Latent Variable Models and Expectation Maximization

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1 Latent Variable Models and Expectation Maximization Oliver Schulte - CMPT 726 Bishop PRML Ch. 9

2 small then only the white circle is seen, and there is no extrema in entropy. in entropy. There There is anis entropy an entropy extrema extrema when when the the µ, Σ) α f trema scale scale is slightly is slightly larger larger than the thanradius the radius of theofbright bright circle, circle, tative p relative to a ref- which which has has In practice In practice this method this method gives gives stable stable identification identification of fea- of fea- to a refnitydensity re arts assumed are assumed to totures turesaover variety a variety of sizes of sizes and copes and copes well with well with intra-class intra-class ckground model model asin(within a range a range r). r). ant to scaling, although experimental tests show that this We as-variability. discussed variability. Theprobabilistic saliency saliency measure measure models is designed is designed attolength betoinvari- invari- is α f K-Means The Expectation Maximization Algorithm EM Example: Gaussian Mixture Models d p r f p, U p ) dp r f er. e finder. 1 p(d θ) (N, f) p(d θ) Learning and thereafter and Parameters thereafter entropy the entropy todecreases Probability theasscale scale increases. increases. Distributions detected tures detected using using The n M. second The second is is variable and the and last the last ll ible possible occlusion occlusion l. shape s the shape and ocpearancthe appearance and and compasses many many of of and oc- obabilistic istic way, this way, this constrained objects objects ll a small covariance) covariance) Assignment ant to scaling, 3: given although fullyexperimental observedtests training show that data, this issetting not entirely not entirely the case thedue casetodue aliasing to aliasing and other and other effects. effects. Note, Note, parameters only monochrome only monochrome θ i forinformation Bayes information nets is used is is used to straight-forward detect to detect and represent features. sent infeatures. many settings not all variables are observed and repre- However, (labelled) in the training data: x i = (x i, h i ) 2.3. e.g Feature Speech Feature representation recognition: have speech signals, but not The phoneme feature The feature detector labels detector identifies identifies regions regions of interest of interest on each on each image. e.g. image. Object The coordinates Therecognition: coordinates of theof centre the have centre give object us give Xus and labels Xthe andsize (car, the size bicycle), of but theof not region thepart region gives labels gives S. Figure S. (wheel, Figure 2 illustrates 2door, illustrates this seat) on this two ontypical two typical images images from the frommotorbike the motorbike dataset. dataset. Unobserved variables are called latent variables figs from Fergus et al. Figure Figure 2: Output 2: Output of theof feature the feature detector detector

3 Latent Variable Models: Pros Statistically powerful, often good predictions. Many applications: Learning with missing data. Clustering: missing cluster label for data points. Principal Component Analysis: data points are generated in linear fashion from a small set of unobserved components. (more later) Matrix Factorization, Recommender Systems: Assign users to unobserved user types, assign items to unobserved item types. Use similarity between user type, item type to predict preference of user for item. Winner of $1M Netflix challenge. If latent variables have an intuitive interpretation (e.g., action movies, factors ), discovers new features.

4 Latent Variable Models: Cons From a user s point of view, like a black box if latent variables don t have an intuitive interpretation. Statistically, hard to guarantee convergence to a correct model with more data (the identifiability problem). Harder computationally, usually no closed form for maximum likelihood estimates. However, the Expectation-Maximization algorithm provides a general-purpose local search algorithm for learning parameters in probabilistic models with latent variables.

5 Outline K-Means The Expectation Maximization Algorithm EM Example: Gaussian Mixture Models

6 Outline K-Means The Expectation Maximization Algorithm EM Example: Gaussian Mixture Models

7 Unsupervised Learning 2 2 (a) 2 2 We will start with an unsupervised learning (clustering) problem: Given a dataset {x 1,..., x N }, each x i R D, partition the dataset into K clusters Intuitively, a cluster is a group of points, which are close together and far from others

8 Distortion Measure 2 (a) (i) Formally, introduce prototypes (or cluster centers) µ k R D Use binary r nk, 1 if point n is in cluster k, otherwise (1-of-K coding scheme again) Find {µ k }, {r nk } to minimize distortion measure: J = N n=1 k=1 K r nk x n µ k 2

9 Minimizing Distortion Measure Minimizing J directly is hard N K J = r nk x n µ k 2 n=1 k=1 However, two things are easy If we know µ k, minimizing J wrt r nk If we know r nk, minimizing J wrt µ k This suggests an iterative procedure Start with initial guess for µ k Iteration of two steps: Minimize J wrt r nk Minimize J wrt µ k Rinse and repeat until convergence

10 Minimizing Distortion Measure Minimizing J directly is hard N K J = r nk x n µ k 2 n=1 k=1 However, two things are easy If we know µ k, minimizing J wrt r nk If we know r nk, minimizing J wrt µ k This suggests an iterative procedure Start with initial guess for µ k Iteration of two steps: Minimize J wrt r nk Minimize J wrt µ k Rinse and repeat until convergence

11 Minimizing Distortion Measure Minimizing J directly is hard N K J = r nk x n µ k 2 n=1 k=1 However, two things are easy If we know µ k, minimizing J wrt r nk If we know r nk, minimizing J wrt µ k This suggests an iterative procedure Start with initial guess for µ k Iteration of two steps: Minimize J wrt r nk Minimize J wrt µ k Rinse and repeat until convergence

12 2 (a) Determining Membership Variables Step 1 in an iteration of K-means is to minimize distortion measure J wrt cluster membership variables r nk (b) 2 2 J = N n=1 k=1 K r nk x n µ k 2 Terms for different data points x n are independent, for each data point set r nk to minimize K r nk x n µ k 2 k=1 Simply set r nk = 1 for the cluster center µ with smallest distance

13 2 (a) Determining Membership Variables Step 1 in an iteration of K-means is to minimize distortion measure J wrt cluster membership variables r nk (b) 2 2 J = N n=1 k=1 K r nk x n µ k 2 Terms for different data points x n are independent, for each data point set r nk to minimize K r nk x n µ k 2 k=1 Simply set r nk = 1 for the cluster center µ with smallest distance

14 2 (a) Determining Membership Variables Step 1 in an iteration of K-means is to minimize distortion measure J wrt cluster membership variables r nk (b) 2 2 J = N n=1 k=1 K r nk x n µ k 2 Terms for different data points x n are independent, for each data point set r nk to minimize K r nk x n µ k 2 k=1 Simply set r nk = 1 for the cluster center µ with smallest distance

15 Determining Cluster Centers Step 2: fix r nk, minimize J wrt the cluster centers µ k 2 (b) K N J = r nk x n µ k 2 switch order of sums k=1 n= (c) So we can minimze wrt each µ k separately Take derivative, set to zero: 2 N r nk (x n µ k ) = n=1 µ k = n r nkx n n r nk i.e. mean of datapoints x n assigned to cluster k

16 Determining Cluster Centers Step 2: fix r nk, minimize J wrt the cluster centers µ k 2 (b) K N J = r nk x n µ k 2 switch order of sums k=1 n= (c) So we can minimze wrt each µ k separately Take derivative, set to zero: 2 N r nk (x n µ k ) = n=1 µ k = n r nkx n n r nk i.e. mean of datapoints x n assigned to cluster k

17 K-means Algorithm Start with initial guess for µ k Iteration of two steps: Minimize J wrt r nk Assign points to nearest cluster center Minimize J wrt µ k Set cluster center as average of points in cluster Rinse and repeat until convergence

18 K-means example 2 (a) 2 2 2

19 K-means example 2 (b) 2 2 2

20 K-means example 2 (c) 2 2 2

21 K-means example 2 (d) 2 2 2

22 K-means example 2 (e) 2 2 2

23 K-means example 2 (f) 2 2 2

24 K-means example 2 (g) 2 2 2

25 K-means example 2 (h) 2 2 2

26 K-means example 2 (i) Next step doesn t change membership stop

27 K-means Convergence Repeat steps until no change in cluster assignments For each step, value of J either goes down, or we stop Finite number of possible assignments of data points to clusters, so we are guaranteed to converge eventually Note it may be a local maximum rather than a global maximum to which we converge

28 K-means Example - Image Segmentation Original image K-means clustering on pixel colour values Pixels in a cluster are coloured by cluster mean Represent each pixel (e.g. 24-bit colour value) by a cluster number (e.g. 4 bits for K = 1), compressed version This technique known as vector quantization Represent vector (in this case from RGB, R 3 ) as a single discrete value

29 K-means Generalized: the set-up Let s generalize the idea. Suppose we have the following set-up. X denotes all observed variables (e.g., data points). Z denotes all latent (hidden, unobserved) variables (e.g., cluster means). J(X, Z θ) where J measures the goodness of an assignment of latent variable models given the data points and parameters θ. e.g., J = -dispersion measure. parameters = assignment of points to clusters. It s easy to maximize J(X, Z θ) wrt θ for fixed Z. It s easy to maximize J(X, Z θ) wrt Z for fixed θ.

30 K-means Generalized: The Algorithm The fact that conditional maximization is simple suggests an iterative algorithm. 1. Guess an initial value for latent variables Z. 2. Repeat until convergence: 2.1 Find best parameter values θ given the current guess for the latent variables. Update the parameter values. 2.2 Find best value for latent variables Z given the current parameter values. Update the latent variable values.

31 Outline K-Means The Expectation Maximization Algorithm EM Example: Gaussian Mixture Models

32 EM Algorithm: The set-up We assume a probabilistic model, specifically the complete-data likelihood function p(x, Z θ). Goodness of the model is the log-likelihood ln p(x, Z θ). Key difference: instead of guessing a single best value for latent variables given current parameter settings, we use the conditional distribution p(z X, θ old ) over latent variables. Given a latent variable distribution, parameter values θ are evaluated by taking the expected goodness ln p(x, Z θ) over all possible latent variable settings.

33 EM Algorithm: The procedure 1. Guess an initial parameter setting θ old. 2. Repeat until convergence: 3. The E-step: Evaluate p(z X, θ old ). (Ideally, find a closed form as a function of Z). 4. The M-step: 4.1 Evaluate the function Q(θ, θ old ) = Z p(z X, θold ) ln p(x, Z θ). 4.2 Maximize Q(θ, θ old ) wrt θ. Update θ old. 5. This procedure is guaranteed to increase at each step. the data log-likelihood ln p(x θ) = Z ln p(x, Z θ). 6. Therefore converges to local log-likelihood maximum. More theoretical analysis in text.

34 Outline K-Means The Expectation Maximization Algorithm EM Example: Gaussian Mixture Models

35 Hard Assignment vs. Soft Assignment 2 (i) In the K-means algorithm, a hard assignment of points to clusters is made However, for points near the decision boundary, this may not be such a good idea Instead, we could think about making a soft assignment of points to clusters

36 Gaussian Mixture Model 1 (a) 1 (b) The Gaussian mixture model (or mixture of Gaussians MoG) models the data as a combination of Gaussians. a: constant density contours. b: marginal probability p(x). c: surface plot. Widely used general approximation for multi-modal distributions.

37 Gaussian Mixture Model 1 (b) 1 (a) Above shows a dataset generated by drawing samples from three different Gaussians

38 Generative Model z 1 (a).5 x.5 1 The mixture of Gaussians is a generative model To generate a datapoint x n, we first generate a value for a discrete variable z n {1,..., K} We then generate a value x n N (x µ k, Σ k ) for the corresponding Gaussian

39 Graphical Model π z n 1 (a).5 x n µ Σ N.5 1 Full graphical model using plate notation Note z n is a latent variable, unobserved Need to give conditional distributions p(z n ) and p(x n z n ) The one-of-k representation is helpful here: z nk {, 1}, z n = (z n1,..., z nk )

40 Graphical Model - Latent Component Variable π z n 1 (a).5 x n µ Σ N.5 1 Use a Bernoulli distribution for p(z n ) i.e. p(z nk = 1) = π k Parameters to this distribution {π k } Must have π k 1 and K k=1 π k = 1 p(z n ) = K k=1 πz nk k

41 Graphical Model - Observed Variable π z n 1 (a).5 x n µ Σ N.5 1 Use a Gaussian distribution for p(x n z n ) Parameters to this distribution {µ k, Σ k } p(x n z nk = 1) = N (x n µ k, Σ k ) K p(x n z n ) = N (x n µ k, Σ k ) z nk k=1

42 Graphical Model - Joint distribution π z n 1 (a).5 x n µ Σ N.5 1 The full joint distribution is given by: p(x, z) = = N p(z n )p(x n z n ) n=1 N K n=1 k=1 π z nk k N (x n µ k, Σ k ) z nk

43 MoG Marginal over Observed Variables The marginal distribution p(x n ) for this model is: p(x n ) = z n p(x n, z n ) = z n p(z n )p(x n z n ) = K π k N (x n µ k, Σ k ) k=1 A mixture of Gaussians

44 MoG Conditional over Latent Variable 1 (b) 1 (c) To apply EM, need the conditional distribution p(z nk = 1 x n, θ) where θ are the model parameters. It is denoted by γ(z nk ) can be computed as: γ(z nk ) p(z nk = 1 x n ) = = p(z nk = 1)p(x n z nk = 1) K j=1 p(z nj = 1)p(x n z nj = 1) π k N (x n µ k, Σ k ) K j=1 π jn (x n µ j, Σ j ) γ(z nk ) is the responsibility of component k for datapoint n

45 MoG Conditional over Latent Variable 1 (b) 1 (c) To apply EM, need the conditional distribution p(z nk = 1 x n, θ) where θ are the model parameters. It is denoted by γ(z nk ) can be computed as: γ(z nk ) p(z nk = 1 x n ) = = p(z nk = 1)p(x n z nk = 1) K j=1 p(z nj = 1)p(x n z nj = 1) π k N (x n µ k, Σ k ) K j=1 π jn (x n µ j, Σ j ) γ(z nk ) is the responsibility of component k for datapoint n

46 MoG Conditional over Latent Variable 1 (b) 1 (c) To apply EM, need the conditional distribution p(z nk = 1 x n, θ) where θ are the model parameters. It is denoted by γ(z nk ) can be computed as: γ(z nk ) p(z nk = 1 x n ) = = p(z nk = 1)p(x n z nk = 1) K j=1 p(z nj = 1)p(x n z nj = 1) π k N (x n µ k, Σ k ) K j=1 π jn (x n µ j, Σ j ) γ(z nk ) is the responsibility of component k for datapoint n

47 EM for Gaussian Mixtures: E-step The complete-data log-likelihood is ln p(x, Z θ) = N n=1 k=1 K z nk [ln π k + lnn (x n µ k, Σ k )]. E step: Calculate responsibilities using current parameters θ old : γ(z nk ) = π kn (x n µ k, Σ k ) K j=1 π jn (x n µ j, Σ j ) The z n vectors assigning data point n to components are independent of each other. Therefore under the posterior distribution p(z nk = 1 x n, θ) the expected value of z nk is γ(z nk ).

48 EM for Gaussian Mixtures: M-step M step: Because of the independence of the component assignments, we can calculate the expectation wrt the component assignments by using the expectations of the component assignments. So Q(θ, θ old ) = N n=1 K k=1 γ(z nk)[ln π k + lnn (x n µ k, Σ k )]. Maximizing Q(θ, θ old ) with respect to the model parameters is more or less straightforward.

49 EM for Gaussian Mixtures II Initialize parameters, then iterate: E step: Calculate responsibilities using current parameters γ(z nk ) = π kn (x n µ k, Σ k ) K j=1 π jn (x n µ j, Σ j ) M step: Re-estimate parameters using these γ(z nk ) N k = N γ(z nk ) n=1 µ k = 1 N γ(z nk )x n N k n=1 Σ k = 1 N γ(z nk )(x n µ N k )(x n µ k ) T k n=1 π k = N k N Think of N k as effective number of points in component k.

50 EM for Gaussian Mixtures II Initialize parameters, then iterate: E step: Calculate responsibilities using current parameters γ(z nk ) = π kn (x n µ k, Σ k ) K j=1 π jn (x n µ j, Σ j ) M step: Re-estimate parameters using these γ(z nk ) N k = N γ(z nk ) n=1 µ k = 1 N γ(z nk )x n N k n=1 Σ k = 1 N γ(z nk )(x n µ N k )(x n µ k ) T k n=1 π k = N k N Think of N k as effective number of points in component k.

51 MoG EM - Example (a) 2 Same initialization as with K-means before Often, K-means is actually used to initialize EM

52 MoG EM - Example (b) 2 Calculate responsibilities γ(z nk )

53 MoG EM - Example (c) 2 Calculate model parameters {π k, µ k, Σ k } using these responsibilities

54 MoG EM - Example (d) 2 Iteration 2

55 MoG EM - Example (e) 2 Iteration 5

56 MoG EM - Example 2 2 Iteration 2 - converged 2 (f) 2

57 EM - Summary EM finds local maximum to likelihood p(x θ) = Z p(x, Z θ) Iterates two steps: E step fills in the missing variables Z (calculates their distribution) M step maximizes expected complete log likelihood (expectation wrt E step distribution)

58 Conclusion Readings: Ch. 9.1, 9.2, 9.4 K-means clustering Gaussian mixture model What about K? Model selection: either cross-validation or Bayesian version (average over all values for K) Expectation-maximization, a general method for learning parameters of models when not all variables are observed

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