Another Walkthrough of Variational Bayes Bevan Jones ML for NLP - - PowerPoint PPT Presentation

▶
another walkthrough of variational bayes
SMART_READER_LITE
LIVE PREVIEW

Another Walkthrough of Variational Bayes Bevan Jones ML for NLP - - PowerPoint PPT Presentation

Mar 23, 2024 96 likes •1.51k views

Another Walkthrough of Variational Bayes Bevan Jones ML for NLP Reading Group The University of Edinburgh 18 th of October, 2012 2 Variational Bayes? Bayes Bayes Theorem But the integral is intractable! Sampling Gibbs,

slide-1
SLIDE 1

Another Walkthrough of Variational Bayes

Bevan Jones ML for NLP Reading Group The University of Edinburgh

18th of October, 2012

slide-2
SLIDE 2

Variational Bayes?

  • Bayes ← Bayes’ Theorem
  • But the integral is intractable!

– Sampling

  • Gibbs, Metropolis Hastings, Slice Sampling, Particle Filters…

– Variational Bayes

  • Change the equations, replacing intractable integrals
  • This involves searching for a good approximation
  • Variational ← Calculus of Variations

– A way of searching through a space of functions for the “best” one

2

slide-3
SLIDE 3

Useful Concepts

  • Probability/Information Theory

– Bayes’ Theorem – Expectations – Jensen’s Inequality – KL Divergence – Conjugacy?

  • Calculus

– Functionals & Functional Derivatives – Lagrange Multipliers

  • Logarithms

3

slide-4
SLIDE 4

Useful Concepts

  • Probability/Information Theory

– Bayes’ Theorem – Expectations – Jensen’s Inequality – KL Divergence – Conjugacy

  • Calculus

– Functionals & Functional Derivatives – Lagrange Multipliers

  • Logarithms

4

slide-5
SLIDE 5

Outline

  • Part I: Principles of Variational Bayes

– The posterior and it’s approximation – Finding the optimal approximation

  • Defining “optimal”
  • Doing the math

– The Mean Field Assumption – An inference procedure – The lower bound

  • Part II: Dirichlet-multinomial case study

– Intractability – The Mean Field Assumption – Dirichlet-multinomial math

5

slide-6
SLIDE 6
  • We have some observed data:
  • We have a model relating latent variable z to the

data:

  • To guess z
  • The problem is one of computing
  • Or just as good

The (Log) Likelihood

6

slide-7
SLIDE 7

Approximating p(z|x)

  • The integral in the expression for p(x) may not be

easily computed

  • But we might be able to get by with an

approximation for p(x, z)

  • We’ll focus on approximating only part of it

7

slide-8
SLIDE 8

Choosing q

  • How to choose q?
  • Ideally, we want the q that is closest to p
  • Define a lower bound on p

– Make this a “function” of q

  • Maximize the lower bound to make it as tight

as possible

– Choose q accordingly

8

slide-9
SLIDE 9

Choosing q

9

  • Define a lower bound F on p
slide-10
SLIDE 10

Choosing q

10

  • Define a lower bound F on p

– Make this a “function” of q

slide-11
SLIDE 11

Choosing q

11

slide-12
SLIDE 12

Choosing q

12

  • Choose q to maximize the lower bound
slide-13
SLIDE 13
  • Jensen’s Inequality

where f is concave

Bounding the Log Likelihood w/ Jensen’s Inequality

13

slide-14
SLIDE 14
  • Jensen’s Inequality

where f is concave

Bounding the Log Likelihood w/ Jensen’s Inequality

14

slide-15
SLIDE 15
  • Jensen’s Inequality

where f is concave

Bounding the Log Likelihood w/ Jensen’s Inequality

15

slide-16
SLIDE 16
  • Jensen’s Inequality

where f is concave

Bounding the Log Likelihood w/ Jensen’s Inequality

16

slide-17
SLIDE 17
  • Jensen’s Inequality

where f is concave

Bounding the Log Likelihood w/ Jensen’s Inequality

17

Apply Jensen’s

slide-18
SLIDE 18
  • Jensen’s Inequality

where f is concave

Bounding the Log Likelihood w/ Jensen’s Inequality

18

slide-19
SLIDE 19
  • We can’t calculate the log likelihood, but we can

compute the lower bound

  • Maximizing F tightens the lower bound on the

likelihood

  • What q maximizes F?
  • If q were a variable we could do this by taking

derivatives and solving for q

The Lower Bound

19

slide-20
SLIDE 20

Functionals: the “Variational” in VB

  • Functional: a kind of “meta-function” that

takes a function as input

  • F[q] is a functional of q
  • Functionals can be optimized like functions

– take the derivative of F[q] with respect to q, – set the derivative to 0, and – solve for q

20

slide-21
SLIDE 21

Derivatives

21

slide-22
SLIDE 22
  • The change in functional as we change its

function argument

Functional Derivatives

22

slide-23
SLIDE 23

Useful Derivatives

23

slide-24
SLIDE 24

Useful Derivatives

24

slide-25
SLIDE 25

Useful Derivatives

25

slide-26
SLIDE 26

Useful Derivatives

26

slide-27
SLIDE 27

Useful Derivatives

27

slide-28
SLIDE 28

Useful Derivatives

28

slide-29
SLIDE 29

Useful Derivatives

29

slide-30
SLIDE 30

Calculating q

  • Use Lagrange multipliers

constraint

30

  • 1. Find derivative wrt
  • 2. Find derivative wrt
  • 3. Solve for and
slide-31
SLIDE 31

Calculating q

  • Use Lagrange multipliers

constraint

31

slide-32
SLIDE 32

Calculating q

  • Use Lagrange multipliers

constraint

32

slide-33
SLIDE 33

Calculating q

  • Use Lagrange multipliers

constraint

33

slide-34
SLIDE 34

Calculating q

  • Use Lagrange multipliers

constraint

34

slide-35
SLIDE 35

Calculating q

  • Use Lagrange multipliers

constraint

35

slide-36
SLIDE 36

Calculating q

  • Use Lagrange multipliers

constraint

36

slide-37
SLIDE 37

Calculating q

  • Use Lagrange multipliers

constraint

37

slide-38
SLIDE 38

Calculating q

38

slide-39
SLIDE 39

Calculating q

39

slide-40
SLIDE 40

Calculating q

40

slide-41
SLIDE 41

Calculating q

41

slide-42
SLIDE 42

Calculating q

42

slide-43
SLIDE 43

Calculating q

43

slide-44
SLIDE 44
  • Maximizing F is minimizing the KL divergence
  • And

KL Divergence: An Alternative View

44

slide-45
SLIDE 45
  • Maximizing F is minimizing the KL divergence

KL Divergence: An Alternative View

45

slide-46
SLIDE 46
  • Maximizing F is minimizing the KL divergence

KL Divergence: An Alternative View

46

slide-47
SLIDE 47

Optimal q

48

slide-48
SLIDE 48

Where are we?

  • We’ve bounded the likelihood (Jensen’s Ineq.)
  • Made this bound tight (Lagrange Multipliers)
  • But the best approximation is no

approximation at all!

  • We need to constrain q so that it’s tractable

49

slide-49
SLIDE 49

Optimal q in an Imperfect World

  • We can’t compute q(z)=p(z|x) directly
  • Instead, constrain the domain of F[q] to some

set of more tractable functions

  • This is usually done by making independence

assumptions

– The mean field assumption: cut all dependencies

50

slide-50
SLIDE 50
  • We have some observed data:
  • We have a model relating latent variables z and θ to the

data:

  • To guess z and θ we need
  • But the integral is hard!
  • Why?

Mean Field Assumption

51

slide-51
SLIDE 51
  • z and θ not independent 
  • Often p(z) is defined in terms θ
  • Example:
  • z = draws from a multinomial (words, POS tags, CFG

rules, …)

  • θ = the weights of those words, POS tags, CFG rules
  • But perhaps things would be tractable if they were

independent

Mean Field Assumption

52

slide-52
SLIDE 52

Mean Field Assumption

53

slide-53
SLIDE 53

The Lower Bound (Again)

54

slide-54
SLIDE 54

The Lower Bound (Again)

55

slide-55
SLIDE 55

The Lower Bound (Again)

56

slide-56
SLIDE 56

The New Lower Bound

Apply independence assumption:

57

slide-57
SLIDE 57

The New Lower Bound

Apply independence assumption:

58

slide-58
SLIDE 58

The New Lower Bound

Apply independence assumption:

59

slide-59
SLIDE 59

The New Lower Bound

Apply independence assumption:

60

slide-60
SLIDE 60
  • The integrals get simpler
  • Even simpler after derivatives

The Benefit of Independence

61

slide-61
SLIDE 61

Optimizing the Lower Bound

62

slide-62
SLIDE 62

Optimal qθ(θ)

  • Use Lagrange multipliers

constraint

63

slide-63
SLIDE 63

Optimal qz(z)

  • Use Lagrange multipliers

constraint

64

slide-64
SLIDE 64

q ≠ p

65

slide-65
SLIDE 65

q ≠ p

66

slide-66
SLIDE 66

Estimating Parameters

  • Now we have our approximation q
  • We need to compute the expectations
  • Use EM-like procedure, alternating between the two

– It was hard to do this for p(z,θ|x) – It’s (hopefully) easy for q(z,θ)

  • if we’ve defined p to make use of conjugacy
  • and if we’ve chosen the right constraint for q

67

slide-67
SLIDE 67

Calculating F

68

slide-68
SLIDE 68
  • As a side effect of inference, we already have
  • It’s the log of the normalization constant for

q(z)

  • So, we really only need two more expectations

Calculating F

69

slide-69
SLIDE 69

Uses for F

  • We can often use F in cases where we would

normally use the log likelihood

– Measuring convergence

  • No guarantee to maximize likelihood, but we do have F
  • Others

– Model selection

  • Choose the model with the highest lower bound

– Selecting the number of clusters

  • Pick the number that gives us the highest lower bound

– Parameter optimization

  • Again, optimize the lower bound w.r.t. the parameters

70

slide-70
SLIDE 70

Conclusion to Part 1

  • VB is conceptually very simple

– Optimization (like 1st year calculus)

  • The challenge

– Choosing the space of approximations to

  • Avoid intractability
  • Closely fit the true distribution

71

slide-71
SLIDE 71

Part II

Case Study: Dirichlet-Multinomial Mixture Model

72

slide-72
SLIDE 72

Dirichlet-Multinomial Mixture Model

φ x z

N

π

K

α β

73

slide-73
SLIDE 73

Dirichlet-Multinomial Mixture Model

φ x z

N

π

K

α β

74

  • Quantity of interest
slide-74
SLIDE 74

Dirichlet-Multinomial Mixture Model

φ x z

N

π

K

α β

75

  • Quantity of interest
slide-75
SLIDE 75
  • Generative model
  • Observed data:
  • What we’re interested in (the posterior)

Model

76

slide-76
SLIDE 76
  • Generative model
  • Observed data:
  • What we’re interested in (the posterior)

Model

77

slide-77
SLIDE 77
  • The posterior probability
  • Intractable integral (marginal likelihood)

The Intractable Integral

78

slide-78
SLIDE 78
  • The marginal likelihood
  • We can compute

The Intractable Integral

79

slide-79
SLIDE 79
  • The marginal likelihood
  • We can compute
  • But what about

The Intractable Integral

80

slide-80
SLIDE 80
  • The marginal likelihood

The Intractable Integral

81

z1 z0 z2 φ z1 z0 z2

… …

slide-81
SLIDE 81
  • The marginal likelihood
  • Integrating out induces dependencies between the ’s
  • There is a similar relationship between and the ’s
  • Calculations become intractable
  • But if we cut the dependencies between , , and …

The Intractable Integral

82

slide-82
SLIDE 82
  • The approximation
  • Then our lower bound

becomes

The Mean Field Assumption

83

slide-83
SLIDE 83

Optimizing F

  • Apply Lagrange multipliers just like before
  • In this case, we have simply replaced z, x, and

θ with vectors

  • The math is exactly the same
  • But we need to find the expectations we

skipped before

– Plug in the Dirichlet and multinomial distributions

84

slide-84
SLIDE 84

Optimal q(z,θ)

85

  • Borrowed from the Mean Field Example
  • See slides 62-63
  • All we need to do is apply the particulars of the

mixture model

slide-85
SLIDE 85

Optimal qθ(θ)

86

  • Factorize qθ(θ)

To get

slide-86
SLIDE 86

Optimal qφ(φ): The Expectation

87

slide-87
SLIDE 87

Optimal qφ(φ): The Expectation

88

slide-88
SLIDE 88

Dirichlet Distribution

89

  • Notable for being the conjugate prior of the

multinomial

slide-89
SLIDE 89

Optimal qφ(φ): The Numerator

90

slide-90
SLIDE 90

Optimal qφ(φ): The Numerator

91

  • Def. of Dirichlet
slide-91
SLIDE 91

Optimal qφ(φ): The Numerator

92

Previous slide

slide-92
SLIDE 92

Optimal qφ(φ): The Numerator

93

slide-93
SLIDE 93

Optimal qφ(φ): The Numerator

94

slide-94
SLIDE 94

Optimal qφ(φ): The Numerator

95

slide-95
SLIDE 95

Optimal qφ(φ): The Normalization

96

slide-96
SLIDE 96

Optimal qφ(φ): The Normalization

97

from previous slide

slide-97
SLIDE 97

Optimal qφ(φ): The Normalization

98

slide-98
SLIDE 98

Optimal qφ(φ): The Normalization

99

  • Def. of Dirichlet
slide-99
SLIDE 99

Optimal qφ(φ): The Normalization

100

slide-100
SLIDE 100

Optimal qφ(φ): Conjugacy Helps

101

slide-101
SLIDE 101

Optimal qφ(φ): Conjugacy Helps

102

slide-102
SLIDE 102

Optimal qφ(φ): Conjugacy Helps

103

slide-103
SLIDE 103

Optimal qφ(φ): Conjugacy Helps

104

slide-104
SLIDE 104

Optimal qφ(φ): Conjugacy Helps

105

slide-105
SLIDE 105

Optimal qφ(φ): Conjugacy Helps

106

slide-106
SLIDE 106

Optimal qφ(φ): Conjugacy Helps

107

  • Def. of Dirichlet
slide-107
SLIDE 107

Optimal qπ(π)

  • q(π) is essentially the same as q(φ)
  • The only difference is that there are multiple

π’s

  • So, q(π) should be a product of Dirichlets

108

slide-108
SLIDE 108

Optimal qπ(π): The Expectation

109

slide-109
SLIDE 109

Optimal qπ(π): The Numerator

110

slide-110
SLIDE 110

Optimal qπ(π): The Numerator

111

Model Def. Previous work

slide-111
SLIDE 111

Optimal qπ(π): The Numerator

112

slide-112
SLIDE 112

Optimal qπ(π): The Denominator

113

slide-113
SLIDE 113

Optimal qπ(π): The Denominator

114

slide-114
SLIDE 114

Optimal qπ(π): The Denominator

115

slide-115
SLIDE 115

Optimal qπ(π): The Denominator

116

  • Def. of Dirichlet
slide-116
SLIDE 116

Optimal qπ(π): The Denominator

117

slide-117
SLIDE 117

Optimal qπ(π): Putting Them Together

118

slide-118
SLIDE 118

Optimal qπ(π): Putting Them Together

119

slide-119
SLIDE 119

Optimal qπ(π): Putting Them Together

120

slide-120
SLIDE 120

Optimal qπ(π): Putting Them Together

121

  • Def. of Dirichlet
slide-121
SLIDE 121

Optimal qπ(π): Putting Them Together

122

slide-122
SLIDE 122

Optimal qz(z)

123

  • Again, from our Mean Field Assumption

Example

  • See slide 63
  • Just apply the assumptions of our model
slide-123
SLIDE 123

Optimal qz(z)

124

  • First, let’s work with the simpler multinomial

distribution

  • Side effect: a kind of estimate for the

multinomial parameter vector

slide-124
SLIDE 124

A Useful Standard Result

  • The expectation under a Dirichlet of the log of

an individual scalar component of a Dirichlet random vector

  • The digamma function

125

slide-125
SLIDE 125

Optimal qz(z): The Expectations

126

slide-126
SLIDE 126

Optimal qz(z): The Expectations

127

Standard Result

slide-127
SLIDE 127

Optimal qz(z): The Expectations

128

slide-128
SLIDE 128

Optimal qz(z): The Expectations

129

slide-129
SLIDE 129

Optimal qz(z): The Expectations

130

slide-130
SLIDE 130
  • Now, let’s work with the product of multinomials
  • Side effect: a kind of set of multinomial

parameter vectors

  • This is essentially the same math required for

HMMs and PCFGs

Optimal qz(z): The Expectations

131

slide-131
SLIDE 131

Optimal qz(z): The Expectations

132

slide-132
SLIDE 132

Optimal qz(z): Putting It Together

133

slide-133
SLIDE 133

Optimal qz(z): Putting It Together

134

slide-134
SLIDE 134

Optimal qz(z): Putting It Together

135

slide-135
SLIDE 135

Implications of Assumption

  • We should get the same result with an even

weaker assumption

136

slide-136
SLIDE 136

Inference

  • “E-Step”: Compute Expected Counts

– Topic counts – Topic-word pair counts

  • “M-Step”: Compute Ratios

– Topic j – Topic-word pair j-k

137

slide-137
SLIDE 137

Calculating F

138

  • Also borrowed from the Mean Field Example
  • See slides 67-68
  • But we adapt it for the mixture model
slide-138
SLIDE 138

Calculating F

139

slide-139
SLIDE 139

Calculating F: The Normalization Constant

140

  • By product of computing
slide-140
SLIDE 140

Conclusion to Part 2

  • Many of the most popular models in NLP are

based on Multinomials

– ngrams, HMMs, PCFGs, …

  • The same math will work there
  • In fact, if you can implement EM, VB only requires

changing a few lines of code

  • Similar to other smoothing techniques, but has a

solid statistical grounding

141