Hierarchical Bayesian Nonparametrics

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1 Hierarchical Bayesian Nonparametrics Micha Elsner April 11, 2013

2 2 For next time We ll tackle a paper: Green, de Marneffe, Bauer and Manning: Multiword Expression Identification with Tree Substitution Grammars: A Parsing tour de force with French You are required to read this paper by next Tuesday You will probably not understand all of it Figure out what you do and do not understand Is it clear what the task is? Is it clear how the results are measured? Which components have to do with features, modeling assumptions, estimation and inference? I also recommend you read (for background): Sag, Baldwin, Bond, Copestake and Flickinger: Multiword expressions: a pain in the neck for NLP Should be much easier, mostly an issues paper, not a models paper

3 Overview Last time: Bayesian methods replace parameters with random variables Specify priors over them Marginalize over them (model average) to get predictive probability Property called conjugacy allows computation of posterior Prior for clustering can be sparse (fewer clusters found than allowed) Extension of this idea allows model with any number of clusters This time: We ll discuss the Chinese Restaurant representation of the DP Use it to construct a language model Compare this model to Kneser-Ney And discuss some extensions 3

4 4 The Dirichlet process Recall: A prior over an infinite list of probabilities, θ θ = θ 1, θ 2... all sum to 1 In a given dataset, we observe items in groups 1... k Finite number Remaining groups have prior weight summing up to A (parameter of DP) If we ve seen documents 1...N which are placed in clusters indicated by C 1:N, with k occupied groups, the posterior is: P(C new = c) #(C 1:N = c) c k P(C new = k + 1) A

5 5 The Chinese Restaurant Standard representation for this is a metaphor: San Francisco Chinese restaurants force strangers to sit together... I think to save space? Imagine a big room full of tables... Some empty, some with people at them Documents are customers in the restaurant Tables are clusters doc 1 doc 3 doc 2 1 2?

6 6 New customers When a new customer (document) enters the restaurant, they have to sit somewhere (be in a cluster): But where? Depends on where people are already sitting Probability of sitting at table 1 2 Probability of sitting at table 2 1 Probability of sitting at another table A New table would get the number 3 But it s any of the infinite tables nobody is sitting at yet We ll just name the one you happen to pick 3 This walkthrough is the generative story for the CRP It doesn t explain an inference procedure! We might get there...

7 7 The CRP vs the DP Two ways of thinking about the same distribution: The CRP represents the posterior given C 1:N That is, our predictions about new data given the old data The DP represents our prior over the parameters θ To make predictions, we marginalize over θ P(C new = c C 1:n ) = theta θ P(C new = c θ)p(θ C 1:n ) P(C new = c θ)p(θ C 1:n ) P(C new = c θ)p(c 1:n θ)dp(θ; A) θ P(C new = c θ)p(c 1:n θ)dp(θ; A) = CRP(C new C 1:n ; A)

8 8 Clustering with the CRP A c i parameters for P(d C) d i N

9 9 Generative story Generative story: Start with A Generate the first document First choose c 1 (which cluster), then generate d i (document words) based on cluster No occupied tables: c1 is in cluster 1 Now generate c 2 Two choices: P(c 2 = 1) 1 (because c 1 is at table 1) Or P(c2 = 2) A (because 2 is the next empty table) Then generate d2 (will be like d 1 if the two are in the same cluster) Choice (sparsity) depends on A Large A: tons of clusters Small A: few clusters Number of clusters grows as we see more data

10 10 Multiple datasets We can extend this to a model of two related datasets: We expect the same clusters to show up in both But with different frequency Sports vs news in two different newspapers Social consciousness vs local consciousness in two towns Word the in two ngram contexts (what we re going to do!)

11 11 Hierarchical Chinese restaurant table 1 in d1 table 1 in d2 table 3 in d2 shared prior over clusters table 2 in d1 table 2 in d2 1 2? doc 1 doc 3 doc 2 dataset 1 doc 3 doc 1 doc 2 dataset 2 doc 4 1 2? 1 2 1?

12 Generative story Customer in restaurant 1 (document from collection 1) Has to sit at a table (find a cluster) Sits at table 1 with probability 2 Two customers already at table 1 That is, if collection 1 already has the sports cluster, new documents likely to be about sports Sits at new table c 3 with probability A 1 If sits at new table, we need to decide what cluster c 3 is going to represent Could be more sports, or something new To find out, take a sample from the prior... Send a customer to the prior If sits at table 1, c3 is part of meta-cluster 1 (sports) If at table 2, part of meta-cluster 2 If new table, new meta-cluster If meta-cluster sports is popular overall (occurs in lots of datasets)... More likely to be popular in dataset 1 as well But dataset 1 can have its own idiosyncracies 12

13 The model Since the number of customers in the prior depends on the table arrangement in the datasets: We now need more latent variables Data: D 1 1:N and D2 1:N Cluster assignments: C 1 1:N, C2 1:N Which table is each customer at? T 1 1:N, T 2 1:N Three A parameters, A 1, A 2, A 0 13

14 14 Predictive probability P dataset 1 (C 1 new = c C 1 1:N, C2 1:M ) #(C1 i = c)+a 1 P prior (C = c; A) Prior is based on table assignments: P prior (c) = #(tables labeled c in datasets 1, 2) #(all tables) We compute this based on the T latent variables

15 15 Okay, so... How this works: Two datasets share a common set of clusters Datasets reflect the common set, but can have their own preferences Customers in upper restaurant count tables in lower restaurant That is, they count types

16 16 Language models Let s apply this insight to LMs Our common prior will be the unigram model Our datasets will be ngram contexts I ll draw this diagram on the board The predictive probability is: P(w i+1 = y w i = x) = #(x, y) + A 1P uni (y) #(x, ) + A 1 Where P uni (prior term) is similar to a type count: P uni (x) = #(customers at table for x) #(customers) The C variables aren t latent in the LM, since they correspond to words The T variables are still latent, which complicates inference

17 17 Recall the Kneser-Ney LM Two insights about productivity: Some contexts more productive than others: San vs green Some words more productive than others: Francisco vs report We can get the second insight as a consequence of this model Productivity of contexts requires Pitman-Yor prior Slight generalization of the Dirichlet Not going to cover

18 18 Kneser-Ney s P uni Intuition: some words appear in many contexts While others are very restricted Francisco (in US English) usually appears after San report appears all over the place ˆP KNuni (b) = a 1if (a b) ever appears 1if (a x) ever appears Proportion of unique word types appearing before b ax

19 19 CRP model vs KN Kneser-Ney uses a P uni based on type counts Our DP model uses a P uni based almost on type counts But some types have extra counts Types with multiple tables in some datasets Why? If a word appears in many contexts... Can raise the prior by placing it at multiple tables Shift around latent variables to make it fit better

20 20 Extensions We understand how to build models over many datasets: Can make an LM with datasets for all kinds of things Document clusters Geographical regions Etc

21 21 Inference Models like this have latent variables for which table each word is at: Makes inference tricky All the words depend on all the other words Because we integrated out the infinite vector θ using the CRP Means it s tricky to do EM Popular way to do model averaging approximately... Approximate expectation by taking samples

22 Gibbs sampling Gibbs sampling To sample from joint distribution over many RVs: P(C 1:N D 1 1:N, D2 1:N, A) Initialize at random Do forever: For each RV in turn, compute distribution conditioned on other RVs: P(C 1 1 C 1 2:N, D 1 1:N...) Sample new value from that distribution: C 1 1 P(C ) Entire state (all rvs) is sample from P 22

23 23 Gibbs vs EM Like EM, Gibbs spends most of time computing marginals on latent variables Unlike EM, Gibbs also needs marginals on the parameters Unlike EM, Gibbs samples instead of maximizing or taking expectations Posterior usually increases on avg. (similar to likelihood) But can decrease slightly due to random chance... No explicit test for convergence Gibbs can escape local maxima But this happens with very low probability

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