Dirichlet Processes: Tutorial and Practical Course

Transcription

1 Dirichlet Processes: Tutorial and Practical Course (updated) Yee Whye Teh Gatsby Computational Neuroscience Unit University College London August 2007 / MLSS Yee Whye Teh (Gatsby) DP August 2007 / MLSS 1 / 90

2 Dirichlet Processes Dirichlet processes (DPs) are a class of Bayesian nonparametric models. Dirichlet processes are used for: Density estimation. Semiparametric modelling. Sidestepping model selection/averaging. I will give a tutorial on DPs, followed by a practical course on implementing DP mixture models in MATLAB. Prerequisites: understanding of the Bayesian paradigm (graphical models, mixture models, exponential families, Gaussian processes) you should know these from previous courses. Other tutorials on DPs: Zoubin Gharamani, UAI Michael Jordan, NIPS Volker Tresp, ICML nonparametric Bayes workshop Workshop on Bayesian Nonparametric Regression, Cambridge, July Yee Whye Teh (Gatsby) DP August 2007 / MLSS 2 / 90

3 Dirichlet Processes Dirichlet processes (DPs) are a class of Bayesian nonparametric models. Dirichlet processes are used for: Density estimation. Semiparametric modelling. Sidestepping model selection/averaging. I will give a tutorial on DPs, followed by a practical course on implementing DP mixture models in MATLAB. Prerequisites: understanding of the Bayesian paradigm (graphical models, mixture models, exponential families, Gaussian processes) you should know these from previous courses. Other tutorials on DPs: Zoubin Gharamani, UAI Michael Jordan, NIPS Volker Tresp, ICML nonparametric Bayes workshop Workshop on Bayesian Nonparametric Regression, Cambridge, July Yee Whye Teh (Gatsby) DP August 2007 / MLSS 2 / 90

4 Dirichlet Processes Dirichlet processes (DPs) are a class of Bayesian nonparametric models. Dirichlet processes are used for: Density estimation. Semiparametric modelling. Sidestepping model selection/averaging. I will give a tutorial on DPs, followed by a practical course on implementing DP mixture models in MATLAB. Prerequisites: understanding of the Bayesian paradigm (graphical models, mixture models, exponential families, Gaussian processes) you should know these from previous courses. Other tutorials on DPs: Zoubin Gharamani, UAI Michael Jordan, NIPS Volker Tresp, ICML nonparametric Bayes workshop Workshop on Bayesian Nonparametric Regression, Cambridge, July Yee Whye Teh (Gatsby) DP August 2007 / MLSS 2 / 90

5 Dirichlet Processes Dirichlet processes (DPs) are a class of Bayesian nonparametric models. Dirichlet processes are used for: Density estimation. Semiparametric modelling. Sidestepping model selection/averaging. I will give a tutorial on DPs, followed by a practical course on implementing DP mixture models in MATLAB. Prerequisites: understanding of the Bayesian paradigm (graphical models, mixture models, exponential families, Gaussian processes) you should know these from previous courses. Other tutorials on DPs: Zoubin Gharamani, UAI Michael Jordan, NIPS Volker Tresp, ICML nonparametric Bayes workshop Workshop on Bayesian Nonparametric Regression, Cambridge, July Yee Whye Teh (Gatsby) DP August 2007 / MLSS 2 / 90

6 Outline 1 Applications 2 Dirichlet Processes 3 Representations of Dirichlet Processes 4 Modelling Data with Dirichlet Processes 5 Practical Course Yee Whye Teh (Gatsby) DP August 2007 / MLSS 3 / 90

7 Outline 1 Applications 2 Dirichlet Processes 3 Representations of Dirichlet Processes 4 Modelling Data with Dirichlet Processes 5 Practical Course Yee Whye Teh (Gatsby) DP August 2007 / MLSS 4 / 90

8 Function Estimation Parametric function estimation (e.g. regression, classification) Data: x = {x 1, x 2,...}, y = {y 1, y 2,...} Model: y i = f (x i w) + N (0, σ 2 ) Prior over parameters Posterior over parameters p(w x, y) = p(w) p(w)p(y x, w) p(y x) Prediction with posteriors p(y x, x, y) = p(y x, w)p(w x, y) dw Yee Whye Teh (Gatsby) DP August 2007 / MLSS 5 / 90

9 Function Estimation Bayesian nonparametric function estimation with Gaussian processes Data: x = {x 1, x 2,...}, y = {y 1, y 2,...} Model: y i = f (x i ) + N (0, σ 2 ) Prior over functions Posterior over functions p(f x, y) = f GP(µ, Σ) p(f )p(y x, f ) p(y x) Prediction with posteriors p(y x, x, y) = p(y x, f )p(f x, y) df Yee Whye Teh (Gatsby) DP August 2007 / MLSS 6 / 90

10 Function Estimation Figure from Carl s lecture. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 7 / 90

11 Density Estimation Parametric density estimation (e.g. Gaussian, mixture models) Data: x = {x 1, x 2,...} Model: x i w F( w) Prior over parameters Posterior over parameters p(w) p(w x) = p(w)p(x w) p(x) Prediction with posteriors p(x x) = p(x w)p(w x) dw Yee Whye Teh (Gatsby) DP August 2007 / MLSS 8 / 90

12 Density Estimation Bayesian nonparametric density estimation with Dirichlet processes Data: x = {x 1, x 2,...} Model: x i F Prior over distributions F DP(α, H) Posterior over distributions p(f x) = p(f)p(x F) p(x) Prediction with posteriors p(x x) = p(x F )p(f x) df = F (x )p(f x) df Not quite correct; see later. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 9 / 90

13 Density Estimation Bayesian nonparametric density estimation with Dirichlet processes Data: x = {x 1, x 2,...} Model: x i F Prior over distributions F DP(α, H) Posterior over distributions p(f x) = p(f)p(x F) p(x) Prediction with posteriors p(x x) = p(x F )p(f x) df = F (x )p(f x) df Not quite correct; see later. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 9 / 90

14 Density Estimation Bayesian nonparametric density estimation with Dirichlet processes Data: x = {x 1, x 2,...} Model: x i F Prior over distributions F DP(α, H) Posterior over distributions p(f x) = p(f)p(x F) p(x) Prediction with posteriors p(x x) = p(x F )p(f x) df = F (x )p(f x) df Not quite correct; see later. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 9 / 90

15 Density Estimation Prior: !15!10! Red: mean density. Blue: median density. Grey: 5-95 quantile. Others: draws. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 10 / 90

16 Density Estimation Posterior: !15!10! Red: mean density. Blue: median density. Grey: 5-95 quantile. Black: data. Others: draws. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 11 / 90

17 Semiparametric Modelling Linear regression model for inferring effectiveness of new medical treatments. y ij = β x ij + b i z ij + ɛ ij y ij is outcome of jth trial on ith subject. x ij, z ij are predictors (treatment, dosage, age, health...). β are fixed-effects coefficients. b i are random-effects subject-specific coefficients. ɛ ij are noise terms. Care about inferring β. If x ij is treatment, we want to determine p(β > 0 x, y). Yee Whye Teh (Gatsby) DP August 2007 / MLSS 12 / 90

18 Semiparametric Modelling y ij = β x ij + b i z ij + ɛ ij Usually we assume Gaussian noise ɛ ij N (0, σ 2 ). Is this a sensible prior? Over-dispersion, skewness,... May be better to model noise nonparametrically, ɛ ij F F DP Also possible to model subject-specific random effects nonparametrically, b i G G DP Yee Whye Teh (Gatsby) DP August 2007 / MLSS 13 / 90

19 Model Selection/Averaging Data: x = {x 1, x 2,...} Models: p(θ k M k ), p(x θ k, M k ) Marginal likelihood p(x M k ) = p(x θ k, M k )p(θ k M k ) dθ k Model selection M = argmax M k p(x M k ) Model averaging p(x x) = M k p(x M k )p(m k x) = M k p(x M k ) p(x M k)p(m k ) p(x) But: is this computationally feasible? Yee Whye Teh (Gatsby) DP August 2007 / MLSS 14 / 90

20 Model Selection/Averaging Data: x = {x 1, x 2,...} Models: p(θ k M k ), p(x θ k, M k ) Marginal likelihood p(x M k ) = p(x θ k, M k )p(θ k M k ) dθ k Model selection M = argmax M k p(x M k ) Model averaging p(x x) = M k p(x M k )p(m k x) = M k p(x M k ) p(x M k)p(m k ) p(x) But: is this computationally feasible? Yee Whye Teh (Gatsby) DP August 2007 / MLSS 14 / 90

21 Model Selection/Averaging Data: x = {x 1, x 2,...} Models: p(θ k M k ), p(x θ k, M k ) Marginal likelihood p(x M k ) = p(x θ k, M k )p(θ k M k ) dθ k Model selection M = argmax M k p(x M k ) Model averaging p(x x) = M k p(x M k )p(m k x) = M k p(x M k ) p(x M k)p(m k ) p(x) But: is this computationally feasible? Yee Whye Teh (Gatsby) DP August 2007 / MLSS 14 / 90

22 Model Selection/Averaging Marginal likelihood is usually extremely hard to compute. p(x M k ) = p(x θ k, M k )p(θ k M k ) dθ k Model selection/averaging is to prevent underfitting and overfitting. But reasonable and proper Bayesian methods should not overfit [Rasmussen and Ghahramani 2001]. Use a really large model M instead, and let the data speak for themselves. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 15 / 90

23 Model Selection/Averaging Clustering How many clusters are there? Yee Whye Teh (Gatsby) DP August 2007 / MLSS 16 / 90

24 Model Selection/Averaging Spike Sorting How many neurons are there? [Görür 2007, Wood et al. 2006a] Yee Whye Teh (Gatsby) DP August 2007 / MLSS 17 / 90

25 Model Selection/Averaging Topic Modelling How many topics are there? [Blei et al. 2004, Teh et al. 2006] Yee Whye Teh (Gatsby) DP August 2007 / MLSS 18 / 90

26 Model Selection/Averaging Grammar Induction How many grammar symbols are there? Figure from Liang. [Liang et al. 2007b, Finkel et al. 2007] Yee Whye Teh (Gatsby) DP August 2007 / MLSS 19 / 90

27 Model Selection/Averaging Visual Scene Analysis How many objects, parts, features? Figure from Sudderth. [Sudderth et al. 2007] Yee Whye Teh (Gatsby) DP August 2007 / MLSS 20 / 90

28 Outline 1 Applications 2 Dirichlet Processes 3 Representations of Dirichlet Processes 4 Modelling Data with Dirichlet Processes 5 Practical Course Yee Whye Teh (Gatsby) DP August 2007 / MLSS 21 / 90

29 Finite Mixture Models A finite mixture model is defined as follows: θ k H π Dirichlet(α/K,..., α/k ) z i π Discrete(π) x i θ z i F( θ z i ) Model selection/averaging over: Hyperparameters in H. Dirichlet parameter α. Number of components K. Determining K hardest. α π z i x i i = 1,..., n H θ k k = 1,..., K Yee Whye Teh (Gatsby) DP August 2007 / MLSS 22 / 90

30 Infinite Mixture Models Imagine that K 0 is really large. If parameters θk and mixing proportions π integrated out, the number of latent variables left does not grow with K no overfitting. At most n components will be associated with data, aka active. Usually, the number of active components is much less than n. This gives an infinite mixture model. Demo: dpm_demo2d Issue 1: can we take this limit K? Issue 2: what is the corresponding limiting model? [Rasmussen 2000] α π z i x i i = 1,..., n H θk k = 1,..., K Yee Whye Teh (Gatsby) DP August 2007 / MLSS 23 / 90

31 Gaussian Processes What are they? A Gaussian process (GP) is a distribution over functions f : X R Denote f GP if f is a GP-distributed random function. For any finite set of input points x 1,..., x n, we require (f (x 1 ),..., f (x n )) to be a multivariate Gaussian. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 24 / 90

32 Gaussian Processes What are they? The GP is parametrized by its mean m(x) and covariance c(x, y) functions: f (x 1 ) m(x 1 ) c(x 1, x 1 )... c(x 1, x n ). N.,..... f (x n ) m(x n ) c(x n, x 1 )... c(x n, x n ) The above are finite dimensional marginal distributions of the GP. A salient property of these marginal distributions is that they are consistent: integrating out variables preserves the parametric form of the marginal distributions above. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 25 / 90

33 Gaussian Processes Visualizing Gaussian Processes. A sequence of input points x 1, x 2, x 3,... dense in X. Draw f (x 1 ) f (x 2 ) f (x 1 ) f (x 3 ) f (x 1 ), f (x 2 ).. Each conditional distribution is Gaussian since (f (x 1 ),..., f (x n )) is Gaussian. Demo: GPgenerate Yee Whye Teh (Gatsby) DP August 2007 / MLSS 26 / 90

34 Dirichlet Processes Start with Dirichlet distributions. A Dirichlet distribution is a distribution over the K -dimensional probability simplex: K = { (π 1,..., π K ) : π k 0, k π k = 1 } We say (π 1,..., π K ) is Dirichlet distributed, with parameters (α 1,..., α K ), if (π 1,..., π K ) Dirichlet(α 1,..., α K ) p(π 1,..., π K ) = Γ( k α k) k Γ(α k) n k=1 π α k 1 k Yee Whye Teh (Gatsby) DP August 2007 / MLSS 27 / 90

35 Dirichlet Processes Examples of Dirichlet distributions. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 28 / 90

36 Dirichlet Processes Agglomerative property of Dirichlet distributions. Combining entries of probability vectors preserves Dirichlet property, for example: (π 1,..., π K ) Dirichlet(α 1,..., α K ) (π 1 + π 2, π 3,..., π K ) Dirichlet(α 1 + α 2, α 3,..., α K ) Generally, if (I 1,..., I j ) is a partition of (1,..., n): i I1 π i,..., i I j π i Dirichlet α i,..., α i i I1 i Ij Yee Whye Teh (Gatsby) DP August 2007 / MLSS 29 / 90

37 Dirichlet Processes Decimative property of Dirichlet distributions. The converse of the agglomerative property is also true, for example if: (π 1,..., π K ) Dirichlet(α 1,..., α K ) (τ 1, τ 2 ) Dirichlet(α 1 β 1, α 1 β 2 ) with β 1 + β 2 = 1, (π 1 τ 1, π 1 τ 2, π 2,..., π K ) Dirichlet(α 1 β 1, α 2 β 2, α 2,..., α K ) Yee Whye Teh (Gatsby) DP August 2007 / MLSS 30 / 90

38 Dirichlet Processes Visualizing Dirichlet Processes A Dirichlet process (DP) is an infinitely decimated Dirichlet distribution: 1 Dirichlet(α) (π 1, π 2 ) Dirichlet(α/2, α/2) π 1 + π 2 = 1 (π 11, π 12, π 21, π 22 ) Dirichlet(α/4, α/4, α/4, α/4) π i1 + π i2 = π i. Each decimation step involves drawing from a Beta distribution (a Dirichlet with 2 components) and multiplying into the relevant entry. Demo: DPgenerate Yee Whye Teh (Gatsby) DP August 2007 / MLSS 31 / 90

39 Dirichlet Processes A Proper but Non-Constructive Definition A probability measure is a function from subsets of a space X to [0, 1] satisfying certain properties. A Dirichlet Process (DP) is a distribution over probability measures. Denote G DP if G is a DP-distributed random probability measure. For any finite set of partitions A 1... A K = X, we require (G(A 1 ),..., G(A K )) to be Dirichlet distributed. A 1 A 4 A 2 A 3 A 5 A 6 Yee Whye Teh (Gatsby) DP August 2007 / MLSS 32 / 90

40 Dirichlet Processes Parameters of the Dirichlet Process A DP has two parameters: Base distribution H, which is like the mean of the DP. Strength parameter α, which is like an inverse-variance of the DP. We write: G DP(α, H) if for any partition (A 1,..., A K ) of X: (G(A 1 ),..., G(A K )) Dirichlet(αH(A 1 ),..., αh(a K )) Yee Whye Teh (Gatsby) DP August 2007 / MLSS 33 / 90

41 Dirichlet Processes Parameters of the Dirichlet Process A DP has two parameters: Base distribution H, which is like the mean of the DP. Strength parameter α, which is like an inverse-variance of the DP. We write: The first two cumulants of the DP: G DP(α, H) Expectation: Variance: E[G(A)] = H(A) V[G(A)] = H(A)(1 H(A)) α + 1 where A is any measurable subset of X. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 33 / 90

42 Dirichlet Processes Existence of Dirichlet Processes A probability measure is a function from subsets of a space X to [0, 1] satisfying certain properties. A DP is a distribution over probability measures such that marginals on finite partitions are Dirichlet distributed. How do we know that such an object exists?!? Kolmogorov Consistency Theorem: [Ferguson 1973]. de Finetti s Theorem: Blackwell-MacQueen urn scheme, Chinese restaurant process, [Blackwell and MacQueen 1973, Aldous 1985]. Stick-breaking Construction: [Sethuraman 1994]. Gamma Process: [Ferguson 1973]. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 34 / 90

43 Dirichlet Processes Existence of Dirichlet Processes A probability measure is a function from subsets of a space X to [0, 1] satisfying certain properties. A DP is a distribution over probability measures such that marginals on finite partitions are Dirichlet distributed. How do we know that such an object exists?!? Kolmogorov Consistency Theorem: [Ferguson 1973]. de Finetti s Theorem: Blackwell-MacQueen urn scheme, Chinese restaurant process, [Blackwell and MacQueen 1973, Aldous 1985]. Stick-breaking Construction: [Sethuraman 1994]. Gamma Process: [Ferguson 1973]. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 34 / 90

44 Dirichlet Processes Existence of Dirichlet Processes A probability measure is a function from subsets of a space X to [0, 1] satisfying certain properties. A DP is a distribution over probability measures such that marginals on finite partitions are Dirichlet distributed. How do we know that such an object exists?!? Kolmogorov Consistency Theorem: [Ferguson 1973]. de Finetti s Theorem: Blackwell-MacQueen urn scheme, Chinese restaurant process, [Blackwell and MacQueen 1973, Aldous 1985]. Stick-breaking Construction: [Sethuraman 1994]. Gamma Process: [Ferguson 1973]. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 34 / 90

45 Dirichlet Processes Existence of Dirichlet Processes A probability measure is a function from subsets of a space X to [0, 1] satisfying certain properties. A DP is a distribution over probability measures such that marginals on finite partitions are Dirichlet distributed. How do we know that such an object exists?!? Kolmogorov Consistency Theorem: [Ferguson 1973]. de Finetti s Theorem: Blackwell-MacQueen urn scheme, Chinese restaurant process, [Blackwell and MacQueen 1973, Aldous 1985]. Stick-breaking Construction: [Sethuraman 1994]. Gamma Process: [Ferguson 1973]. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 34 / 90

46 Dirichlet Processes Existence of Dirichlet Processes A probability measure is a function from subsets of a space X to [0, 1] satisfying certain properties. A DP is a distribution over probability measures such that marginals on finite partitions are Dirichlet distributed. How do we know that such an object exists?!? Kolmogorov Consistency Theorem: [Ferguson 1973]. de Finetti s Theorem: Blackwell-MacQueen urn scheme, Chinese restaurant process, [Blackwell and MacQueen 1973, Aldous 1985]. Stick-breaking Construction: [Sethuraman 1994]. Gamma Process: [Ferguson 1973]. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 34 / 90

47 Outline 1 Applications 2 Dirichlet Processes 3 Representations of Dirichlet Processes 4 Modelling Data with Dirichlet Processes 5 Practical Course Yee Whye Teh (Gatsby) DP August 2007 / MLSS 35 / 90

48 Dirichlet Processes Representations of Dirichlet Processes Suppose G DP(α, H). G is a (random) probability measure over X. We can treat it as a distribution over X. Let θ 1,..., θ n G be a random variable with distribution G. We saw in the demo that draws from a Dirichlet process seem to be discrete distributions. If so, then: G = π k δ θ k k=1 and there is positive probability that θ i s can take on the same value θ k for some k, i.e. the θ i s cluster together. In this section we are concerned with representations of Dirichlet processes based upon both the clustering property and the sum of point masses. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 36 / 90

49 Dirichlet Processes Representations of Dirichlet Processes Suppose G DP(α, H). G is a (random) probability measure over X. We can treat it as a distribution over X. Let θ 1,..., θ n G be a random variable with distribution G. We saw in the demo that draws from a Dirichlet process seem to be discrete distributions. If so, then: G = π k δ θ k k=1 and there is positive probability that θ i s can take on the same value θ k for some k, i.e. the θ i s cluster together. In this section we are concerned with representations of Dirichlet processes based upon both the clustering property and the sum of point masses. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 36 / 90

50 Dirichlet Processes Representations of Dirichlet Processes Suppose G DP(α, H). G is a (random) probability measure over X. We can treat it as a distribution over X. Let θ 1,..., θ n G be a random variable with distribution G. We saw in the demo that draws from a Dirichlet process seem to be discrete distributions. If so, then: G = π k δ θ k k=1 and there is positive probability that θ i s can take on the same value θ k for some k, i.e. the θ i s cluster together. In this section we are concerned with representations of Dirichlet processes based upon both the clustering property and the sum of point masses. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 36 / 90

51 Posterior Dirichlet Processes Sampling from a Dirichlet Process Suppose G is Dirichlet process distributed: G DP(α, H) G is a (random) probability measure over X. We can treat it as a distribution over X. Let θ G be a random variable with distribution G. We are interested in: p(θ) = p(θ G)p(G) dg p(g θ) = p(θ G)p(G) p(θ) Yee Whye Teh (Gatsby) DP August 2007 / MLSS 37 / 90

52 Posterior Dirichlet Processes Conjugacy between Dirichlet Distribution and Multinomial Consider: (π 1,..., π K ) Dirichlet(α 1,..., α K ) z (π 1,..., π K ) Discrete(π 1,..., π K ) z is a multinomial variate, taking on value i {1,..., n} with probability π i. Then: ( z Discrete Pα1 i α i,..., ) P α K i α i (π 1,..., π K ) z Dirichlet(α 1 + δ 1 (z),..., α K + δ K (z)) where δ i (z) = 1 if z takes on value i, 0 otherwise. Converse also true. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 38 / 90

53 Posterior Dirichlet Processes Fix a partition (A 1,..., A K ) of X. Then (G(A 1 ),..., G(A K )) Dirichlet(αH(A 1 ),..., αh(a K )) P(θ A i G) = G(A i ) Using Dirichlet-multinomial conjugacy, P(θ A i ) = H(A i ) (G(A 1 ),..., G(A K )) θ Dirichlet(αH(A 1 )+δ θ (A 1 ),..., αh(a K )+δ θ (A K )) The above is true for every finite partition of X. In particular, taking a really fine partition, p(θ)dθ = H(dθ) Also, the posterior G θ is also a Dirichlet process: ( G θ DP α + 1, αh + δ ) θ α + 1 Yee Whye Teh (Gatsby) DP August 2007 / MLSS 39 / 90

54 Posterior Dirichlet Processes Fix a partition (A 1,..., A K ) of X. Then (G(A 1 ),..., G(A K )) Dirichlet(αH(A 1 ),..., αh(a K )) P(θ A i G) = G(A i ) Using Dirichlet-multinomial conjugacy, P(θ A i ) = H(A i ) (G(A 1 ),..., G(A K )) θ Dirichlet(αH(A 1 )+δ θ (A 1 ),..., αh(a K )+δ θ (A K )) The above is true for every finite partition of X. In particular, taking a really fine partition, p(θ)dθ = H(dθ) Also, the posterior G θ is also a Dirichlet process: ( G θ DP α + 1, αh + δ ) θ α + 1 Yee Whye Teh (Gatsby) DP August 2007 / MLSS 39 / 90

55 Posterior Dirichlet Processes Fix a partition (A 1,..., A K ) of X. Then (G(A 1 ),..., G(A K )) Dirichlet(αH(A 1 ),..., αh(a K )) P(θ A i G) = G(A i ) Using Dirichlet-multinomial conjugacy, P(θ A i ) = H(A i ) (G(A 1 ),..., G(A K )) θ Dirichlet(αH(A 1 )+δ θ (A 1 ),..., αh(a K )+δ θ (A K )) The above is true for every finite partition of X. In particular, taking a really fine partition, p(θ)dθ = H(dθ) Also, the posterior G θ is also a Dirichlet process: ( G θ DP α + 1, αh + δ ) θ α + 1 Yee Whye Teh (Gatsby) DP August 2007 / MLSS 39 / 90

56 Posterior Dirichlet Processes Fix a partition (A 1,..., A K ) of X. Then (G(A 1 ),..., G(A K )) Dirichlet(αH(A 1 ),..., αh(a K )) P(θ A i G) = G(A i ) Using Dirichlet-multinomial conjugacy, P(θ A i ) = H(A i ) (G(A 1 ),..., G(A K )) θ Dirichlet(αH(A 1 )+δ θ (A 1 ),..., αh(a K )+δ θ (A K )) The above is true for every finite partition of X. In particular, taking a really fine partition, p(θ)dθ = H(dθ) Also, the posterior G θ is also a Dirichlet process: ( G θ DP α + 1, αh + δ ) θ α + 1 Yee Whye Teh (Gatsby) DP August 2007 / MLSS 39 / 90

57 Posterior Dirichlet Processes G DP(α, H) θ G G θ H ( ) G θ DP α + 1, αh+δ θ α+1 Yee Whye Teh (Gatsby) DP August 2007 / MLSS 40 / 90

58 Blackwell-MacQueen Urn Scheme First sample: θ 1 G G G DP(α, H) θ 1 H G θ 1 DP(α + 1, αh+δ θ 1 α+1 ) Second sample: θ 2 θ 1, G G G θ 1 DP(α + 1, αh+δ θ 1 α+1 ) θ 2 θ 1 αh+δ θ 1 α+1 G θ 1, θ 2 DP(α + 2, αh+δ θ 1 +δ θ2 α+2 ) n th sample θ n θ 1:n 1, G G θ n θ 1:n 1 αh+p n 1 i=1 δ θ i α+n 1 G θ 1:n 1 DP(α + n 1, αh+p n 1 i=1 δ θ i G θ 1:n DP(α + n, αh+ α+n 1 ) P n i=1 δ θ i α+n ) Yee Whye Teh (Gatsby) DP August 2007 / MLSS 41 / 90

59 Blackwell-MacQueen Urn Scheme First sample: θ 1 G G G DP(α, H) θ 1 H G θ 1 DP(α + 1, αh+δ θ 1 α+1 ) Second sample: θ 2 θ 1, G G G θ 1 DP(α + 1, αh+δ θ 1 α+1 ) θ 2 θ 1 αh+δ θ 1 α+1 G θ 1, θ 2 DP(α + 2, αh+δ θ 1 +δ θ2 α+2 ) n th sample θ n θ 1:n 1, G G θ n θ 1:n 1 αh+p n 1 i=1 δ θ i α+n 1 G θ 1:n 1 DP(α + n 1, αh+p n 1 i=1 δ θ i G θ 1:n DP(α + n, αh+ α+n 1 ) P n i=1 δ θ i α+n ) Yee Whye Teh (Gatsby) DP August 2007 / MLSS 41 / 90

60 Blackwell-MacQueen Urn Scheme First sample: θ 1 G G G DP(α, H) θ 1 H G θ 1 DP(α + 1, αh+δ θ 1 α+1 ) Second sample: θ 2 θ 1, G G G θ 1 DP(α + 1, αh+δ θ 1 α+1 ) θ 2 θ 1 αh+δ θ 1 α+1 G θ 1, θ 2 DP(α + 2, αh+δ θ 1 +δ θ2 α+2 ) n th sample θ n θ 1:n 1, G G θ n θ 1:n 1 αh+p n 1 i=1 δ θ i α+n 1 G θ 1:n 1 DP(α + n 1, αh+p n 1 i=1 δ θ i G θ 1:n DP(α + n, αh+ α+n 1 ) P n i=1 δ θ i α+n ) Yee Whye Teh (Gatsby) DP August 2007 / MLSS 41 / 90

61 Blackwell-MacQueen Urn Scheme Blackwell-MacQueen urn scheme produces a sequence θ 1, θ 2,... with the following conditionals: θ n θ 1:n 1 αh + n 1 i=1 δ θ i α + n 1 Picking balls of different colors from an urn: Start with no balls in the urn. with probability α, draw θ n H, and add a ball of that color into the urn. With probability n 1, pick a ball at random from the urn, record θ n to be its color, return the ball into the urn and place a second ball of same color into urn. Blackwell-MacQueen urn scheme is like a representer for the DP a finite projection of an infinite object. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 42 / 90

62 Exchangeability and de Finetti s Theorem Starting with a DP, we constructed Blackwell-MacQueen urn scheme. The reverse is possible using de Finetti s Theorem. Since θ i are iid G, their joint distribution is invariant to permutations, thus θ 1, θ 2,... are exchangeable. Thus a distribution over measures must exist making them iid. This is the DP. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 43 / 90

63 Chinese Restaurant Process Draw θ 1,..., θ n from a Blackwell-MacQueen urn scheme. They take on K < n distinct values, say θ 1,..., θ K. This defines a partition of 1,..., n into K clusters, such that if i is in cluster k, then θ i = θ k. Random draws θ 1,..., θ n from a Blackwell-MacQueen urn scheme induces a random partition of 1,..., n. The induced distribution over partitions is a Chinese restaurant process (CRP). Yee Whye Teh (Gatsby) DP August 2007 / MLSS 44 / 90

64 Chinese Restaurant Process Generating from the CRP: First customer sits at the first table. Customer n sits at: Table k with probability at table k. A new table K + 1 with probability n k α+n 1 where n k is the number of customers α α+n 1. Customers integers, tables clusters. The CRP exhibits the clustering property of the DP. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 45 / 90

65 Chinese Restaurant Process Generating from the CRP: First customer sits at the first table. Customer n sits at: Table k with probability at table k. A new table K + 1 with probability n k α+n 1 where n k is the number of customers α α+n 1. Customers integers, tables clusters. The CRP exhibits the clustering property of the DP. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 45 / 90

66 Chinese Restaurant Process To get back from the CRP to Blackwell-MacQueen urn scheme, simply draw θ k H for k = 1,..., K, then for i = 1,..., n set θ i = θ z i where z i is the table that customer i sat at. The CRP teases apart the clustering property of the DP, from the base distribution. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 46 / 90

67 Stick-breaking Construction Returning to the posterior process: G DP(α, H) θ G G θ H G θ DP(α + 1, αh+δ θ α+1 ) Consider a partition (θ, X\θ) of X. We have: (G(θ), G(X\θ)) Dirichlet((α + 1) αh+δ θ α+1 (θ), (α + 1) αh+δ θ α+1 (X\θ)) = Dirichlet(1, α) G has a point mass located at θ: G = βδ θ + (1 β)g with β Beta(1, α) and G is the (renormalized) probability measure with the point mass removed. What is G? Yee Whye Teh (Gatsby) DP August 2007 / MLSS 47 / 90

68 Stick-breaking Construction Returning to the posterior process: G DP(α, H) θ G G θ H G θ DP(α + 1, αh+δ θ α+1 ) Consider a partition (θ, X\θ) of X. We have: (G(θ), G(X\θ)) Dirichlet((α + 1) αh+δ θ α+1 (θ), (α + 1) αh+δ θ α+1 (X\θ)) = Dirichlet(1, α) G has a point mass located at θ: G = βδ θ + (1 β)g with β Beta(1, α) and G is the (renormalized) probability measure with the point mass removed. What is G? Yee Whye Teh (Gatsby) DP August 2007 / MLSS 47 / 90

69 Stick-breaking Construction Returning to the posterior process: G DP(α, H) θ G G θ H G θ DP(α + 1, αh+δ θ α+1 ) Consider a partition (θ, X\θ) of X. We have: (G(θ), G(X\θ)) Dirichlet((α + 1) αh+δ θ α+1 (θ), (α + 1) αh+δ θ α+1 (X\θ)) = Dirichlet(1, α) G has a point mass located at θ: G = βδ θ + (1 β)g with β Beta(1, α) and G is the (renormalized) probability measure with the point mass removed. What is G? Yee Whye Teh (Gatsby) DP August 2007 / MLSS 47 / 90

70 Stick-breaking Construction Returning to the posterior process: G DP(α, H) θ G G θ H G θ DP(α + 1, αh+δ θ α+1 ) Consider a partition (θ, X\θ) of X. We have: (G(θ), G(X\θ)) Dirichlet((α + 1) αh+δ θ α+1 (θ), (α + 1) αh+δ θ α+1 (X\θ)) = Dirichlet(1, α) G has a point mass located at θ: G = βδ θ + (1 β)g with β Beta(1, α) and G is the (renormalized) probability measure with the point mass removed. What is G? Yee Whye Teh (Gatsby) DP August 2007 / MLSS 47 / 90

71 Stick-breaking Construction Currently, we have: G DP(α, H) θ G G DP(α + 1, αh+δ θ α+1 ) G = βδ θ + (1 β)g θ H β Beta(1, α) Consider a further partition (θ, A 1,..., A K ) of X: (G(θ), G(A 1 ),..., G(A K )) =(β, (1 β)g (A 1 ),..., (1 β)g (A K )) The agglomerative/decimative property of Dirichlet implies: (G (A 1 ),..., G (A K )) Dirichlet(αH(A 1 ),..., αh(a K )) G DP(α, H) Yee Whye Teh (Gatsby) DP August 2007 / MLSS 48 / 90

72 Stick-breaking Construction Currently, we have: G DP(α, H) θ G G DP(α + 1, αh+δ θ α+1 ) G = βδ θ + (1 β)g θ H β Beta(1, α) Consider a further partition (θ, A 1,..., A K ) of X: (G(θ), G(A 1 ),..., G(A K )) =(β, (1 β)g (A 1 ),..., (1 β)g (A K )) Dirichlet(1, αh(a 1 ),..., αh(a K )) The agglomerative/decimative property of Dirichlet implies: (G (A 1 ),..., G (A K )) Dirichlet(αH(A 1 ),..., αh(a K )) G DP(α, H) Yee Whye Teh (Gatsby) DP August 2007 / MLSS 48 / 90

73 Stick-breaking Construction Currently, we have: G DP(α, H) θ G G DP(α + 1, αh+δ θ α+1 ) G = βδ θ + (1 β)g θ H β Beta(1, α) Consider a further partition (θ, A 1,..., A K ) of X: (G(θ), G(A 1 ),..., G(A K )) =(β, (1 β)g (A 1 ),..., (1 β)g (A K )) Dirichlet(1, αh(a 1 ),..., αh(a K )) The agglomerative/decimative property of Dirichlet implies: (G (A 1 ),..., G (A K )) Dirichlet(αH(A 1 ),..., αh(a K )) G DP(α, H) Yee Whye Teh (Gatsby) DP August 2007 / MLSS 48 / 90

74 Stick-breaking Construction We have: where G DP(α, H) G = β 1 δ θ 1 + (1 β 1 )G 1 G = β 1 δ θ 1 + (1 β 1 )(β 2 δ θ 2 + (1 β 2 )G 2 ). G = π k δ θ k k=1 π k = β k k 1 i=1 (1 β i) β k Beta(1, α) θ k H This is the stick-breaking construction. Demo: SBgenerate Yee Whye Teh (Gatsby) DP August 2007 / MLSS 49 / 90

75 Stick-breaking Construction Starting with a DP, we showed that draws from the DP looks like a sum of point masses, with masses drawn from a stick-breaking construction. The steps are limited by assumptions of regularity on X and smoothness on H. [Sethuraman 1994] started with the stick-breaking construction, and showed that draws are indeed DP distributed, under very general conditions. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 50 / 90

76 Dirichlet Processes Representations of Dirichlet Processes Posterior Dirichlet process: G DP(α, H) θ G G Blackwell-MacQueen urn scheme: Chinese restaurant process: θ n θ 1:n 1 αh + n 1 i=1 δ θ i α + n 1 p(customer n sat at table k past) = i=1 θ H ( ) G θ DP α + 1, αh+δ θ α+1 { nk n 1+α α n 1+α if occupied table if new table Stick-breaking construction: k 1 π k = β k (1 β i ) β k Beta(1, α) θk H G = π k δ θ k k=1 Yee Whye Teh (Gatsby) DP August 2007 / MLSS 51 / 90

77 Outline 1 Applications 2 Dirichlet Processes 3 Representations of Dirichlet Processes 4 Modelling Data with Dirichlet Processes 5 Practical Course Yee Whye Teh (Gatsby) DP August 2007 / MLSS 52 / 90

78 Density Estimation Recall our approach to density estimation with Dirichlet processes: G DP(α, H) x i G The above does not work. Why? Problem: G is a discrete distribution; in particular it has no density! Solution: Convolve the DP with a smooth distribution: G DP(α, H) F x ( ) = F ( θ)dg(θ) x i F x G = π k δ θ k k=1 F x ( ) = π k F ( θk ) k=1 x i F x Yee Whye Teh (Gatsby) DP August 2007 / MLSS 53 / 90

79 Density Estimation Recall our approach to density estimation with Dirichlet processes: G DP(α, H) x i G The above does not work. Why? Problem: G is a discrete distribution; in particular it has no density! Solution: Convolve the DP with a smooth distribution: G DP(α, H) F x ( ) = F ( θ)dg(θ) x i F x G = π k δ θ k k=1 F x ( ) = π k F ( θk ) k=1 x i F x Yee Whye Teh (Gatsby) DP August 2007 / MLSS 53 / 90

80 Density Estimation Recall our approach to density estimation with Dirichlet processes: G DP(α, H) x i G The above does not work. Why? Problem: G is a discrete distribution; in particular it has no density! Solution: Convolve the DP with a smooth distribution: G DP(α, H) F x ( ) = F ( θ)dg(θ) x i F x G = π k δ θ k k=1 F x ( ) = π k F ( θk ) k=1 x i F x Yee Whye Teh (Gatsby) DP August 2007 / MLSS 53 / 90

81 Density Estimation !15!10! F ( µ, Σ) is Gaussian with mean µ, covariance Σ. H(µ, Σ) is Gaussian-inverse-Wishart conjugate prior. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 54 / 90

82 Density Estimation !15!10! F ( µ, Σ) is Gaussian with mean µ, covariance Σ. H(µ, Σ) is Gaussian-inverse-Wishart conjugate prior. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 54 / 90

83 Clustering Recall our approach to density estimation: G = π k δ θ k DP(α, H) k=1 F x ( ) = π k F ( θk ) k=1 Above model equivalent to: x i F x z i Discrete(π) θ i = θ z i x i z i F ( θ i ) = F ( θ z i ) This is simply a mixture model with an infinite number of components. This is called a DP mixture model. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 55 / 90

84 Clustering DP mixture models are used in a variety of clustering applications, where the number of clusters is not known a priori. They are also used in applications in which we believe the number of clusters grows without bound as the amount of data grows. DPs have also found uses in applications beyond clustering, where the number of latent objects is not known or unbounded. Nonparametric probabilistic context free grammars. Visual scene analysis. Infinite hidden Markov models/trees. Haplotype inference.... In many such applications it is important to be able to model the same set of objects in different contexts. This corresponds to the problem of grouped clustering and can be tackled using hierarchical Dirichlet processes. [Teh et al. 2006] Yee Whye Teh (Gatsby) DP August 2007 / MLSS 56 / 90

85 Grouped Clustering Yee Whye Teh (Gatsby) DP August 2007 / MLSS 57 / 90

86 Grouped Clustering Yee Whye Teh (Gatsby) DP August 2007 / MLSS 57 / 90

87 Hierarchical Dirichlet Processes H Hierarchical Dirichlet process: G 0 γ, H DP(γ, H) G j α, G 0 DP(α, G 0 ) θ ji G j G j γ α G 0 G j θji i = 1,..., n j = 1,..., J Yee Whye Teh (Gatsby) DP August 2007 / MLSS 58 / 90

88 Hierarchical Dirichlet Processes G 0 γ, H DP(γ, H) G 0 = β k δ θ k β γ Stick(γ) k=1 G j α, G 0 DP(α, G 0 ) G j = π jk δ θ k π j α, β DP(α, β) k=1 G 0 θ k H H G 1 G 2 Yee Whye Teh (Gatsby) DP August 2007 / MLSS 59 / 90

89 Summary Dirichlet process is just a glorified Dirichlet distribution. Draws from a DP are probability measures consisting of a weighted sum of point masses. Many representations: Blackwell-MacQueen urn scheme, Chinese restaurant process, stick-breaking construction. DP mixture models are mixture models with countably infinite number of components. I have not delved into: Applications. Generalizations, extensions, other nonparametric processes. Inference: MCMC sampling, variational approximation. Also see the tutorial material from Ghaharamani, Jordan and Tresp. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 60 / 90

90 Summary Dirichlet process is just a glorified Dirichlet distribution. Draws from a DP are probability measures consisting of a weighted sum of point masses. Many representations: Blackwell-MacQueen urn scheme, Chinese restaurant process, stick-breaking construction. DP mixture models are mixture models with countably infinite number of components. I have not delved into: Applications. Generalizations, extensions, other nonparametric processes. Inference: MCMC sampling, variational approximation. Also see the tutorial material from Ghaharamani, Jordan and Tresp. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 60 / 90

91 Bayesian Nonparametrics Parametric models can only capture a bounded amount of information from data, since they have bounded complexity. Real data is often complex and the parametric assumption is often wrong. Nonparametric models allow relaxation of the parametric assumption, bringing significant flexibility to our models of the world. Nonparametric models can also often lead to model selection/averaging behaviours without the cost of actually doing model selection/averaging. Nonparametric models are gaining popularity, spurred by growth in computational resources and inference algorithms. In addition to DPs, HDPs and their generalizations, other nonparametric models include Indian buffet processes, beta processes, tree processes... [Tutorials at Workshop on Bayesian Nonparametric Regression, Isaac Newton Institute, Cambridge, July 2007] Yee Whye Teh (Gatsby) DP August 2007 / MLSS 61 / 90

92 Bayesian Nonparametrics Parametric models can only capture a bounded amount of information from data, since they have bounded complexity. Real data is often complex and the parametric assumption is often wrong. Nonparametric models allow relaxation of the parametric assumption, bringing significant flexibility to our models of the world. Nonparametric models can also often lead to model selection/averaging behaviours without the cost of actually doing model selection/averaging. Nonparametric models are gaining popularity, spurred by growth in computational resources and inference algorithms. In addition to DPs, HDPs and their generalizations, other nonparametric models include Indian buffet processes, beta processes, tree processes... [Tutorials at Workshop on Bayesian Nonparametric Regression, Isaac Newton Institute, Cambridge, July 2007] Yee Whye Teh (Gatsby) DP August 2007 / MLSS 61 / 90

93 Bayesian Nonparametrics Parametric models can only capture a bounded amount of information from data, since they have bounded complexity. Real data is often complex and the parametric assumption is often wrong. Nonparametric models allow relaxation of the parametric assumption, bringing significant flexibility to our models of the world. Nonparametric models can also often lead to model selection/averaging behaviours without the cost of actually doing model selection/averaging. Nonparametric models are gaining popularity, spurred by growth in computational resources and inference algorithms. In addition to DPs, HDPs and their generalizations, other nonparametric models include Indian buffet processes, beta processes, tree processes... [Tutorials at Workshop on Bayesian Nonparametric Regression, Isaac Newton Institute, Cambridge, July 2007] Yee Whye Teh (Gatsby) DP August 2007 / MLSS 61 / 90

94 Bayesian Nonparametrics Parametric models can only capture a bounded amount of information from data, since they have bounded complexity. Real data is often complex and the parametric assumption is often wrong. Nonparametric models allow relaxation of the parametric assumption, bringing significant flexibility to our models of the world. Nonparametric models can also often lead to model selection/averaging behaviours without the cost of actually doing model selection/averaging. Nonparametric models are gaining popularity, spurred by growth in computational resources and inference algorithms. In addition to DPs, HDPs and their generalizations, other nonparametric models include Indian buffet processes, beta processes, tree processes... [Tutorials at Workshop on Bayesian Nonparametric Regression, Isaac Newton Institute, Cambridge, July 2007] Yee Whye Teh (Gatsby) DP August 2007 / MLSS 61 / 90

95 Bayesian Nonparametrics Parametric models can only capture a bounded amount of information from data, since they have bounded complexity. Real data is often complex and the parametric assumption is often wrong. Nonparametric models allow relaxation of the parametric assumption, bringing significant flexibility to our models of the world. Nonparametric models can also often lead to model selection/averaging behaviours without the cost of actually doing model selection/averaging. Nonparametric models are gaining popularity, spurred by growth in computational resources and inference algorithms. In addition to DPs, HDPs and their generalizations, other nonparametric models include Indian buffet processes, beta processes, tree processes... [Tutorials at Workshop on Bayesian Nonparametric Regression, Isaac Newton Institute, Cambridge, July 2007] Yee Whye Teh (Gatsby) DP August 2007 / MLSS 61 / 90

96 Outline 1 Applications 2 Dirichlet Processes 3 Representations of Dirichlet Processes 4 Modelling Data with Dirichlet Processes 5 Practical Course Yee Whye Teh (Gatsby) DP August 2007 / MLSS 62 / 90

97 Exploring the Dirichlet Process Before using DPs, it is important to understand its properties, so that we understand what prior assumptions we are imposing on our models. In this practical course we shall work towards implementing a DP mixture model to cluster NIPS papers, thus the relevant properties are the clustering properties of the DP. Consider the Chinese restaurant process representation of DPs: First customer sits at the first table. Customer n sits at: Table k with probability at table k. A new table K + 1 with probability n k α+n 1 where n k is the number of customers α α+n 1. How does number of clusters K scale as a function of α and of n (on average)? How does the number n k of customers sitting around table k depend on k and n (on average)? Yee Whye Teh (Gatsby) DP August 2007 / MLSS 63 / 90

98 Exploring the Dirichlet Process Before using DPs, it is important to understand its properties, so that we understand what prior assumptions we are imposing on our models. In this practical course we shall work towards implementing a DP mixture model to cluster NIPS papers, thus the relevant properties are the clustering properties of the DP. Consider the Chinese restaurant process representation of DPs: First customer sits at the first table. Customer n sits at: Table k with probability at table k. A new table K + 1 with probability n k α+n 1 where n k is the number of customers α α+n 1. How does number of clusters K scale as a function of α and of n (on average)? How does the number n k of customers sitting around table k depend on k and n (on average)? Yee Whye Teh (Gatsby) DP August 2007 / MLSS 63 / 90

99 Exploring the Dirichlet Process Before using DPs, it is important to understand its properties, so that we understand what prior assumptions we are imposing on our models. In this practical course we shall work towards implementing a DP mixture model to cluster NIPS papers, thus the relevant properties are the clustering properties of the DP. Consider the Chinese restaurant process representation of DPs: First customer sits at the first table. Customer n sits at: Table k with probability at table k. A new table K + 1 with probability n k α+n 1 where n k is the number of customers α α+n 1. How does number of clusters K scale as a function of α and of n (on average)? How does the number n k of customers sitting around table k depend on k and n (on average)? Yee Whye Teh (Gatsby) DP August 2007 / MLSS 63 / 90

100 Exploring the Dirichlet Process Before using DPs, it is important to understand its properties, so that we understand what prior assumptions we are imposing on our models. In this practical course we shall work towards implementing a DP mixture model to cluster NIPS papers, thus the relevant properties are the clustering properties of the DP. Consider the Chinese restaurant process representation of DPs: First customer sits at the first table. Customer n sits at: Table k with probability at table k. A new table K + 1 with probability n k α+n 1 where n k is the number of customers α α+n 1. How does number of clusters K scale as a function of α and of n (on average)? How does the number n k of customers sitting around table k depend on k and n (on average)? Yee Whye Teh (Gatsby) DP August 2007 / MLSS 63 / 90

101 Exploring the Pitman-Yor Process Sometimes the assumptions embedded in using DPs to model data are inappropriate. The Pitman-Yor process is a generalization of the DP that often has more appropriate properties. It has two parameters: d and α with 0 d < 1 and α > d. When d = 0 the Pitman-Yor process reduces to a DP. It also has a Chinese restaurant process representation: First customer sits at the first table. Customer n sits at: Table k with probability n k d where n α+n 1 k is the number of customers at table k. A new table K + 1 with probability α+kd. α+n 1 How does K scale as a function of n, α and d (on average)? Yee Whye Teh (Gatsby) DP August 2007 / MLSS 64 / 90

102 Exploring the Pitman-Yor Process Sometimes the assumptions embedded in using DPs to model data are inappropriate. The Pitman-Yor process is a generalization of the DP that often has more appropriate properties. It has two parameters: d and α with 0 d < 1 and α > d. When d = 0 the Pitman-Yor process reduces to a DP. It also has a Chinese restaurant process representation: First customer sits at the first table. Customer n sits at: Table k with probability n k d where n α+n 1 k is the number of customers at table k. A new table K + 1 with probability α+kd. α+n 1 How does K scale as a function of n, α and d (on average)? Yee Whye Teh (Gatsby) DP August 2007 / MLSS 64 / 90

103 Exploring the Pitman-Yor Process Sometimes the assumptions embedded in using DPs to model data are inappropriate. The Pitman-Yor process is a generalization of the DP that often has more appropriate properties. It has two parameters: d and α with 0 d < 1 and α > d. When d = 0 the Pitman-Yor process reduces to a DP. It also has a Chinese restaurant process representation: First customer sits at the first table. Customer n sits at: Table k with probability n k d where n α+n 1 k is the number of customers at table k. A new table K + 1 with probability α+kd. α+n 1 How does K scale as a function of n, α and d (on average)? Yee Whye Teh (Gatsby) DP August 2007 / MLSS 64 / 90

104 Exploring the Pitman-Yor Process Sometimes the assumptions embedded in using DPs to model data are inappropriate. The Pitman-Yor process is a generalization of the DP that often has more appropriate properties. It has two parameters: d and α with 0 d < 1 and α > d. When d = 0 the Pitman-Yor process reduces to a DP. It also has a Chinese restaurant process representation: First customer sits at the first table. Customer n sits at: Table k with probability n k d where n α+n 1 k is the number of customers at table k. A new table K + 1 with probability α+kd. α+n 1 How does K scale as a function of n, α and d (on average)? Yee Whye Teh (Gatsby) DP August 2007 / MLSS 64 / 90

105 Dirichlet Process Mixture Models We model a data set x 1,..., x n using the following model: G DP(α, H) θ i G G x i θ i F( θ i ) for i = 1,..., n Each θ i is a latent parameter modelling x i, while G is the unknown distribution over parameters modelled using a DP. This is the basic DP mixture model. Implement a DP mixture model. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 65 / 90

106 Dirichlet Process Mixture Models We model a data set x 1,..., x n using the following model: G DP(α, H) θ i G G x i θ i F( θ i ) for i = 1,..., n Each θ i is a latent parameter modelling x i, while G is the unknown distribution over parameters modelled using a DP. This is the basic DP mixture model. Implement a DP mixture model. Yee Whye Teh (Gatsby) DP August 2007 / MLSS 65 / 90

107 Dirichlet Process Mixture Models Infinite Limit of Finite Mixture Models Different representations lead to different inference algorithms for DP mixture models. The most common are based on the Chinese restaurant process and on the stick-breaking construction. Here we shall work with the Chinese restaurant process representation, which, incidentally, can also be derived as the infinite limit of finite mixture models. A finite mixture model is defined as follows: θk H π Dirichlet(α/K,..., α/k ) z i π Discrete(π) x i θz i F ( θz i ) for k = 1,..., K for i = 1,..., n Yee Whye Teh (Gatsby) DP August 2007 / MLSS 66 / 90

108 Dirichlet Process Mixture Models Infinite Limit of Finite Mixture Models Different representations lead to different inference algorithms for DP mixture models. The most common are based on the Chinese restaurant process and on the stick-breaking construction. Here we shall work with the Chinese restaurant process representation, which, incidentally, can also be derived as the infinite limit of finite mixture models. A finite mixture model is defined as follows: θk H π Dirichlet(α/K,..., α/k ) z i π Discrete(π) x i θz i F ( θz i ) for k = 1,..., K for i = 1,..., n Yee Whye Teh (Gatsby) DP August 2007 / MLSS 66 / 90

109 Dirichlet Process Mixture Models Infinite Limit of Finite Mixture Models Different representations lead to different inference algorithms for DP mixture models. The most common are based on the Chinese restaurant process and on the stick-breaking construction. Here we shall work with the Chinese restaurant process representation, which, incidentally, can also be derived as the infinite limit of finite mixture models. A finite mixture model is defined as follows: θk H π Dirichlet(α/K,..., α/k ) z i π Discrete(π) x i θz i F ( θz i ) for k = 1,..., K for i = 1,..., n Yee Whye Teh (Gatsby) DP August 2007 / MLSS 66 / 90

110 Dirichlet Process Mixture Models Infinite Limit of Finite Mixture Models Different representations lead to different inference algorithms for DP mixture models. The most common are based on the Chinese restaurant process and on the stick-breaking construction. Here we shall work with the Chinese restaurant process representation, which, incidentally, can also be derived as the infinite limit of finite mixture models. A finite mixture model is defined as follows: θk H π Dirichlet(α/K,..., α/k ) z i π Discrete(π) x i θz i F ( θz i ) for k = 1,..., K for i = 1,..., n Yee Whye Teh (Gatsby) DP August 2007 / MLSS 66 / 90