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

description. Again, we ignore this technicality in our terminology. The term is motivated by the fact that´,,1,661,00derives@‰yy,,isinCHAPTER 6. EFFICIENT PARSING WITH STOCHASTIC CONTEXT-FREE GRAMMARS 150Definition 6.9 Given a string 1 , .1the probabilities of all paths that.6{G , the outer probability´@_,6)¸.—£ ˜M0 of an Earley state is the sum of¡start with the initial state,¡generate the prefix 1 0 £££1 ,16?¡pass through6)¸ £j@$˜ , for some@,generate the suffix 1 @ £££1ED ?¡1 starting with state6)¸@>£ ˜ ,end in the final state.¡Outer probabilities complement inner probabilities in that they refer to precisely to those partsof complete paths generating not covered by the corresponding inner probabilityŠ@7,6)¸ .—£ ˜M0 . A1potentially confusing aspect of this definition is that the choice of the production ) ¸outer probability associated with state6)¸a.of the RHS: all states sharing the ) same",˜ and will identical´@have ..>˜ is not part of the.M£ ˜ . In fact, the definition makes no reference to the first part.—£ ˜M0 is the probability that an Earley parser operating as a string generatoryields the prefix 1 0å…å…å6?¸Intuitively,´@_,6)1 and theD ?å…å…å suffix 1, while passing state6)¸throughis independent of.). As was the case for 1 forward is actually an expectation of the number@ probabilities,´.M£ ˜ at position H (whichof such states in set H , as unit production cycles can lead to paths that have more than one state fitting thisreduces to the ‘outer probability’ of )as defined in Baker (1979) if the dot is in final position.6.5.2.1 Computing expected production countsBefore going into the details of computing outer probabilities we briefly describe their use inobtaining the expected rule counts needed for the E-step in grammar estimation.Let ¸ .%. )string . Alternatively, 1complete Earley path generating . Let production 1 ¸.along a path‰.)denote the expected number of uses of production ) ¸ 0) ¸ .w. 0 is the expected number of times that ) ¸ . .) ¸the derivation ofused for prediction in a.%.‰ë0 be the number of occurrences of predicted states with0_.%.‰ë0) ¸.w.1) ¸y 06derives+-,‰K.9 1C 1 ‰1C10_.%.‰ë0) ¸1+-,9 1 +-,‰ ?91)z@ C1@C?0å…å…å of all paths passing through H : @ ) ¸£.. Inner and outer probabilities have been defined such that this1)z@+-,9 0å…å…å@A?C10F£C1C1Summation is over all predicted states using ) ¸.. +-,9 :^0 is the sum of the probabilities”' å/+-,9C1C1