The dissertation of Andreas Stolcke is approved: University of ...

More documents

Recommendations

Info

CHAPTER 2. FOUNDATIONS 22¡An explicit, probabilistic bias expressed by the prior +-, 450 .An implicit, heuristic bias as part of the choice of the topology of the search space (the search bias).¡Even if the merging operator(s) are defined such that all models in the space are reachable from theinitialmodel, this bias is still significant due to the local and sequential nature of the posterior probabilitymaximization.We will return to the question of how to relax the search bias by using less constrained search methods(Section 3.4.5).The remaining section of this chapter present some of the common mathematical tools needed ininstantiating the Bayesian model merging approach in the domain of probabilistic grammars. The followingchapters 3, 4 and 5 each describe one such instantiation.2.5.4 Minimum Description LengthBayesian inference based on posterior probabilities has an alternative formulation in terms offinformation-theoretic concepts. The dualism between the two formulations is useful both for a deeperunderstanding of the underlying principles, and for the construction of prior distributions (see Section 2.5.6below).The maximization of+-,+-,+-,4|_)g0w6450450 )À.implicit in Bayesian model inference is equivalent to minimizingL log +-, 4|_)h0'6|L log +-, 450ÁL log +-, )/. 450F£Information theory tells us that the negative logarithm of the probability of a discrete event b is the optimalcode word length for communicating an instance of b , so as to minimize the average code length of arepresentative message.description lengths.Accordingly, the terms in the above equation can be interpreted as message orSpecifically, L log +9, 450 is the description length of the model under the prior distribution;L log +-, )/. 450 corresponds to a description of the data ) using 4 as the model on which code lengthsare based. The negative log of the joint probability can therefore be interpreted as the total description lengthof model and data.Inference or estimation by minimum description length (MDL) (Rissanen 1983; Wallace & Freeman1987) is thus equivalent to, and a useful alternative conceptualization of posterior probability maximization. 99 The picture become somewhat more complex when distributions over continuous spaces are involved. For those the correspondingMDL formulation has to consider the optimal granularity of the discrete encoding. We will can conveniently avoid this complication asthe only formal use of description lengths will be to devise priors for discrete objects, namely, grammar structures.
Æ,È0ÈNÉ[[[N 1=00£CHAPTER 2. FOUNDATIONS 232.5.5 Structure vs. parameter priorsA grammatical model is here described in two stages:1. A model structure or topology is specified as a set of states, nonterminals, transitions, productions, etc.(depending on the type of model). These elements represent discrete choices as to which derivationsfrom the grammar can have non-zero probability.2. Conditional on a given structure, the model’s continuous parameters are specified. These are typicallymultinomial parameters.4TÃ—and the parameter part . The model prior +9, 450 can therefore be written asÄ U 4TÃWe will write 4Â60 to describe the decomposition of model 4 into the structure partUÄ. 4TÃE0Even this framework leaves some room for choice: as discussed earlier, one may choose to makethe structure specification very unconstrained, e.g., by allowing all probability parameters to take on non-zerovalues, effectively pushing the structure specification into the parameter choice. Examples of this will bediscussed in the following chapters.+-,+-,+-,;UÄ450w64TÃE02.5.5.1 Priors for multinomial parametersSince the continuous parameters in the grammar types dealt with in this thesis are all from multinomialdistributions, it is convenient to discuss some standard priors for this type of distribution at thispoint.Each multinomial represents a discrete, finite probabilistic choice of some event. Let ¢ be thenumber of choices in ST6,2U U =1£££ 0 a multinomial and, the probability parameters associated with eachchoice (only 1 of these parameters are W @ U @6 free since 1).¢JLA standard prior for multinomials is the Dirichlet distribution11@ (2.14)+-,SE0%6Å-,;ÆÆ =UÇ ^ ?where 1£££normalizing constant Æ 1£££7Å9,2ÆÆ—=1£££\@YX1are parameters of the prior which can be given an intuitive interpretation (see below). The0 is the ¢ -dimensional Beta function,Æ =,2Æ,2Æ—=Å-,;ÆÆ =10Á[[[0 6|W @ Æ—@is the total prior weight.One important reason for the use of the Dirichlet prior in the case of multinomial parameters(Cheeseman et al. 1988; Cooper & Herskovits 1992; Buntine 1992) is its mathematical expediency. It is1£££È ,2Æ 0%6Æ =The prior weights Æw@ determine the bias embodied in the prior: the prior expectation of U@ is Ç$^Ç0 , where
Page 1 and 2: The dissertation of Andreas Stolcke
Page 3 and 4: Bayesian Learning of Probabilistic
Page 5 and 6: iAcknowledgmentsLife and work in Be
Page 7 and 8: iiiContentsList of FiguresList of T
Page 9 and 10: CONTENTSv4.5.4 Summary and Discussi
Page 14 and 15: CHAPTER 1. INTRODUCTION 2Instance-b
Page 16 and 17: CHAPTER 1. INTRODUCTION 4A.0.830.33
Page 18 and 19: CHAPTER 1. INTRODUCTION 6the ¨ 0 l
Page 20 and 21: ..1 £££1; 450,1 £££1; 450CHAP
Page 22 and 23: VU=@U@@=U===UCHAPTER 2. FOUNDATIONS
Page 24 and 25: ,,vv,v,v,,directly. However, note t
Page 26 and 27: 4@@@@-@b@6@˜--@@@0@@@@@CHAPTER 2.
Page 28 and 29: 6tt,u ·¥¸¹u ,10ºtu ,2 10Yt ¸
Page 30 and 31: CHAPTER 2. FOUNDATIONS 18As more da
Page 32 and 33: CHAPTER 2. FOUNDATIONS 20Global mod
Page 36 and 37: ÊS==66@N,ÆÆ=NÆ00ÆÊ=S=N0Æ=#@0
Page 38 and 39: 666CHAPTER 2. FOUNDATIONS 262.5.7 P
Page 40 and 41: It, u¦¸¹u Ù 0w6¬tt,_, u Ù 0
Page 42 and 43: , uu!¸¹u Ù 0c6,u ,ÔÔ0 ö1 ö1
Page 44 and 45: CHAPTER 3. HIDDEN MARKOV MODELS 32T
Page 46 and 47: CHAPTER 3. HIDDEN MARKOV MODELS 34R
Page 48 and 49: 4ÿ= ê•4TÃE0&Ò¢¡? •ç1 Lht
Page 50 and 51: 6666ò U1ò +9,9. 4 20+-,¡ . 4 10C
Page 52 and 53: 2. For each candidate"I!computeLet"
Page 54 and 55: 6\“ç%&ät\“ç tè ä, u¦¸¹u
Page 56 and 57: , u1 ¸¼u Ù 0 and , u3 ¸¹u Ù 0
Page 58 and 59: CHAPTER 3. HIDDEN MARKOV MODELS 46l
Page 60 and 61: CHAPTER 3. HIDDEN MARKOV MODELS 48c
Page 62 and 63: CHAPTER 3. HIDDEN MARKOV MODELS 50d
Page 64 and 65: CHAPTER 3. HIDDEN MARKOV MODELS 520
Page 66 and 67: correlation between initial and fin
Page 68 and 69: ,CHAPTER 3. HIDDEN MARKOV MODELS 56
Page 72 and 73: ,CHAPTER 3. HIDDEN MARKOV MODELS 60
Page 74 and 75: CHAPTER 3. HIDDEN MARKOV MODELS 62b
Page 76 and 77: CHAPTER 3. HIDDEN MARKOV MODELS 64t
Page 78 and 79: CHAPTER 3. HIDDEN MARKOV MODELS 66t
Page 80 and 81: CHAPTER 3. HIDDEN MARKOV MODELS 68s
Page 84 and 85:
CHAPTER 3. HIDDEN MARKOV MODELS 723
Page 86 and 87:
CHAPTER 3. HIDDEN MARKOV MODELS 74b
Page 88 and 89:
domain. 3 In short, we will leave o
Page 90 and 91:
,,,,,£CHAPTER 4. STOCHASTIC CONTEX
Page 92 and 93:
9 ¸)Ô ¸ 9 ¸Ô 1 2 £££;,ÔÔC
Page 94 and 95:
¸= ¸= ¸.1.2¸¸) 1_) 20&6#=,,,,
Page 96 and 97:
CHAPTER 4. STOCHASTIC CONTEXT-FREE
Page 98 and 99:
Page 100 and 101:
Page 102 and 103:
Page 104 and 105:
==Ì==ÌCHAPTER 4. STOCHASTIC CONTE
Page 106 and 107:
,= ===I¸theybÜ„thiscg„\\ ¸¸
Page 108 and 109:
Page 110 and 111:
Page 112 and 113:
Page 114 and 115:
Page 116 and 117:
104Chapter 5Probabilistic Attribute
Page 118 and 119:
,1makingandCHAPTER 5. PROBABILISTIC
Page 120 and 121:
CHAPTER 5. PROBABILISTIC ATTRIBUTE
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
Page 128 and 129:
Page 130 and 131:
Page 132 and 133:
Page 134 and 135:
122Chapter 6Efficient parsing with
Page 136 and 137:
1z1CHAPTER 6. EFFICIENT PARSING WIT
Page 138 and 139:
and each state in set ( -ÏH ):6)
Page 140 and 141:
NPDetVTVI P CHAPTER 6. EFFICIENT PA
Page 142 and 143:
CHAPTER 6. EFFICIENT PARSING WITH S
Page 144 and 145:
) In particular, the string probabi
Page 146 and 147:
H : d=:6) ¸ÆÆÆ:6) ¸ .V=i£Ù )
Page 148 and 149:
©) 6#=©,©,©,,ÆNNö,²NL++and>+
Page 150 and 151:
) The probabilistic unit-production
Page 152 and 153:
¸0 ¸ 29¸¸ 99W9 [;t£ ? 1 ?1u 6
Page 154 and 155:
The forward and inner probabilities
Page 156 and 157:
9 itself²²NN++9 ¸0ÌLL++??1£,=C
Page 158 and 159:
,,by nonterminals. Multiplying this
Page 160 and 161:
6CHAPTER 6. EFFICIENT PARSING WITH
Page 162 and 163:
description. Again, we ignore this
Page 164 and 165:
for all pairs of states d =¸+= Š
Page 166 and 167:
,9 ¸0 : 0¸ £j9¸ A )z9£CHAPTER
Page 168 and 169:
Page 170 and 171:
are then summed over all nontermina
Page 172 and 173:
+CHAPTER 6. EFFICIENT PARSING WITH
Page 174 and 175:
1CHAPTER 6. EFFICIENT PARSING WITH
Page 176 and 177:
Page 178 and 179:
= ¸Let t,t) ¸ .V=i£,t,t,6666yyyy
Page 180 and 181:
168Chapter 7-grams from Stochastic
Page 182 and 183:
CHAPTER 7. -GRAMS FROM STOCHASTIC
Page 184 and 185:
)ÅÆÅÅÅÆÅÅÅÆÅÅÅÆÅÅÅ
Page 186 and 187:
-grams CCHAPTER 7. -GRAMS FROM ST
Page 188 and 189:
,?Ó,tÌ?L A0,I 1N I N A A 2 A 3 N
Page 190 and 191:
,,CHAPTER 7. -GRAMS FROM STOCHASTI
Page 192 and 193:
Consider the following problem: sta
Page 194 and 195:
CHAPTER 8. FUTURE DIRECTIONS 1828.2
Page 196 and 197:
184BibliographyAHO, ALFRED V., RAVI
Page 198 and 199:
BIBLIOGRAPHY 186DAGAN, IDO, FERNAND
Page 200 and 201:
BIBLIOGRAPHY 188——, & ——. 1
Page 202 and 203:
BIBLIOGRAPHY 190QUINLAN, J. ROSS, &
Page 204:
BIBLIOGRAPHY 192WALLACE, C. S., & P
show all

The dissertation of Andreas Stolcke is approved: University of ...

Create successful ePaper yourself

Delete template?

Save as template?