1.61 𝑔1

01 A, B +

1 𝑔1

10 A, B +

  • 2.3  𝑔1

11 A, B

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SLIDE 135

Example: Using Indicator Function

A B value

  • 3.4

1

  • 1.61

1 1 1

  • 2.3

𝑥1  𝑔1(A, B)

B C value

  • 4.61

1 1 1 1

  • 4.61

𝑥2  𝑔2(B, C)

𝑥1 = 1 𝑔1 A, B = value 𝑥2 = −4.61 𝑔2 B, C = 1 if B = C 0 else 𝑔2

00 B, C = 1 if

B = 0, C = 0 𝑔2

01 B, C = 1 if

B = 0, C = 1 𝑔2

10 B, C = 1 if

B = 1, C = 0 𝑔2

11 B, C = 1 if

B = 1, C = 1 𝑔2 B, C = 𝑔2

00 B, C + 𝑔2 11 B, C

𝑔1 A, B =

  • 3.4  𝑔1

00 A, B +

  • 1.61 𝑔1

01 A, B +

1 𝑔1

10 A, B +

  • 2.3  𝑔1

11 A, B

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SLIDE 136

Example: Using Indicator Function

A B value

  • 3.4

1

  • 1.61

1 1 1

  • 2.3

𝑥1  𝑔1(A, B)

B C value

  • 4.61

1 1 1 1

  • 4.61

𝑥2  𝑔2(B, C)

𝑥1 = 1 𝑔1 A, B = value 𝑥2 = −4.61 𝑔2 B, C = 1 if B = C 0 else 𝑔2 B, C = 𝑔2

00 B, C + 𝑔2 11 B, C

𝑔1 A, B =

  • 3.4  𝑔1

00 A, B +

  • 1.61 𝑔1

01 A, B +

1 𝑔1

10 A, B +

  • 2.3  𝑔1

11 A, B

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SLIDE 137

Example: Using Indicator Function

A B value

  • 3.4

1

  • 1.61

1 1 1

  • 2.3

𝑥1  𝑔1(A, B)

B C value

  • 4.61

1 1 1 1

  • 4.61

𝑥2  𝑔2(B, C)

𝑥1 = 1 𝑔1 A, B = value 𝑥2 = −4.61 𝑔2 B, C = 1 if B = C 0 else 𝑔2 B, C = 𝑔2

00 B, C + 𝑔2 11 B, C

𝑔1 A, B =

  • 3.4  𝑔1

00 A, B +

  • 1.61 𝑔1

01 A, B +

1 𝑔1

10 A, B +

  • 2.3  𝑔1

11 A, B

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SLIDE 138

Example: Using Indicator Function

A B value

  • 3.4

1

  • 1.61

1 1 1

  • 2.3

𝑥1  𝑔1(A, B)

B C value

  • 4.61

1 1 1 1

  • 4.61

C D value 1

  • 4.61

1

  • 4.61

1 1 D A value

  • 4.61

1 1 1 1

  • 4.61

𝑥2  𝑔2(B, C) 𝑥3  𝑔3(C, D) 𝑥4  𝑔4(D, A)

𝑥1 = 1 𝑔1 A, B = value 𝑥2 = −4.61 𝑔2 B, C = 1 if B = C 0 else 𝑥3 = −4.61 𝑔3 C, D = 1 if C  D 0 else 𝑥4 = −4.61 𝑔4 D, A = 1 if D = A 0 else 𝑔3

00 C, D = 1 if

C = 0, D = 0 𝑔3

01 C, D = 1 if

C = 0, D = 1 𝑔3

10 C, D = 1 if

C = 1, D = 0 𝑔3

11 C, D = 1 if

C = 1, D = 1 𝑔4

00 D, A = 1 if

D = 0, A = 0 𝑔4

01 C, D = 1 if

D = 0, A = 1 𝑔4

10 C, D = 1 if

D = 1, A = 0 𝑔4

11 C, D = 1 if

D = 1, A = 1 𝑔2 B, C = 𝑔2

00 B, C + 𝑔2 11 B, C

𝑔1 A, B =

  • 3.4  𝑔1

00 A, B +

  • 1.61 𝑔1

01 A, B +

1 𝑔1

10 A, B +

  • 2.3  𝑔1

11 A, B

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SLIDE 139

Example: Using Indicator Function

A B value

  • 3.4

1

  • 1.61

1 1 1

  • 2.3

𝑥1  𝑔1(A, B)

B C value

  • 4.61

1 1 1 1

  • 4.61

C D value 1

  • 4.61

1

  • 4.61

1 1 D A value

  • 4.61

1 1 1 1

  • 4.61

𝑥2  𝑔2(B, C) 𝑥3  𝑔3(C, D) 𝑥4  𝑔4(D, A)

𝑥1 = 1 𝑔1 A, B = value 𝑥2 = −4.61 𝑔2 B, C = 1 if B = C 0 else 𝑥3 = −4.61 𝑔3 C, D = 1 if C  D 0 else 𝑥4 = −4.61 𝑔4 D, A = 1 if D = A 0 else 𝑔3 C, D = 𝑔3

01 C, D + 𝑔3 10 C, D

𝑔4 D, A = 𝑔4

00 D, A + 𝑔4 11 D, A )

𝑔2 B, C = 𝑔2

00 B, C + 𝑔2 11 B, C

𝑔1 A, B =

  • 3.4  𝑔1

00 A, B +

  • 1.61 𝑔1

01 A, B +

1 𝑔1

10 A, B +

  • 2.3  𝑔1

11 A, B

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SLIDE 140

Example: Using Indicator Function

A B value

  • 3.4

1

  • 1.61

1 1 1

  • 2.3

𝑥1  𝑔1(A, B)

B C value

  • 4.61

1 1 1 1

  • 4.61

C D value 1

  • 4.61

1

  • 4.61

1 1 D A value

  • 4.61

1 1 1 1

  • 4.61

𝑥2  𝑔2(B, C) 𝑥3  𝑔3(C, D) 𝑥4  𝑔4(D, A)

𝑥1 = 1 𝑔1 A, B = value 𝑥2 = −4.61 𝑔2 B, C = 1 if B = C 0 else 𝑥3 = −4.61 𝑔3 C, D = 1 if C  D 0 else 𝑥4 = −4.61 𝑔4 D, A = 1 if D = A 0 else 𝑔3 C, D = 𝑔3

01 C, D + 𝑔3 10 C, D

𝑔4 D, A = 𝑔4

00 D, A + 𝑔4 11 D, A )

𝑔2 B, C = 𝑔2

00 B, C + 𝑔2 11 B, C

( ) = 1 ( ) 𝑓𝑦𝑞−

𝑥𝑗  𝑔𝑗(D𝑗)

𝑙 𝑗 =1 𝑔1 A, B =

  • 3.4  𝑔1

00 A, B +

  • 1.61 𝑔1

01 A, B +

1 𝑔1

10 A, B +

  • 2.3  𝑔1

11 A, B

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SLIDE 141

Example: Using Indicator Function

A B value

  • 3.4

1

  • 1.61

1 1 1

  • 2.3

𝑥1  𝑔1(A, B)

B C value

  • 4.61

1 1 1 1

  • 4.61

C D value 1

  • 4.61

1

  • 4.61

1 1 D A value

  • 4.61

1 1 1 1

  • 4.61

𝑥2  𝑔2(B, C) 𝑥3  𝑔3(C, D) 𝑥4  𝑔4(D, A)

𝑥1 = 1 𝑔1 A, B = value 𝑥2 = −4.61 𝑔2 B, C = 1 if B = C 0 else 𝑥3 = −4.61 𝑔3 C, D = 1 if C  D 0 else 𝑥4 = −4.61 𝑔4 D, A = 1 if D = A 0 else 𝑔1 A, B =

  • 3.4  𝑔1

00 A, B +

  • 1.61 𝑔1

01 A, B +

1 𝑔1

10 A, B +

  • 2.3  𝑔1

11 A, B

𝑔3 C, D = 𝑔3

01 C, D + 𝑔3 10 C, D

𝑔4 D, A = 𝑔4

00 D, A + 𝑔4 11 D, A )

𝑔2 B, C = 𝑔2

00 B, C + 𝑔2 11 B, C

(A, B, C, D) = 𝑓𝑦𝑞−SP Z(A, B, C, D)

SP = −3.4 3.4  𝑔1

00 A, B + −1.6

1.61  𝑔1

01 A, B + 1  𝑔1 10 A, B + −2.

2.3  𝑔1

11 A, B + −4.61  𝑔2 00 B, C +

−4.61  𝑔2

11 B, C + −𝟓. 𝟕𝟐  𝑔3 01 C, D + −𝟓. 𝟕𝟐  𝑔3 10 C, D +−4.61  𝑔4 00 D, A + −4.61  𝑔4 11 D, A )

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SLIDE 142

Graphical Models: So far

In the last segment we learned about log-linear models and indicator functions. Log-linear models are important because they allow to capture factors more concisely, the indicator functions can be directly extracted from data, and the weights can be learned (to fit the

  • ptimization objective, discussed later).

So far we have been discussing the joint probability distribution ( ). We next focus on conditional distributions.

slide-143
SLIDE 143

Conditional Random Field

   

(  ) 1 (

)  2 ( )  …  m ( ) ( ) ( )  1 ( )  2 ( )  …  m ( )

slide-144
SLIDE 144

(  ) 1 (

)  2 ( )  …  m ( ) ( ) ( )  1 ( )  2 ( )  …  m ( )

Conditional Random Field

slide-145
SLIDE 145

Conditional Random Field: Example

slide-146
SLIDE 146

Conditional Random Field: Example

slide-147
SLIDE 147

Conditional Random Field: Example

slide-148
SLIDE 148

Conditional Random Field: Example

slide-149
SLIDE 149

Conditional Random Field: Example

slide-150
SLIDE 150

Conditional Random Field: Example

slide-151
SLIDE 151

Conditional Random Field: Example

1

t (

) 2

t (

)

slide-152
SLIDE 152

Conditional Random Field: Example

1

t (

) 2

t (

)

slide-153
SLIDE 153

Conditional Random Field: Example

(  )

1

1 (

)  …  7

6 (

)  2

1 (

)  …  2

7 (

)  1

1 (

)  …  7

6 (

)  2

1 (

)  …  2

7 (

)

slide-154
SLIDE 154

Conditional Random Field: Example

(  )

1

1 (

)  …  7

6 (

)  2

1 (

)  …  2

7 (

)  1

1 (

)  …  7

6 (

)  2

1 (

)  …  2

7 (

)

slide-155
SLIDE 155

𝑄 𝑧 𝑦 =

1 𝑎(𝑦) exp

(𝒙𝑈𝒈(𝑧, 𝑦))

Conditional Random Field: Log-Linear Form

slide-156
SLIDE 156

i t r 𝑧 𝑗, 𝑠 𝑦 𝑢

𝑄 𝑧 𝑦 =

1 𝑎(𝑦) exp

(𝒙𝑈𝒈(𝑧, 𝑦))

slide-157
SLIDE 157

first town w Pencho Portland 0.1 James Portland 0.3 first last w Pencho Smith 0.7 James Chandra 0.4

𝑔1 𝑔2 𝑔3 𝑔4 first town last 𝑧 𝑦

𝑄 𝑧 𝑦 =

1 𝑎(𝑦) exp

(𝒙𝑈𝒈(𝑧, 𝑦))

slide-158
SLIDE 158

𝑄 first, last | town = exp ( 0.1 ∗ f1 + 0.3 ∗ f2 + 0.7 ∗ f3 + 0.4 ∗ f4) 𝑎(town)

𝑄 𝑧 𝑦 =

1 𝑎(𝑦) exp

(𝒙𝑈𝒈(𝑧, 𝑦))

first town w Pencho Portland 0.1 James Portland 0.3 first last w Pencho Smith 0.7 James Chandra 0.4

𝑔1 𝑔2 𝑔3 𝑔4 first town last 𝑧 𝑦

slide-159
SLIDE 159
slide-160
SLIDE 160

Queries: MAP vs. Max Marginals

(𝑧1, … , 𝑧𝑜)MAP 𝑄(𝑧1, … , 𝑧𝑜)

𝑧1, … , 𝑧𝑜

(𝑧1, … , 𝑧𝑜)ML (𝑧1

ML, … , 𝑧𝑜ML)

𝑧1

ML

𝑄(𝑧1)

𝑧1

𝑧n

ML

𝑄(𝑧𝑜)

𝑧𝑜 ……

A B val 0.2 1 0.3 1 0.15 1 1 0.35

slide-161
SLIDE 161

Queries: MAP vs. Max Marginals

(𝑧1, … , 𝑧𝑜)MAP 𝑄(𝑧1, … , 𝑧𝑜)

𝑧1, … , 𝑧𝑜

(𝑧1, … , 𝑧𝑜)ML (𝑧1

ML, … , 𝑧𝑜ML)

𝑧1

ML

𝑄(𝑧1)

𝑧1

𝑧n

ML

𝑄(𝑧𝑜)

𝑧𝑜 ……

A B val 0.2 1 0.3 1 0.15 1 1 0.35

(1,1) MAP (0,1)

ML

A 𝑄(A) 𝑄(A, B) A, B B 𝑄(B)

slide-162
SLIDE 162

 3 1 2

slide-163
SLIDE 163

0 1 2 3 4 B val 0.99 1 0.01

0 (B)

S B val 0.5 1 0.1 1 0.5 1 1 0.9

1 (S,B)

A B val 0.99 1 0.1 1 0.01 1 1 0.9

2 (A,B)

G S val 0.7 1 0.4 1 0.3 1 1 0.6

3 (G,S)

D S val 0.8 1 0.3 1 0.2 1 1 0.7

4 (D,S)

(B, S  G, D, A) 0 (B)  1 (S, B)  2 (A, B)  3 (G, S)  4 (D, S)

Z(G, D, A) B,S (G, D, A)  0 (B)  1 (S, B)  2 (A, B)  3 (G, S)  4 (D, S) P(B=0 | G = 1, D = 0, A = 1)

slide-164
SLIDE 164

0 1 2 3 4 B val 0.99 1 0.01

0 (B)

S B val 0.5 1 0.1 1 0.5 1 1 0.9

1 (S,B)

A B val 0.99 1 0.1 1 0.01 1 1 0.9

2 (A,B)

G S val 0.7 1 0.4 1 0.3 1 1 0.6

3 (G,S)

D S val 0.8 1 0.3 1 0.2 1 1 0.7

4 (D,S)

P(B=0  G, D, A) =  P(B=0, S  G, D, A) = P(B, S=0  G, D, A) + P(B, S=1  G, D, A)

S

P(B=0  G=1, D=0, A=1) = P(B=0, S = 0G=1, D=0, A=1) + P(B=0, S=1G=1, D=0, A=1) P(B=0 | G = 1, D = 0, A = 1)

slide-165
SLIDE 165

0 1 2 3 4 B val 0.99 1 0.01

0 (B)

S B val 0.5 1 0.1 1 0.5 1 1 0.9

1 (S,B)

A B val 0.99 1 0.1 1 0.01 1 1 0.9

2 (A,B)

G S val 0.7 1 0.4 1 0.3 1 1 0.6

3 (G,S)

D S val 0.8 1 0.3 1 0.2 1 1 0.7

4 (D,S)

(G=1, D=0, A=1)

0 (0)  1 (0, 0)  2 (1, 0)  3 (1, 0)  4 (0, 0) + 0 (1)  1 (0, 1)  2 (1, 1)  3 (1, 0)  4 (0, 0) + 0 (0)  1 (1, 0)  2 (1, 0)  3 (1, 1)  4 (0, 1) + 0 (1)  1 (1, 1)  2 (1, 1)  3 (1, 1)  4 (0, 1) + = 0.99  0.5  0.01  0.3  0.8 + 0.01  0.1  0.9  0.3  0.8 + 0.99  0.5  0.01  0.6  0.3 + 0.01  0.9  0.9  0.6  0.7 = 0.001188 + 0.000216 + 0.000891 + 0.003402 = 0.005697

P(B=0 | G = 1, D = 0, A = 1)

slide-166
SLIDE 166

0 1 2 3 4 B val 0.99 1 0.01

0 (B)

S B val 0.5 1 0.1 1 0.5 1 1 0.9

1 (S,B)

A B val 0.99 1 0.1 1 0.01 1 1 0.9

2 (A,B)

G S val 0.7 1 0.4 1 0.3 1 1 0.6

3 (G,S)

D S val 0.8 1 0.3 1 0.2 1 1 0.7

4 (D,S) P(B=0, S = 0G=1, D=0, A=1)

= 0 (B)  1 (S, B)  2 (A, B)  3 (G, S)  4 (D, S)/Z = = 0 (0)  1 (0, 0)  2 (1, 0)  3 (1, 0)  4 (0, 0)/Z = 0.99  0.5  0.01  0.3  0.8)/Z = 0.001188/0.005697

P(B=0, S = 1G=1, D=0, A=1)

= 0 (B)  1 (S, B)  2 (A, B)  3 (G, S)  4 (D, S)/Z = = 0 (0)  1 (1, 0)  2 (1, 0)  3 (1, 1)  4 (0, 1)/Z = 0.99  0.5  0.01  0.6  0.3)/Z = 0.000891/0.005697

+

0.002079/0.005697 P(B=0 | G = 1, D = 0, A = 1)

slide-167
SLIDE 167
slide-168
SLIDE 168

2 1 3 4

1 x(x) 2 x(x) 3 x(x) x 4(x) x 4(x) = 1 x(x)  2 x(x)  3 x(x) A ( )   X(x)

v : Val  MsgVal   v: Val  (MsgVal, (V  Val)) B ( ) C ( )   X(x) =  (( )  A ( )  B ( )  C ( ) )

slide-169
SLIDE 169

v   v(v) = (v) v  v  (v) = 1

  • X

(X) =    v(v) ( X)

slide-170
SLIDE 170

0 1 2 3 4 B val 0.99 1 0.01

0 (B)

S B val 0.5 1 0.1 1 0.5 1 1 0.9

1 (S,B)

A B val 0.99 1 0.1 1 0.01 1 1 0.9

2 (A,B)

G S val 0.7 1 0.4 1 0.3 1 1 0.6

3 (G,S)

D S val 0.8 1 0.3 1 0.2 1 1 0.7

4 (D,S)

P(B=0 | G = 1, D = 0, A = 1)

slide-171
SLIDE 171

0 1 2 3 4 B val 0.99 1 0.01

0 (B)

S B val 0.5 1 0.1 1 0.5 1 1 0.9

1 (S,B)

A B val 0.99 1 0.1 1 0.01 1 1 0.9

2 (A,B)

G S val 0.7 1 0.4 1 0.3 1 1 0.6

3 (G,S)

D S val 0.8 1 0.3 1 0.2 1 1 0.7

4 (D,S)

1 1

P(B=0 | G = 1, D = 0, A = 1)

slide-172
SLIDE 172

0 1 2 3 4

G 3(G = 0) = 0 G 3(G = 1) = 1 D 4(D = 0) = 1 D 4(D = 1) = 0 A 2(A = 0) = 0 A 2(A = 1) = 1 3 S(S = 0) =  3(G, S = 0)  G 3(G)

G

3 S(S = 1) =  3(G, S = 1) G 3(G)

G

4 S(S = 0) =  4(D, S = 0) D 4(D)

D

4 S(S = 1) =  4(D, S = 1)  D 4(D)

D

2 B(B = 0) =  2(A, B = 0)  A 2(A)

A

2 B(B = 1) =  2(A, B = 1) A 2(A)

A

0 B(B = 0) = 0(B = 0) 0 B(B = 1) = 0(B = 1) 1 B(B = 0) =  1(S, B = 0)  S 1(S)

S

1 B(B = 1) =  1(S, B = 1) S 1(S)

S

S 1(S = 0) = 3 S(S = 0)  4 S(S = 0) S 1(S = 1) = 3 S(S = 1)  4 S(S = 1)

P(B=0 | G = 1, D = 0, A = 1)

1 1

slide-173
SLIDE 173

0 1 2 3 4

P(B = 0 | G=1, D=0, A=1) =0 B(B = 0) 1 B(B = 0)  2 B(B = 0)

= 0(B = 0)  (  1(S, B = 0)  S 1(S) )  (  2(A, B = 0)  A 2(A) ) = 0(B = 0)  (  1(S, B = 0)  (3 S(S)  4 S(S)))  ( 2(A, B = 0)  (A 2(A) ) = 0(B = 0)  (  1(S, B = 0)  (( 3(G, S)  G 3(G))  ( 4(D, S) D 4(D))))

 (  2(A, B = 0)  (A 2(A))

S A S A S A G D

P(B=0 | G = 1, D = 0, A = 1)

1 1

slide-174
SLIDE 174

0 1 2 3 4

P(B = 0 | G=1, D=0, A=1) =

= 0.99  (1(S=0, B = 0)  (3(G=0, S=0)  G 3(G=0) + 3(G=1, S=0)  G 3(G=1))  (4(D=0, S=0) D 4(D=0) + 4(D=1, S=0) D 4(D=1))) + (1(S=1, B = 0)  (3(G=0, S=1)  G 3(G=0) + 3(G=1, S=1)  G 3(G=1))  (4(D=0, S=1) D 4(D=0) + 4(D=1, S=1) D 4(D=1)))  (2(A=0, B = 0)  (A 2(A=0) + 2(A=1, B = 0)  (A 2(A=1))) = 0.99  ((0.5  (0.7  0 + 0.3  1)  (0.8  1 + 0.2  0) + (0.5  (0.4  0 + 0.6  1)  (0.3  1 + 0.7  0))  (0.99  0 + 0.01  1) = 0.99  (0.50.30.8 + 0.50.60.3)  0.01 = 0.002079 B val 0.99 1 0.01

0 (B)

S B val 0.5 1 0.1 1 0.5 1 1 0.9

1 (S,B)

A B val 0.99 1 0.1 1 0.01 1 1 0.9

2 (A,B)

G S val 0.7 1 0.4 1 0.3 1 1 0.6

3 (G,S)

D S val 0.8 1 0.3 1 0.2 1 1 0.7

4 (D,S)

P(B=0 | G = 1, D = 0, A = 1)

1 1

slide-175
SLIDE 175

𝑦 𝑧 𝑧1 𝑧2 𝑧𝑜 𝑄 𝑧 𝑦 𝑧1 𝑧2 𝑧𝑜

𝑄 𝑧 𝑦 =

1 𝑎(𝑦) exp

(𝒙𝑈𝒈(𝑧, 𝑦))

slide-176
SLIDE 176

𝑦 𝑧 𝑧1 𝑧2 𝑧𝑜 𝑄 𝑧 𝑦 𝑧1 𝑧2 𝑧𝑜

𝑄 𝑧 𝑦 =

1 𝑎(𝑦) exp

(𝒙𝑈𝒈(𝑧, 𝑦))

𝑧best 𝑄 𝑧 𝑦 𝑧  𝑦 = 𝒙𝑈𝒈(𝑧, 𝑦) 𝑧  𝑦

slide-177
SLIDE 177

𝑦 𝑧 𝑧1 𝑧2 𝑧𝑜 𝑄 𝑧 𝑦 𝑧1 𝑧2 𝑧𝑜

𝑧best 𝑄 𝑧 𝑦 𝑧  𝑦 = 𝒙𝑈𝒈(𝑧, 𝑦) 𝑧  𝑦

𝑄 𝑧 𝑦 =

1 𝑎(𝑦) exp

(𝒙𝑈𝒈(𝑧, 𝑦))

slide-178
SLIDE 178

𝑧1

best

𝑄 𝑧1 𝑦 𝑧n

best

𝑄 𝑧𝑜 𝑦

…………..

𝑧best = (𝑧1

best, … , 𝑧n

best)

slide-179
SLIDE 179

𝑧1

best

𝑄 𝑧1 𝑦 𝑧n

best

𝑄 𝑧𝑜 𝑦

…………..

𝑧best = (𝑧1

best, … , 𝑧n

best)

slide-180
SLIDE 180

function chunkData(e, t) var n = []; var r = e.length; var i = 0; for (; i < r; i += t) if (i + t < r) n.push(e.substring(i, i + t)); else n.push(e.substring(i, r)); return n;

slide-181
SLIDE 181

function chunkData(e, t) var n = []; var r = e.length; var i = 0; for (; i < r; i += t) if (i + t < r) n.push(e.substring(i, i + t)); else n.push(e.substring(i, r)); return n;

t r i

length

i t r

length

… …

slide-182
SLIDE 182

t r i

length

i t r

length

i t w i step 0.5 j step 0.4 i r w i len 0.6 i length 0.3 r length w length length 0.5 len length 0.3

… … i t r

length function chunkData(e, t) var n = []; var r = e.length; var i = 0; for (; i < r; i += t) if (i + t < r) n.push(e.substring(i, i + t)); else n.push(e.substring(i, r)); return n;

slide-183
SLIDE 183

t r i

length

i t r

length

i t w i step 0.5 j step 0.4 i r w i len 0.6 i length 0.3 r length w length length 0.5 len length 0.3

… … i t r

length

𝒙𝑈𝒈(𝑗, 𝑢, 𝑠, 𝑚𝑓𝑜𝑕𝑢ℎ)

function chunkData(e, t) var n = []; var r = e.length; var i = 0; for (; i < r; i += t) if (i + t < r) n.push(e.substring(i, i + t)); else n.push(e.substring(i, r)); return n;

slide-184
SLIDE 184

t r i

length

i t r

length

i t w i step 0.5 j step 0.4 i r w i len 0.6 i length 0.3 r length w length length 0.5 len length 0.3

… … i t r length i len step

𝒙𝑈𝒈(𝑗, 𝑢, 𝑠, 𝑚𝑓𝑜𝑕𝑢ℎ)

function chunkData(e, t) var n = []; var r = e.length; var i = 0; for (; i < r; i += t) if (i + t < r) n.push(e.substring(i, i + t)); else n.push(e.substring(i, r)); return n;

slide-185
SLIDE 185

t r i

length

i t r

length

i t w i step 0.5 j step 0.4 i r w i len 0.6 i length 0.3 r length w length length 0.5 len length 0.3

… …

function chunkData(str, step) var colNames = []; var len = str.length; var i = 0; for (; i < len; i += step) if (i + step < len) colNames.push(str.substring(i, i + step)); else colNames.push(str.substring(i, len)); return colNames;

i t r length i len step

𝒙𝑈𝒈(𝑗, 𝑢, 𝑠, 𝑚𝑓𝑜𝑕𝑢ℎ)

function chunkData(e, t) var n = []; var r = e.length; var i = 0; for (; i < r; i += t) if (i + t < r) n.push(e.substring(i, i + t)); else n.push(e.substring(i, r)); return n;

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𝒙𝑈

𝑄 𝑧 𝑦 =

1 𝑎(𝑦) exp

(𝒙𝑈𝒈(𝑧, 𝑦))

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SLIDE 188

∀j ∀y Σ 𝑥ifi(x(j),y(j)) ≥ Σ 𝑥ifi(x(j),y) + 𝚬(y,y(j))

𝒙𝑈

𝑄 𝑧 𝑦 =

1 𝑎(𝑦) exp

(𝒙𝑈𝒈(𝑧, 𝑦))

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SLIDE 189

∀j ∀y Σ 𝑥ifi(x(j),y(j)) ≥ Σ 𝑥ifi(x(j),y) + 𝚬(y,y(j))

𝒙𝑈

𝑄 𝑧 𝑦 =

1 𝑎(𝑦) exp

(𝒙𝑈𝒈(𝑧, 𝑦))

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SLIDE 190

∀j ∀y Σ 𝑥ifi(x(j),y(j)) ≥ Σ 𝑥ifi(x(j),y) + 𝚬(y,y(j))

𝒙𝑈

𝑎(𝑦)

𝑄 𝑧 𝑦 =

1 𝑎(𝑦) exp

(𝒙𝑈𝒈(𝑧, 𝑦))

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SLIDE 191
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SLIDE 192
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SLIDE 193
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SLIDE 194

…

var n = []; var r = e.length; var i = 0; for (; i < r; i += t) if (i + t < r) n.push(e.subs(i, i + t)); else n.push(e.subs(i, r)); return n; var colNames = []; var len = str.length; var i = 0; for (; i < len; i += step) if (i + step < len) colNames.push(str.subs(i, i + step)); else colNames.push(str.subs(i, len)); return colNames;

150MB Time: milliseconds

Names: 63% Types: 81% (helps typechecking) 𝑄 𝑧 𝑦

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SLIDE 195

∈