Bayesian Nonparametrics

Transcription

1 Bayesian Nonparametrics Lorenzo Rosasco Class 18 April 11, 2011

2 About this class Goal To give an overview of some of the basic concepts in Bayesian Nonparametrics. In particular, to discuss Dirichelet processes and their several characterizations and properties.

3 Plan Parametrics, nonparametrics and priors A reminder on distributions Dirichelet processes Definition Stick Breaking Pólya Urn Scheme and Chinese processes

4 References and Acknowledgments This lecture heavily draws (sometimes literally) from the list of references below, which we suggest as further readings. Figures are taken either from Sudderth PhD thesis or Teh Tutorial. Main references/sources: See also: Yee Whye Teh, Tutorial in the Machine Learning Summer School, and his notes Dirichelet Processes. Erik Sudderth, PhD Thesis. Gosh and Ramamoorthi, Bayesian Nonparametrics, (book). Zoubin Ghahramani, Tutorial ICML. Michael Jordan, Nips Tutorial. Rasmussen, Williams, Gaussian Processes for Machine Learning, (book). Ferguson, paper in Annals of Statistics. Sethuraman, paper in Statistica Sinica. Berlinet, Thomas-Agnan, RKHS in Probability and Statistics, (book). Thanks to Dan, Rus and Charlie for various discussions.

5 Parametrics vs Nonparametrics We can illustrate the difference between the two approaches considering the following prototype problems. 1 function estimation 2 density estimation

6 (Parametric) Function Estimation Data, S = (X, Y ) = (x i, y i ) n i=1 Model, y i = f θ (x i ) + ɛ i, e.g. f θ (x) = θ, x and ɛ N (0, σ 2 ), σ > 0. prior θ P(θ) posterior P(θ X, Y ) = prediction P(y x, X, Y ) = P(θ)P(Y X, θ) P(Y X) P(y x, θ)p(θ X, Y )dθ

7 (Parametric) Density Estimation Data, S = (x i ) n i=1 Model, x i F θ prior θ P(θ) posterior prediction P(θ X) = P(θ)P(X θ) P(X) P(x X) = P(x θ)p(θ X)dθ

8 Nonparametrics: a Working Definition In the above models the number of parameters available for learning is fixed a priori. Ideally the more data we have, the more parameters we would like to explore. This is in essence the idea underlying nonparametric models.

9 The Right to a Prior A finite sequence is exchangeable if its distribution does not change under permutation of the indices. A sequence is infinitely exchangeable if any finite subsequence is exchangeable. De Finetti s Theorem If the random variables (x i ) i=1 are infinitely exchangeable, then there exists some space Θ and a corresponding distribution p(θ), such that the joint distribution of n observations is given by: n P(x 1,..., x n ) = P(θ) P(x i θ)dθ. Θ i=1

10 Question The previous classical result is often advocated as a justification for considering (possibly infinite dimensional) priors. Can we find computationally efficient nonparametric models? We already met one when we considered the Bayesian interpretation of regularization...

11 Reminder: Stochastic Processes Stochastic Process A family (X t ) : (Ω, P) R, t T, of random variables over some index set T. Note that: X t (ω), ω Ω, is a number, X t ( ) is a random variable, X ( ) (ω) : T R is a function and is called sample path.

12 Gaussian Processes GP(f 0, K ), Gaussian Process (GP) with mean f 0 and covariance function K A family (G x ) x X of random variables over X such that: for any x 1,..., x n in X, G x1,..., G xn is a multivariate Gaussian. We can define the mean f 0 : X R of the GP from the mean f 0 (x 1 ),..., f 0 (x n ) and the covariance function K : X X R settting K (x i, x j ) equal to the corresponding entries of covariance matrix. Then K is symm., pos. def. function. A sample path of the GP can be thought of as a random function f GP(f 0, K ).

13 (Nonparametric) Function Estimation Data, S = (X, Y ) = (x i, y i ) n i=1 Model, y i = f (x i ) + ɛ i prior f GP(f 0, K ) posterior P(f X, Y ) = prediction P(y x, X, Y ) = P(f )P(Y X, f ) P(Y X) P(y x, f )P(f X, Y )df We have seen that the last equation can be computed in closed form.

14 (Nonparametric) Density Estimation Dirichelet Processes (DP) will give us a way to build nonparametric priors for density estimation. Data, S = (x i ) n i=1 Model, x i F prior F DP(α, H) posterior prediction P(F X) = P(F)P(X F) P(X) P(x X) = P(x F)P(F X)dF

15 Plan Parametrics, nonparametrics and priors A reminder on distributions Dirichelet processes Definition Stick Breaking Pólya Urn Scheme and Chinese processes

16 Dirichelet Distribution It is a distribution over the K-dimensional simplex S K, i.e. x R K such that K i=1 x i = 1 and x i 0 for all i. The Dirichelet distribution is given by P(x) = P(x 1,..., x K ) = Γ( K i=1 αi ) K i=1 Γ(αi ) K (x i ) αi 1 where α = (α 1,..., α K ) is a parameter vector and Γ is the Gamma function. We write x Dir(α), i.e. x 1,..., x K Dir(α 1,..., α K ). i=1

17 Dirichelet Dirichlet Distribution Processes Examples of Dirichlet distributions. university-log Yee Whye Teh (Gatsby) DP August 2007 / MLSS 28 / 80

18 Reminder: Gamma Function and Beta Distribution The Gamma function γ(z) = 0 t z 1 e t dt. It is possible to prove that Γ(z + 1) = zγ(z). Beta Distribution Special case of the Dirichelet distribution given by K = 2. P(x α, β) = Γ(α + β) Γ(α) + Γ(β) x (α 1) (1 x) (β 1) Note that here x [0, 1] whereas for the Dirichelet distribution we would have x = (x 1, x 2 ) with x 1, x 2 > 0 and x 1 + x 2 = 1.

19 Beta Distribution " = 1, # = 1 " = 2, # = 2 " = 5, # = 3 " = 4, # = " = 1.0, # = 1.0 " = 1.0, # = 3.0 " = 1.0, # = 0.3 " = 0.3, # = p(! ",#) p(! ",#) ! ! For large parameters the distribution is unimodal. For small parameters it favors biased binomial distributions.! 2! 2

20 Properties of the Dirichelet Distribution Note that the K -simplex S K can be seen as the space of probabilities of a discrete (categorical) random variable with K possible values. Let α 0 = K i=1 αi. Expectation Variance Covariance E[x i ] = αi α 0. V[x i ] = αi (α 0 α i ) α 2 0 (α 0 + 1). Cov(x i, x j ) = α i α j α 2 0 (α 0 + 1).

21 Properties of the Dirichelet Distribution Aggregation: let (x 1,..., x K ) Dir(α 1,..., α K ) then (x 1 + x 2,..., x K ) Dir(α 1 + α 2,..., α K ). More generally, aggregation of any subset of the categories produces a Dirichelet distribution with parameters summed as above. The marginal distribution of any single component of a Dirichelet distribution follows a beta distribution.

22 Conjugate Priors Let X F and F P( α) = P α. P(F X, α) = P(F, α)p(x F, α) P(X, α) We say that P(F, α) is a conjugate prior for the likelihood P(X F, α) if, for any X and α, the posterior distribution P(F X, α) is in the same family of the prior. Moreover in this case the prior and the posterior distributions are then called conjugate distributions. The Dirichelet distribution is conjugate to the multinomial distribution

23 Multinomial Distribution Let X have values in {1,..., K }. Given π 1,..., π K define the probability mass function, multinomial distribution P(X π 1,..., π K ) = K i=1 π δ i (X) i. Given n observations the total probability of all possible sequences of length n taking those values is P(x 1,..., x n π 1,..., π K ) = n! K i=1 C i! K i=1 π C i i, where C i = n j=1 δ i(x j ) (C i is the number of observations with value i). For K = 2 this is just the binomial distribution.

24 Conjugate Posteriors and Predictions Given n observations S = x 1,..., x n from a multinomial distribution P( θ) with a Dirichelet prior P(θ α) we have P(θ S, α) P(θ α)p(s θ) K (θ i ) (α i +C i 1) Dir(α 1 + C 1,..., α K + C K ) i=1 where C i is the number of observations with value i.

25 Plan Parametrics, nonparametrics and priors A reminder on distributions Dirichelet processes Definition Stick Breaking Pólya Urn Scheme and Chinese processes

26 Dirichelet Processes Given a space X we denote with F a distribution on X and with F(X) the set of all possible distributions on X. Informal Description A Dirichelet process (DP) will be a distribution over F(X). A sample from a DP can be seen as a (random) probability distribution on X.

27 Dirichelet Processes (cont.) A partition of X is a collection of subsets B 1,..., B N is such that, if B i B j =, i j and N i=1 B i = X. Definition (Existence Theorem) Let α > 0 and H a probability distribution on X. One can prove that there exists a unique distribution DP(α, H) on F(X) such that, if F DP(α, H) and B 1,..., B N is a partition of X then (F(B 1 ),..., F(B N )) Dir(αH(B 1 ),..., αh(b N )). The above result is proved (Ferguson 73) using Kolmogorov s Consistency theorem (Kolmogorov 33).

28 Dirichelet Processes Illustrated Sec Dirichlet Processes 97 T 1 T 2 T 3 ~ T 1 ~ T 4 ~ T 5 ~ T 2 ~ T 3 Figure Dirichlet processes induce Dirichlet distributions on every finite, measurable partition. Left: An example base measure H on a bounded, two dimensional space Θ (darker regions have higher probability). Center: A partition with K = 3 cells. The weight that a random measure G DP(α,H) assigns to these cells follows a Dirichlet distribution (see eq. (2.166)). We shade each cell T k according to its mean E[G(T k)] = H(T k). Right: Another partition with K = 5 cells. The consistency of G implies, for example, that (G(T 1)+G(T 2)) and G( e T 1) follow identical beta distributions.

29 Dirichelet Processes (cont.) The previous definition is the one giving the name to the process. It is in fact also possible to show that a Dirichelet process corresponds to a stochastic process where the sample paths are probability distributions on X.

30 Properties of Dirichelet Processes Hereafter F DP(α, H) and A is a measurable set in X. Expectation: E[F(A)] = αh(a). Variance: V[F(A)] = H(A)(1 H(A)) α+1

31 Properties of Dirichelet Processes (cont.) Posterior and Conjugacy: let x F and consider a fixed partition B 1,..., B N, then P(F(B 1 ),..., F(B N ) x B k ) = Dir(αH(B 1 ),..., αh(b k ) + 1,..., αh(b N )). It is possible to prove that if S = (x 1,..., x n ) F, and F DP(α, H), then ( ( )) 1 n P(F S, α, H) = DP α + n, αh + δ xi n + α i=1

32 Plan Parametrics, nonparametrics and priors A reminder on distributions Dirichelet processes Definition Stick Breaking Pólya Urn Scheme and Chinese processes

33 A Qualitative Reasoning From the form of the posterior we have that ( E(F (A) S, α, H) = 1 αh(a) + n + α If α < and n one can argue that ) n δ xi (A). i=1 E(F(A) S, α, H) = π i δ xi (A) i=1 where (π i ) i=1 is the sequence corresponding to the limit lim n C i /n of the empirical frequencies of the observations (x i ) i=1. If the posterior concentrates about its mean the above reasoning suggests that the obtained distribution is discrete.

34 Stick Breaking Construction Explicit construction of a DP. Let α > 0, (π i ) i=1 such that i 1 i 1 π i = β i (1 β j ) = β i (1 π j ) j=1 where β i Beta(1, α), for all i. Let H be a distribution on X and define where θ i H, for all i. F = π i δ θi i=1 j=1

35 Stick Breaking Construction (cont.) it is possible to prove (Sethuraman 94) that the previous construction returns a DP and conversely a Dirichelet process is discrete almost surely.

36 Stick Breaking Construction: Interpretation Sec Dirichlet Processes 101 β 1 π 1 π β β 1 1 β 2! k ! k β3 π 3 π 4 1 β3 β 4 1 β 4 β k k ! k 0.2! k 0.2 π k k α =1 α =5 Figure Sequential stick breaking construction of the infinite set of mixture weights π GEM(α) corresponding to a measure G DP(α,H). Left: The first weight π 1 Beta(1, α). Each subsequent weight π k (red) is some random proportion β k (blue) of the remaining, unbroken stick of probability mass. Right: The first K = 20 weights generated by four random stick breaking constructions (two with α = 1, two with α = 5). Note that the weights π k do not monotonically decrease. The weights π partition a unit-length stick in an infinite set: the i-th weight is a random proportion β i of the stick remaining after sampling the first i 1 weights. discrete parameters {θ k } k=1. For a given α and dataset size N, there are strong bounds on the accuracy of particular finite truncations of this stick breaking process [147], which are often used in approximate computational methods [29, 147, 148, 289]. Several other stick breaking processes have been proposed which sample the proportions β from different distributions [147, 148, 233]. For example, the two parameter

37 The Role of the Strength Parameter Note that E[β i ] = 1/(1 + α). for small α, the first few components will have all the mass. for large α, F approaches the distribution H assigning uniform weights to the samples θ i.

38 Plan Parametrics, nonparametrics and priors A reminder on distributions Dirichelet processes Definition Stick Breaking Pólya Urn Scheme and Chinese processes

39 Pólya Urn Scheme The observation that a sample from a DP is discrete allows to simplify the form of the prediction distribution, ( E(F(A) S, α, H) = 1 αh(a) + n + α ) K N i δ xi (A). where N i are the number of observations with value i. In fact, It is possible to prove (Blackwell and MacQueen 94) that if the base measure has a density h, then ( P(x S, α, H) = 1 αh(x ) + n + α i=1 ) K N i δ xi (x ). i=1

40 Chinese Restaurant Processes The previous prediction distribution gives a distribution over partitions. Pitman and Dubins called it Chinese Restaurant Processes (CRP) inspired by the seemingly infinite seating capacity of restaurants in San Francisco s Chinatown.

41 Chinese Restaurant Processes (cont.) There is an infinite (countable) set of tables. First customer sits in the first table. Customer n sits at table k with probability n k α + n + 1, where n k is the number of customers at table k. Customer n sits at table k + 1 with probability α α + n + 1.

42 Number of Clusters and Strength Parameter It is possible to prove (Antoniak 77??) that the number of clusters K grows as α log n as we increase the number of observations n.

43 Dirichelet Process Mixture The clustering effect in DP arises from assuming that there are multiple observations having the same values. This is hardly the case in practice. Dirichelet Process Mixture (DPM) The above observation suggests to consider the following model, F DP(α, H), θ i F and x i G( θ i ). Usually G is a distribution in the exponential family and H = H(λ) a corresponding conjugate prior.

44 Dirichelet Process Mixture CRP give another representation of the DPM. Let z i denote the unique cluster associated to x i, then z i π and x i G(θ zi ). If we marginalize the indicator variables z i s we obtain an infinite mixture model P(x π, θ 1, θ 2,... ) = π i f (x θ i ) i=1

45 Dirichelet Process and Model Selection Rather than choosing a finite number of components K, the DP use the stick breaking construction to adapt the number of clusters to the data. The complexity of the model is controlled by the strength parameter α.

46 Conclusions DP provide a framework for nonparametric inference. Different characterizations shed light on different properties. DP mixtures allow to adapt the number of components to the number of samples......but the complexity of the model is controlled by the strength parameter α. Neither the posterior distribution nor the prediction distribution can be found analytically approximate inference is needed- see next class.

47 What about Generalizatoin Bounds? Note that ideally X F and F DP(α, H ) for some α, H and we can compute the posterior P = P(F X, α, H ). In practice we have only samples S = (x 1,..., x n ) F and have to choose α, H to compute P n = P(F S, α, H). [Consistency] Does P n approximate P (in some suitable sense)? [Model Selection] How should we choose α (and H)?