Another Walkthrough of Variational Bayes. Bevan Jones Machine Learning Reading Group Macquarie University
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1 Another Walkthrough of Variational Bayes Bevan Jones Machine Learning Reading Group Macquarie University
2 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
3 Useful Concepts Probability/Information Theory Bayes Theorem Expectations Jensen s Inequality KL Divergence Calculus Functionals & Functional Derivatives Lagrange Multipliers Logarithms 3
4 Outline The true likelihood Approximating the posterior The lower bound and a definition for best Finding the optimal approximation Functionals & functional derivatives Connection to KL divergence The Mean-field approximation An inference procedure Dirichlet-multinomial example 4
5 The (Log) Likelihood 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 5
6 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 6
7 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 7
8 Bounding the Log Likelihood w/ Jensen s Inequality Jensen s Inequality where f is concave 8
9 Bounding the Log Likelihood w/ Jensen s Inequality Jensen s Inequality where f is concave 9
10 Bounding the Log Likelihood w/ Jensen s Inequality Jensen s Inequality where f is concave 10
11 Bounding the Log Likelihood w/ Jensen s Inequality Jensen s Inequality where f is concave 11
12 The Lower Bound 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 12
13 Functionals: the Variational in VB Functional: a kind of meta-function that takes a function as input We can view F[q] as a functional of q Calculus of functionals parallels that of functions Then, we can take the derivative of F[q] with respect to q, set it to 0, and solve for q 13
14 Derivatives 14
15 Functional Derivatives The change in functional as we change its function argument 15
16 Useful Derivatives 16
17 Useful Derivatives 17
18 Useful Derivatives 18
19 Useful Derivatives 19
20 Useful Derivatives 20
21 Calculating q Use Lagrange multipliers constraint 21
22 Calculating q 22
23 Calculating q 23
24 Calculating q 24
25 KL Divergence: An Alternative View Maximizing F is minimizing the KL divergence And 25
26 Optimal q The best q(z) is p(z x) 26
27 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 27
28 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 28
29 Example 2: Mean Field Assumption 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! Apply the mean field assumption 29
30 The New Lower Bound 30
31 The New Lower Bound 31
32 The New Lower Bound 32
33 The New Lower Bound Apply mean field assumption 33
34 The Benefit of Independence The integrals get simpler In fact, these go away 34
35 Optimizing the Lower Bound 35
36 Optimal q θ (θ) Use Lagrange multipliers constraint 36
37 Optimal q z (z) Use Lagrange multipliers constraint 37
38 The Approximation q p 38
39 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 39
40 Calculating F 40
41 Calculating F 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 41
42 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 42
43 Worked Example Dirichlet-Multinomial Mixture Model 43
44 Dirichlet-Multinomial Mixture Model α φ β z π K x N 44
45 The Intractable Integral 45
46 The Mean Field Assumption 46
47 Optimizing F Apply Lagrange multipliers just like example 2 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 47
48 Optimal q(z,θ) Borrowed from example 2 See slides All we need to do is apply the particulars of the Mixture model 48
49 Optimal q θ (θ) 49
50 Optimal q φ (φ): The Expectation 50
51 Dirichlet Distribution 51
52 Optimal q φ (φ): The Numerator 52
53 Optimal q φ (φ): The Normalization 53
54 Optimal q φ (φ): Conjugacy Helps 54
55 Optimal q π (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 55
56 Optimal q π (π): The Expectation 56
57 Optimal q π (π): The Numerator 57
58 Optimal q π (π): The Denominator 58
59 Optimal q π (π): Putting Them Together 59
60 A Useful Standard Result The digamma function The expectation under a Dirichlet of the log of an individual component of a Dirichlet random variable 60
61 Optimal q z (z) Again, borrowed from example 2 See slides Here, we plug in the model definition 61
62 Optimal q z (z) First, let s work with the simpler multinomial distribution Side effect: a kind of estimate for the multinomial parameter vector 62
63 Optimal q z (z): The Expectations 63
64 Optimal q z (z): The Expectations Now, let s work with the product of multinomials Side effect: a kind of set of multinomial parameter vectors This is essentialy the same math required for HMMs and PCFGs 64
65 Optimal q z (z): The Expectations 65
66 Optimal q z (z): Putting It Together 66
67 Implications of Assumption We should get the same result with an even weaker assumption 67
68 Inference E-Step : Expected Counts Topic counts Topic-word pair counts M-Step : The Proportions Topic j Topic-word pair j-k 68
69 Calculating F Also borrowed from example 2 See slides But we adapt it for the mixture model 69
70 Calculating F 70
71 Calculating F: The Normalization Constant By product of computing 71
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