1、. . . . . . . . . . . . . . . . . . . . . . . . . . . .24 6.3A Stick-breaking Approach to IBP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25 1Introduction to the Dirichlet Distribution An example of a pmf is an ordinary six-sided die - to sample the pmf you roll the die and prod
2、uce a number from one to six. But real dice are not exactly uniformly weighted, due to the laws of physics and the reality of manufacturing. A bag of 100 real dice is an example of a random pmf - to sample this random pmf you put your hand in the bag and draw out a die, that is, you draw a pmf. A ba
3、g of dice manufactured using a crude process 100 years ago will likely have probabilities that deviate wildly from the uniform pmf, whereas a bag of state-of-the-art dice used by Las Vegas casinos may have barely perceptible imperfections. We can model the randomness of pmfs with the Dirichlet distr
4、ibution. One application area where the Dirichlet has proved to be particularly useful is in modeling the distribu- tion of words in text documents 9. If we have a dictionary containing k possible words, then a particular document can be represented by a pmf of length k produced by normalizing the e
5、mpirical frequency of its words. A group of documents produces a collection of pmfs, and we can fi t a Dirichlet distribution to capture the variability of these pmfs. Diff erent Dirichlet distributions can be used to model documents by diff erent authors or documents on diff erent topics. In this s
6、ection, we describe the Dirichlet distribution and some of its properties. In Sections 1.2 and 1.4, we illustrate common modeling scenarios in which the Dirichlet is frequently used: fi rst, as a conjugate prior for the multinomial distribution in Bayesian statistics, and second, in the context of t
7、he compound Dirichlet (a.k.a. P olya distribution), which fi nds extensive use in machine learning and natural language processing. Then, in Section 2, we discuss how to generate realizations from the Dirichlet using three methods: urn-drawing, stick-breaking, and transforming Gamma random variables
8、. In Sections 3 and 6, we delve into Bayesian non-parametric statistics, introducing the Dirichlet process, the Chinese restaurant process, and the Indian buff et process. 1.1 Defi nition of the Dirichlet Distribution A pmf with k components lies on the (k 1)-dimensional probability simplex, which i
9、s a surface in Rk denoted by k and defi ned to be the set of vectors whose k components are non-negative and sum to 1, that is k= q Rk| Pk i=1qi = 1,qi 0 for i = 1,2,.,k. While the set klies in a k-dimensional space, k is itself a (k 1)-dimensional object. As an example, Fig. 1 shows the two-dimensi
10、onal probability simplex for k = 3 events lying in three-dimensional Euclidean space. Each point q in the simplex can be thought of as a probability mass function in its own right. This is because each component of q is non-negative, and the components sum to 1. The Dirichlet distribution can be tho
11、ught of as a probability distribution over the (k 1)-dimensional probability simplex k; that is, as a distribution over pmfs of length k. Dirichlet distribution:Let Q = Q1,Q2,.,Qk be a random pmf, that is Qi 0 for i = 1,2,.,k and Pk i=1Qi = 1. In addition, suppose that = 1,2,.,k, with i 0 for each i
12、, and let 0= Pk i=1i. Then, Q is said to have a Dirichlet distribution with parameter , which we denote by Q Dir(), if it has1 f(q;) = 0 if q is not a pmf, and if q is a pmf then f(q;) = (0) Qk i=1(i) k Y i=1 qi1 i ,(1) 1The density of the Dirichlet is positive only on the simplex, which as noted pr
13、eviously, is a (k 1)-dimensional object living in a k-dimensional space. Because the density must satisfy P(Q A) = R Af(q;)d(q) for some measure , we must restrict the measure to being over a (k 1)-dimensional space; otherwise, integrating over a (k 1)-dimensional subset of a k-dimensional space wil
14、l always give an integral of 0. Furthermore, to have a=(儀匀罚儁儁儀讀缁弋H缀栞礄圀椀褂嬃霃鞃霃攃罣噞晒搀漀挀愀昀搀攀攀挀昀戀攀攀愀攀最椀昀罣噞晒搀漀挀尀尀戀搀攀挀搀搀愀挀搀挀刀挀樀吀焀伀昀夀堀洀圀猀夀稀攀最夀礀礀攀瘀洀挀愀夀琀倀吀瘀夀稀儀罣嘀愀挀挀昀昀愀搀搀:栀艹艹艹u艹葠絶晹攀琀眀娀夀栀吀刀昀砀渀最夀唀儀匀砀爀娀爀稀焀洀一礀搀猀砀氀瘀洀漀栀瘀欀漀欀儀焀最罣噞晒噧晒葨罣鑞畞 攠臿羉祏r葓鹎啓f葭筟膘媍潧綂葙啎祏齢Sf葭罣扜扜珿罓豻呜疂耀舀(磬. . . . . . . . . . . . . . . . . . . . . . . .
15、. . . .24 6.3A Stick-breaking Approach to IBP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25 1Introduction to the Dirichlet Distribution An example of a pmf is an ordinary six-sided die - to sample the pmf you roll the die and produce a number from one to six. But real dice are
16、not exactly uniformly weighted, due to the laws of physics and the reality of manufacturing. A bag of 100 real dice is an example of a random pmf - to sample this random pmf you put your hand in the bag and draw out a die, that is, you draw a pmf. A bag of dice manufactured using a crude process 100
17、 years ago will likely have probabilities that deviate wildly from the uniform pmf, whereas a bag of state-of-the-art dice used by Las Vegas casinos may have barely perceptible imperfections. We can model the randomness of pmfs with the Dirichlet distribution. One application area where the Dirichle
18、t has proved to be particularly useful is in modeling the distribu- tion of words in text documents 9. If we have a dictionary containing k possible words, then a particular document can be represented by a pmf of length k produced by normalizing the empirical frequency of its words. A group of docu
19、ments produces a collection of pmfs, and we can fi t a Dirichlet distribution to capture the variability of these pmfs. Diff erent Dirichlet distributions can be used to model documents by diff erent authors or documents on diff erent topics. In this section, we describe the Dirichlet distribution a
20、nd some of its properties. In Sections 1.2 and 1.4, we illustrate common modeling scenarios in which the Dirichlet is frequently used: fi rst, as a conjugate prior for the multinomial distribution in Bayesian statistics, and second, in the context of the compound Dirichlet (a.k.a. P olya distributio
21、n), which fi nds extensive use in machine learning and natural language processing. Then, in Section 2, we discuss how to generate realizations from the Dirichlet using three methods: urn-drawing, stick-breaking, and transforming Gamma random variables. In Sections 3 and 6, we delve into Bayesian no
22、n-parametric statistics, introducing the Dirichlet process, the Chinese restaurant process, and the Indian buff et process. 1.1 Defi nition of the Dirichlet Distribution A pmf with k components lies on the (k 1)-dimensional probability simplex, which is a surface in Rk denoted by k and defi ned to b
23、e the set of vectors whose k components are non-negative and sum to 1, that is k= q Rk| Pk i=1qi = 1,qi 0 for i = 1,2,.,k. While the set klies in a k-dimensional space, k is itself a (k 1)-dimensional object. As an example, Fig. 1 shows the two-dimensional probability simplex for k = 3 events lying
24、in three-dimensional Euclidean space. Each point q in the simplex can be thought of as a probability mass function in it=m(儀匀汀缸儁儁儀讀缁%H缀渀翺堀椀脂漃澃漃礃贃锃锃锃暕颋癓呑搀漀挀挀搀攀挀挀昀挀戀搀搀搀攀愀最椀昀暕颋癓呑搀漀挀尀尀戀攀昀戀挀愀愀愀挀稀昀匀刀琀爀攀猀昀焀最攀夀圀砀稀砀欀礀栀戀甀夀琀稀愀儀椀嘀猀夀洀欀昀搀颋癓吀搀愀攀攀搀挀攀昀戀挀攀缀栀艹6艹葠絶晹儀砀焀堀搀甀琀瀀唀甀爀攀氀樀倀眀欀稀欀栀夀渀搀砀渀戀夀娀猀瘀焀吀礀戀搀爀稀圀暕瞋颀呖呗须瞋詟瑢昰歔饘呗须瞋煓
25、甀殘顙畿形畣彾颋呓须琀昀頀谀鐀洀欀匀疂耀舀(磬. . . . . . . . . . . . . . . . . . . . . . . . . . . .24 6.3A Stick-breaking Approach to IBP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25 1Introduction to the Dirichlet Distribution An example of a pmf is an ordinary six-sided die - to sample the pmf yo
26、u roll the die and produce a number from one to six. But real dice are not exactly uniformly weighted, due to the laws of physics and the reality of manufacturing. A bag of 100 real dice is an example of a random pmf - to sample this random pmf you put your hand in the bag and draw out a die, that i
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