Abstract:
Hidden Markov Model (HMM) parameters are updated using update equations based on growth transformation optimization of a minimum classification error objective function. Using the list of N-best competitor word sequences obtained by decoding the training data with the current-iteration HMM parameters, the current HMM parameters are updated iteratively. The updating procedure involves using weights for each competitor word sequence that can take any positive real value. The updating procedure is further extended to the case where a decoded lattice of competitors is used. In this case, updating the model parameters relies on determining the probability for a state at a time point based on the word that spans the time point instead of the entire word sequence. This word-bound span of time is shorter than the duration of the entire word sequence and thus reduces the computing time.

Description:
BACKGROUND 
       [0001]    Hidden Markov Models (HMMs) are a well established framework for a variety of pattern recognition applications, including, most prominently, speech recognition applications. Hidden Markov Models consist of interconnected states where each state is represented by a Gaussian distribution or by a mixture of Gaussians. Speech units, such as phonemes, are associated with one or more HMM states. Typically, the means and variances of the distributions for the HMMs are learned from training data. 
         [0002]    One technique for training HMM parameters is to use a maximum likelihood criterion based on an Expectation-Maximization algorithm. Under this technique, the parameters are adjusted to maximize the likelihood of a set of training data. However, due to data sparseness, maximum likelihood does not produce HMM parameters that are ideal for data that is not well-represented in the training data. 
         [0003]    Another method of training HMM parameters is known as discriminative training. In discriminative training, the goal is to set the HMM parameters so that the HMM is able to discriminate between a correct word sequence and one or more incorrect word sequences. 
         [0004]    One specific form of discriminative training is known as minimum classification error (MCE) training. In MCE training, the HMM parameters are trained by optimizing an objective function that is closely related to classification errors, where a classification error is the selection of an incorrect word sequence instead of a correct word sequence. Although MCE training has been performed before, conventional MCE optimization has been based on a sequential gradient-decent based technique named Generalized Probabilistic Decent (GPD), which optimizes the MCE objective function as a highly complex function of the HMM parameters. Such gradient-based techniques often require special and delicate care for tuning the parameter-dependent learning rate. 
         [0005]    Another form of discriminative training is known as maximization of mutual information (MMI). Under MMI, an objective function related to the mutual information is optimized using one of a set of optimization techniques. One of these techniques is known as Growth Transformation (GT) or Extended Baum-Welch (EBW). However, GT/EBW was developed for rational functions such as mutual information. Because MCE does not provide a rational function, growth transformation/extended Baum-Welch optimization has not been applied to minimum classification error training. 
         [0006]    The discussion above is merely provided for general background information and is not intended to be used as an aid in determining the scope of the claimed subject matter. 
       SUMMARY 
       [0007]    Model parameters are updated using update equations based on growth transformation optimization of a minimum classification error objective function. Using a decoded list of N-best competitor word sequences, updating the model parameters involves using weights for each competitor word sequence that can be any positive real value. Using a decoded lattice of competitors, updating the model parameters relies on determining the probability for a state at a time point based on the word that spans the time point instead of the entire word sequence. 
         [0008]    This Summary is provided to introduce a selection of concepts in a simplified form that are further described below in the Detailed Description. This Summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used as an aid in determining the scope of the claimed subject matter. The claimed subject matter is not limited to implementations that solve any or all disadvantages noted in the background. 
     
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0009]      FIG. 1  is a block diagram of a discriminative training system. 
           [0010]      FIG. 2  is a flow diagram of a method of discriminative training. 
           [0011]      FIG. 3  is a method of updating model parameters using N-Best competitors. 
           [0012]      FIG. 4  is a method of updating model parameters using a lattice of competitors. 
           [0013]      FIG. 5  is a lattice of competitor word sequences. 
           [0014]      FIG. 6  is a lattice of states for an arc of the lattice of  FIG. 5 . 
           [0015]      FIG. 7  is a general computing environment in which embodiments may be practiced. 
       
    
    
     DETAILED DESCRIPTION 
       [0016]      FIG. 1  provides a block diagram of a general system for performing discriminative training to train a set of acoustic models.  FIG. 2  provides a flow diagram of the discriminative training of  FIG. 1 . 
         [0017]    In step  200  of  FIG. 2 , parameter values for a set of baseline acoustic models  100  are set. Such baseline acoustic models can be formed using maximum likelihood training as is known in the art. At step  202 , a decoder  102  decodes each of a set of training utterances  104  using baseline acoustic models  100  to form at least one possible word sequence that can be represented by the training utterance. In addition, decoder  102  provides a separate probability for each word sequence, where the probability describes the probability of the training utterance given the word sequence. 
         [0018]    The word sequences identified for an utterance by decoder  102  are listed as competitors  106 , since each word sequence is competing against the other word sequences to be identified as the correct word sequence for the utterance. Competitors  106  may be in the form of a list of N-Best word sequences or may be in the form of a lattice of word sequences. A lattice of word sequences is a compact representation of N-best word sequences because it can use a single word arc to represent the occurrence of a word in multiple word sequences. 
         [0019]    Competitor word sequences  106  are provided to discriminative trainer  108  along with training utterances  104 , baseline acoustic model  100  and a true transcript  110  of each utterance. At step  204 , discriminative trainer  108  uses minimum classification error discriminative training to update the acoustic model parameters of baseline acoustic model  100  based on competitors  106 , true transcript  110 , training utterances  104  and baseline acoustic model  100 . In particular, discriminative trainer  108  uses update equations that are formed by optimizing a minimum classification error objective function using a growth transform optimization technique as discussed in more detail below. Using the update equations, discriminative trainer  108  produces updated acoustic models  112 . 
         [0020]    At step  206 , the method determines if the parameters of acoustic models  112  have converged. If they have not converged, updated acoustic models  112  are used by decoder  102  to again decode training utterances  104  to identify a new set of competitor word sequences  106  by returning to step  202 . Discriminative trainer  108  then updates acoustic model  112  using the new competitors  106 , the previous acoustic model  112 , true transcript  110  and training utterances  104 . 
         [0021]    When the model parameters of acoustic models  112  have converged at step  206 , the process ends at step  208  and the updated acoustic models  112  are provided as the final acoustic models, which may be used for speech recognition. 
         [0022]    In the following, for utterance r, S r  denotes the true transcript; “s r , s r  S r ” denotes a competitor word sequence, and s r  without special annotation denotes any possible word sequence of utterance r, which can be either the true transcript S r  or any competitor word sequence. 
         [0023]    The minimum classification error discriminative training used in step  204  above is based on the following misclassification measure: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       d 
                       r 
                     
                      
                     
                       ( 
                       
                         
                           X 
                           r 
                         
                         , 
                         Λ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         - 
                         log 
                       
                        
                       
                           
                       
                        
                       
                         
                           p 
                           Λ 
                         
                          
                         
                           ( 
                           
                             
                               X 
                               r 
                             
                             , 
                             
                               S 
                               r 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       log 
                        
                       
                         
                           ∑ 
                           
                             
                               s 
                               r 
                             
                             , 
                             
                               
                                 s 
                                 r 
                               
                               ≠ 
                               
                                 S 
                                 r 
                               
                             
                           
                         
                          
                         
                           
                             w 
                              
                             
                               ( 
                               
                                 s 
                                 r 
                               
                               ) 
                             
                           
                           · 
                           
                             
                               p 
                               Λ 
                             
                              
                             
                               ( 
                               
                                 
                                   X 
                                   r 
                                 
                                 , 
                                 
                                   s 
                                   r 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   EQ 
                   . 
                   
                       
                   
                    
                   1 
                 
               
             
           
         
       
     
         [0024]    where d r (X r ,Λ) is the misclassification measure for utterance r based on a sequence of T feature vectors, X r =[x 1 , . . . x t  . . . x T ], which represents the utterance and based on acoustic model parameters Λ, p Λ (X r ,S r ) is the joint probability of utterance feature vectors X r  and true transcript S r  based on acoustic model parameters Λ, p Λ (X r ,s r ) given the restriction s r  S r  is the joint probability of utterance feature vectors X r  and a competitor word sequence s r , w(s r ) given the restriction s r  S r , is a non-negative weighting factor for competitor word sequence s r , and the summation in the second term on the right is taken over all competitor word sequences that are not the true transcript. In the following sections, for the true transcript S r , a value of 1 is assigned to w(s r ) where s r =S r . 
         [0025]    Using the misclassification measure of EQ. 1, a loss function for an N-Best version of MCE becomes: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       l 
                       r 
                     
                      
                     
                       ( 
                       
                         
                           d 
                           r 
                         
                          
                         
                           ( 
                           
                             
                               X 
                               r 
                             
                             , 
                             Λ 
                           
                           ) 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           
                             s 
                             r 
                           
                           , 
                           
                             
                               s 
                               r 
                             
                             ≠ 
                             
                               S 
                               r 
                             
                           
                         
                       
                        
                       
                         
                           w 
                            
                           
                             ( 
                             
                               s 
                               r 
                             
                             ) 
                           
                         
                          
                         
                           
                             p 
                             Λ 
                           
                            
                           
                             ( 
                             
                               
                                 X 
                                 r 
                               
                               , 
                               
                                 s 
                                 r 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         ∑ 
                         
                           s 
                           r 
                         
                       
                        
                       
                         
                           w 
                            
                           
                             ( 
                             
                               s 
                               r 
                             
                             ) 
                           
                         
                          
                         
                           
                             p 
                             Λ 
                           
                            
                           
                             ( 
                             
                               
                                 X 
                                 r 
                               
                               , 
                               
                                 s 
                                 r 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   EQ 
                   . 
                   
                       
                   
                    
                   2 
                 
               
             
           
         
       
     
         [0026]    where the loss function l r (d r (X r ,Λ)) is for a single utterance r. The loss function has a value between zero and one and tends to have values that are either very close to zero or very close to one. 
         [0027]    The loss function over the entire set of R training utterances becomes: 
         [0000]    
       
         
           
             
               
                 
                   
                     L 
                      
                     
                       ( 
                       Λ 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         r 
                         = 
                         1 
                       
                       R 
                     
                      
                     
                       
                         l 
                         r 
                       
                        
                       
                         ( 
                         
                           
                             d 
                             r 
                           
                            
                           
                             ( 
                             
                               
                                 X 
                                 r 
                               
                               , 
                               Λ 
                             
                             ) 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   EQ 
                   . 
                   
                       
                   
                    
                   3 
                 
               
             
           
         
       
     
         [0028]    where L(Λ) has values between zero and R. 
         [0029]    Minimizing the loss function of EQ. 3 is equivalent to maximizing the following objective function: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           O 
                            
                           
                             ( 
                             Λ 
                             ) 
                           
                         
                         = 
                         
                           
                             R 
                             - 
                             
                               L 
                                
                               
                                 ( 
                                 Λ 
                                 ) 
                               
                             
                           
                           = 
                             
                            
                           
                             
                               ∑ 
                               
                                 r 
                                 = 
                                 1 
                               
                               R 
                             
                              
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                   
                                     
                                       ∑ 
                                       
                                         
                                           s 
                                           r 
                                         
                                         , 
                                         
                                           
                                             s 
                                             r 
                                           
                                           ≠ 
                                           
                                             S 
                                             r 
                                           
                                         
                                       
                                     
                                      
                                     
                                       
                                         w 
                                          
                                         
                                           ( 
                                           
                                             s 
                                             r 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         
                                           p 
                                           Λ 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               X 
                                               r 
                                             
                                             , 
                                             
                                               s 
                                               r 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   
                                     
                                       ∑ 
                                       
                                         s 
                                         r 
                                       
                                     
                                      
                                     
                                       
                                         w 
                                          
                                         
                                           ( 
                                           
                                             s 
                                             r 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         
                                           p 
                                           Λ 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               X 
                                               r 
                                             
                                             , 
                                             
                                               s 
                                               r 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             ∑ 
                             
                               r 
                               = 
                               1 
                             
                             R 
                           
                            
                           
                             
                               
                                 w 
                                  
                                 
                                   ( 
                                   
                                     S 
                                     r 
                                   
                                   ) 
                                 
                               
                                
                               
                                 
                                   p 
                                   Λ 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       X 
                                       r 
                                     
                                     , 
                                     
                                       S 
                                       r 
                                     
                                   
                                   ) 
                                 
                               
                             
                             
                               
                                 ∑ 
                                 
                                   s 
                                   r 
                                 
                               
                                
                               
                                 
                                   w 
                                    
                                   
                                     ( 
                                     
                                       s 
                                       r 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     p 
                                     Λ 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         X 
                                         r 
                                       
                                       , 
                                       
                                         s 
                                         r 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   EQ 
                   . 
                   
                       
                   
                    
                   4 
                 
               
             
           
         
       
     
         [0030]    Note that the right hand term in EQ. 5 is a sum of rational functions rather than a rational function. As such, it is not amenable to growth transformation/extended Baum-Welch optimization. 
         [0031]    To allow for growth transformation/extended Baum-Welch optimization, embodiments reformulate the objective function of EQ. 4 into a true rational function as: 
         [0000]    
       
         
           
             
               
                 
                   
                     O 
                      
                     
                       ( 
                       Λ 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             
                               ∑ 
                               
                                 s 
                                 1 
                               
                             
                              
                             
                               … 
                                
                               
                                   
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     s 
                                     R 
                                   
                                 
                                  
                                 
                                   
                                     ∏ 
                                     
                                       r 
                                       = 
                                       1 
                                     
                                     R 
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       w 
                                        
                                       
                                         ( 
                                         
                                           s 
                                           r 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       
                                         p 
                                         Λ 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             X 
                                             1 
                                           
                                            
                                           … 
                                            
                                           
                                               
                                           
                                            
                                           
                                             X 
                                             
                                               R 
                                               , 
                                             
                                           
                                            
                                           
                                             s 
                                             1 
                                           
                                            
                                           … 
                                            
                                           
                                               
                                           
                                            
                                           
                                             s 
                                             R 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 r 
                                 = 
                                 1 
                               
                               R 
                             
                              
                             
                               c 
                                
                               
                                 ( 
                                 
                                   s 
                                   r 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         ∑ 
                         
                           s 
                           1 
                         
                       
                        
                       
                         … 
                          
                         
                             
                         
                          
                         
                           
                             ∑ 
                             
                               s 
                               R 
                             
                           
                            
                           
                             
                               ∏ 
                               
                                 r 
                                 = 
                                 1 
                               
                               R 
                             
                              
                             
                                 
                             
                              
                             
                               
                                 w 
                                  
                                 
                                   ( 
                                   
                                     s 
                                     r 
                                   
                                   ) 
                                 
                               
                                
                               
                                 
                                   p 
                                   Λ 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       X 
                                       1 
                                     
                                      
                                     … 
                                      
                                     
                                         
                                     
                                      
                                     
                                       X 
                                       
                                         R 
                                         , 
                                       
                                     
                                      
                                     
                                       s 
                                       1 
                                     
                                      
                                     … 
                                      
                                     
                                         
                                     
                                      
                                     
                                       s 
                                       R 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   EQ 
                   . 
                   
                       
                   
                    
                   5 
                 
               
             
           
         
       
     
         [0032]    where Σ s     1    . . . Σ s     R    denotes nested summations, with each summand based on a different combination of competitors for the R utterances. For example, if there were only two utterances, and each utterance had two competitors and one true transcript such that s 1 =[s 1,1 ,s 1,2 ,S 1 ] and s 2 =[s 2,1 ,s 2,2 ,S 2 ], nine summands would be computed using the following combinations (s 1,1 ,s 2,1 ), (s 1,1 ,s 2,2 ), (s 1,1 ,S 2 ), (s 1,2 ,s 2,1 ), (s 1,2 ,s 2,2 ), (s 1,2 ,S 2 ), (S 1 ,s 2,1 ), (S 1 ,s 2,2 ), (S 1 ,S 2 ). In equation 5, s r  denotes the competitor or true transcript for utterance r for the current summand, p Λ (X 1  . . . X R ,s 1  . . . s R ) is the joint probability of all R training utterances and the corresponding set of word sequences for the current summand, and 
         [0000]        C ( s   r )=δ( s   r   ,S   r )  EQ. 6 
         [0033]    where δ(s r ,S r ) is the Kronechker delta function that equals one if s r =S r  and zero otherwise. 
         [0034]    Under growth transformation optimization, instead of optimizing the objective function of EQ. 5 directly, an auxiliary function is optimized where the auxiliary function is defined as: 
         [0000]        F (Λ;Λ′)= G (Λ)− O (Λ′) H (Λ)+ D   EQ. 7 
         [0035]    where F(Λ;Λ′) is the auxiliary function, G(Λ) is the numerator of the objective function, i.e., 
         [0000]    
       
         
           
             
               
                 G 
                  
                 
                   ( 
                   Λ 
                   ) 
                 
               
               = 
               
                 
                   ∑ 
                   
                     s 
                     1 
                   
                 
                  
                 
                   … 
                    
                   
                     
                       ∑ 
                       
                         s 
                         R 
                       
                     
                      
                     
                       
                         ∏ 
                         
                           r 
                           = 
                           1 
                         
                         R 
                       
                        
                       
                           
                       
                        
                       
                         
                           w 
                            
                           
                             ( 
                             
                               s 
                               r 
                             
                             ) 
                           
                         
                          
                         
                           
                             p 
                             Λ 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   X 
                                   1 
                                 
                                  
                                 … 
                                  
                                 
                                     
                                 
                                  
                                 
                                   X 
                                   R 
                                 
                               
                               , 
                               
                                 
                                   s 
                                   1 
                                 
                                  
                                 … 
                                  
                                 
                                     
                                 
                                  
                                 
                                   s 
                                   R 
                                 
                               
                             
                             ) 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               r 
                               = 
                               1 
                             
                             R 
                           
                            
                           
                             c 
                              
                             
                               ( 
                               
                                 s 
                                 r 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
             
             , 
           
         
       
     
       H(Λ) is the denominator of the objective function, i.e., 
       [0036]    
       
         
           
             
               
                 H 
                  
                 
                   ( 
                   Λ 
                   ) 
                 
               
               = 
               
                 
                   ∑ 
                   
                     s 
                     1 
                   
                 
                  
                 
                     
                 
                  
                 
                   … 
                    
                   
                       
                   
                    
                   
                     
                       ∑ 
                       
                         s 
                         R 
                       
                     
                      
                     
                       
                         ∏ 
                         
                           r 
                           = 
                           1 
                         
                         R 
                       
                        
                       
                         
                           w 
                            
                           
                             ( 
                             
                               s 
                               r 
                             
                             ) 
                           
                         
                          
                         
                           
                             p 
                             Λ 
                           
                            
                           
                             ( 
                             
                               
                                 
                                   X 
                                   1 
                                 
                                  
                                 
                                     
                                 
                                  
                                 … 
                                  
                                 
                                     
                                 
                                  
                                 
                                   X 
                                   R 
                                 
                               
                               , 
                               
                                 
                                   s 
                                   1 
                                 
                                  
                                 
                                     
                                 
                                  
                                 … 
                                  
                                 
                                     
                                 
                                  
                                 
                                   s 
                                   R 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
             
             , 
           
         
       
     
       O(Λ′) is the objective function using the previous parameter set Λ′ and D is a quantity that is independent of the parameter set. 
       [0037]    The auxiliary function of EQ. 7 can be used to form a further auxiliary function that guarantees an increase in the auxiliary function when the further auxiliary function increases. This further auxiliary function is then maximized by taking the derivative of the further auxiliary function with respect to the mean and variance of each state of the Hidden Markov Model resulting in update equations of: 
         [0000]    
       
         
           
             
               
                 
                   
                     μ 
                     1 
                   
                   = 
                   
                     
                       
                         
                           ∑ 
                           
                             r 
                             = 
                             1 
                           
                           R 
                         
                          
                         
                           
                             ∑ 
                             
                               t 
                               = 
                               1 
                             
                             
                               T 
                               r 
                             
                           
                            
                           
                             Δ 
                              
                             
                                 
                             
                              
                             
                               γ 
                                
                               
                                 ( 
                                 
                                   i 
                                   , 
                                   r 
                                   , 
                                   t 
                                 
                                 ) 
                               
                             
                              
                             
                               x 
                               t 
                             
                           
                         
                       
                       + 
                       
                         
                           D 
                           i 
                         
                          
                         
                           μ 
                           i 
                           ′ 
                         
                       
                     
                     
                       
                         
                           ∑ 
                           
                             r 
                             = 
                             1 
                           
                           R 
                         
                          
                         
                           
                             ∑ 
                             
                               t 
                               = 
                               1 
                             
                             
                               T 
                               r 
                             
                           
                            
                           
                             Δ 
                              
                             
                                 
                             
                              
                             
                               γ 
                                
                               
                                 ( 
                                 
                                   i 
                                   , 
                                   r 
                                   , 
                                   t 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       + 
                       
                         D 
                         i 
                       
                     
                   
                 
               
               
                 
                   EQ 
                   . 
                   
                       
                   
                    
                   8 
                 
               
             
             
               
                 
                   
                     ∑ 
                     i 
                   
                    
                   
                     = 
                     
                       
                         
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     r 
                                     = 
                                     1 
                                   
                                   R 
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       t 
                                       = 
                                       1 
                                     
                                     
                                       T 
                                       r 
                                     
                                   
                                    
                                   
                                     [ 
                                     
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       
                                         γ 
                                          
                                         
                                           ( 
                                           
                                             i 
                                             , 
                                             r 
                                             , 
                                             t 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             x 
                                             r 
                                           
                                           - 
                                           
                                             μ 
                                             i 
                                           
                                         
                                         ) 
                                       
                                        
                                       
                                         
                                           ( 
                                           
                                             
                                               x 
                                               r 
                                             
                                             - 
                                             
                                               μ 
                                               i 
                                             
                                           
                                           ) 
                                         
                                         T 
                                       
                                     
                                     ] 
                                   
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   D 
                                   i 
                                 
                                  
                                 
                                   E 
                                   i 
                                   ′ 
                                 
                               
                               + 
                               
                                 
                                   
                                     D 
                                     i 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         μ 
                                         i 
                                       
                                       - 
                                       
                                         μ 
                                         i 
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     ( 
                                     
                                       
                                         μ 
                                         i 
                                       
                                       - 
                                       
                                         μ 
                                         i 
                                         ′ 
                                       
                                     
                                     ) 
                                   
                                   T 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             ∑ 
                             
                               r 
                               = 
                               1 
                             
                             R 
                           
                            
                           
                             
                               ∑ 
                               
                                 t 
                                 = 
                                 1 
                               
                               
                                 T 
                                 r 
                               
                             
                              
                             
                               Δ 
                                
                               
                                   
                               
                                
                               
                                 γ 
                                  
                                 
                                   ( 
                                   
                                     i 
                                     , 
                                     r 
                                     , 
                                     t 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         + 
                         
                           D 
                           i 
                         
                       
                     
                   
                 
               
               
                 
                   EQ 
                   . 
                   
                       
                   
                    
                   9 
                 
               
             
           
         
       
     
         [0038]    where μ i  is the mean for state i, T r  is the number of time points or frames of utterance r, x t  is the feature vector for utterance r at time point t, μ i ′ is the mean for state i for the previous version of the acoustic model, Σ i  is the variance for state i, Σ i ′ is the variance for the previous model for state i, and where: 
         [0000]    
       
         
           
             
               
                 
                   
                     Δγ 
                      
                     
                       ( 
                       
                         i 
                         , 
                         r 
                         , 
                         t 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         s 
                         r 
                       
                     
                      
                     
                       
                         w 
                          
                         
                           ( 
                           
                             s 
                             r 
                           
                           ) 
                         
                       
                        
                       
                         
                           p 
                           
                             Λ 
                             ′ 
                           
                         
                          
                         
                           ( 
                           
                             
                               s 
                               r 
                             
                              
                             
                               X 
                               r 
                             
                           
                           ) 
                         
                       
                        
                       
                         
                           
                             Q 
                              
                             
                               ( 
                               
                                 Λ 
                                 ′ 
                               
                               ) 
                             
                           
                           
                             
                               ∑ 
                               
                                 s 
                                 r 
                               
                             
                              
                             
                               
                                 w 
                                  
                                 
                                   ( 
                                   
                                     s 
                                     r 
                                   
                                   ) 
                                 
                               
                                
                               
                                 
                                   p 
                                   
                                     Λ 
                                     ′ 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       s 
                                       r 
                                     
                                      
                                     
                                       X 
                                       r 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         [ 
                         
                           
                             C 
                              
                             
                               ( 
                               
                                 s 
                                 r 
                               
                               ) 
                             
                           
                           - 
                           
                             
                               
                                 p 
                                 
                                   Λ 
                                   ′ 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     S 
                                     r 
                                   
                                    
                                   
                                     X 
                                     r 
                                   
                                 
                                 ) 
                               
                             
                             
                               
                                 ∑ 
                                 
                                   s 
                                   r 
                                 
                               
                                
                               
                                 
                                   w 
                                    
                                   
                                     ( 
                                     
                                       s 
                                       r 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     p 
                                     
                                       Λ 
                                       ′ 
                                     
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         s 
                                         r 
                                       
                                        
                                       
                                         X 
                                         r 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                        
                       
                         
                           γ 
                           
                             i 
                             , 
                             r 
                             , 
                             
                               s 
                               r 
                             
                           
                         
                          
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   EQ 
                   . 
                   
                       
                   
                    
                   10 
                 
               
             
             
               
                 
                   
                     D 
                     i 
                   
                   = 
                   
                     E 
                     · 
                     
                       
                         ∑ 
                         
                           r 
                           = 
                           1 
                         
                         R 
                       
                        
                       
                         
                           
                             Q 
                              
                             
                               ( 
                               
                                 Λ 
                                 ′ 
                               
                               ) 
                             
                           
                           
                             
                               ∑ 
                               
                                 s 
                                 r 
                               
                             
                              
                             
                               
                                 w 
                                  
                                 
                                   ( 
                                   
                                     s 
                                     r 
                                   
                                   ) 
                                 
                               
                                
                               
                                 
                                   p 
                                   
                                     Λ 
                                     ′ 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       s 
                                       r 
                                     
                                      
                                     
                                       X 
                                       r 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                          
                         
                           
                             
                               p 
                               
                                 Λ 
                                 ′ 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   S 
                                   r 
                                 
                                  
                                 
                                   X 
                                   r 
                                 
                               
                               ) 
                             
                           
                           
                             
                               ∑ 
                               
                                 s 
                                 r 
                               
                             
                              
                             
                               
                                 w 
                                  
                                 
                                   ( 
                                   
                                     s 
                                     r 
                                   
                                   ) 
                                 
                               
                                
                               
                                 
                                   p 
                                   
                                     Λ 
                                     ′ 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       s 
                                       r 
                                     
                                      
                                     
                                       X 
                                       r 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                          
                         
                           
                             ∑ 
                             
                               s 
                               r 
                             
                           
                            
                           
                             
                               w 
                                
                               
                                 ( 
                                 
                                   s 
                                   r 
                                 
                                 ) 
                               
                             
                              
                             
                               
                                 p 
                                 
                                   Λ 
                                   ′ 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     s 
                                     r 
                                   
                                    
                                   
                                     X 
                                     r 
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               
                                 ∑ 
                                 t 
                               
                                
                               
                                 
                                   γ 
                                   
                                     i 
                                     , 
                                     r 
                                     , 
                                     
                                       s 
                                       r 
                                     
                                   
                                 
                                  
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   EQ 
                   . 
                   
                       
                   
                    
                   11 
                 
               
             
           
         
       
     
         [0039]    where p Λ′ (s r |X r ) is the probability of word sequence candidate s r  given utterance X r  using previous model parameters Λ′, γ i,r,s     r   (t) is the occupation probability of being in HMM state i at time point t given utterance feature vectors X r  and word sequence candidate s r , E is a factor controlling the learning rate and: 
         [0000]    
       
         
           
             
               
                 
                   
                     Q 
                      
                     
                       ( 
                       
                         Λ 
                         ′ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∏ 
                       
                         r 
                         = 
                         1 
                       
                       R 
                     
                      
                     
                       
                         ∑ 
                         
                           s 
                           r 
                         
                       
                        
                       
                         
                           w 
                            
                           
                             ( 
                             
                               s 
                               r 
                             
                             ) 
                           
                         
                          
                         
                           
                             p 
                             
                               Λ 
                               ′ 
                             
                           
                            
                           
                             ( 
                             
                               
                                 s 
                                 r 
                               
                                
                               
                                 X 
                                 r 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   EQ 
                   . 
                   
                       
                   
                    
                   12 
                 
               
             
           
         
       
     
         [0040]    Thus, Q(Λ′) is the product of weighted sums, where a weighted sum is formed for each utterance. In equations 10 and 11, Q(Λ′) is divided by 
         [0000]    
       
         
           
             
               ∑ 
               
                 s 
                 r 
               
             
              
             
               
                 w 
                  
                 
                   ( 
                   
                     s 
                     r 
                   
                   ) 
                 
               
                
               
                 
                   p 
                   Λ 
                 
                 · 
                 
                   ( 
                   
                     
                       s 
                       r 
                     
                      
                     
                       X 
                       r 
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    to form a term that is equal to the product of weighted sums for all of the utterances except the current utterance r. 
         [0041]    As shown in EQS. 10-12, for N-Best MCE optimized through growth transformation optimization, the probabilities of candidate word sequences are weighted by a weight w(s r ) that is non-negative and that is different than one. This weighting allows candidate word sequences in addition to the best candidate word sequence to influence the updating of the mean and variance. As a result, it is thought that the mean and variance are more discriminative. 
         [0042]      FIG. 3  provides a flow diagram of a method of updating model parameters using N-Best update EQS. 8-12 above. In step  300 , an utterance is selected and at step  302  weights for each word sequence of the selected utterance are determined. Under one embodiment, uniform weights are applied by setting the weight to 
         [0000]    
       
         
           
             
               1 
               N 
             
             , 
           
         
       
     
         [0000]    where N is the number of competing candidate word sequences for each utterance (not including the true transcript). In other embodiments, the weight can be set as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         w 
                          
                         
                           ( 
                           
                             s 
                             r 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           
                             p 
                             
                               Λ 
                               ′ 
                             
                           
                            
                           
                             ( 
                             
                               
                                 X 
                                 r 
                               
                                
                               
                                 s 
                                 r 
                               
                             
                             ) 
                           
                         
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             N 
                           
                            
                           
                             
                               p 
                               
                                 Λ 
                                 ′ 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   X 
                                   r 
                                 
                                  
                                 
                                   s 
                                   
                                     r 
                                     , 
                                     i 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     , 
                     
                       sr 
                        
                       
                           
                       
                        
                       Sr 
                     
                   
                    
                   
                     
 
                   
                    
                   where 
                 
               
               
                 
                   EQ 
                   . 
                   
                       
                   
                    
                   13 
                 
               
             
             
               
                 
                   
                     
                       p 
                       
                         Λ 
                         ′ 
                       
                     
                      
                     
                       ( 
                       
                         
                           X 
                           r 
                         
                         , 
                         
                           s 
                           r 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         p 
                         
                           Λ 
                           ′ 
                         
                       
                        
                       
                         ( 
                         
                           
                             X 
                             r 
                           
                            
                           
                             s 
                             r 
                           
                         
                         ) 
                       
                     
                      
                     
                       p 
                        
                       
                         ( 
                         
                           s 
                           r 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   EQ 
                   . 
                   
                       
                   
                    
                   14 
                 
               
             
           
         
       
     
         [0043]    where N is the number of competitor word sequences for utterance r, s r,i  is the ith competitor word sequence for utterance r, p Λ′ (X r |s r ) is the probability of utterance feature vectors X r  given the previous parameter set Λ′ and word sequence s r , which is provided by decoder  102  when forming the N-Best competitors  106 , and p(s r ) is the language model probability of word sequence s r . Note that for the true transcript S r , its weight is always set to 1 by design. 
         [0044]    At step  304  a word sequence from the N-Best list of word sequences for the selected utterance is selected. 
         [0045]    At step  306 , the language model probability of the selected word sequence is determined, if it was not already determined to set the weights for the word sequences at step  302 . At step  308 , the occupation probability, γ i,r,s (t), of each state and each time point in the selected utterance is determined for the selected word sequence. Under one embodiment, this occupation probability is determined by forming a lattice of possible states for the word sequence and using a forward-backward algorithm to determine the occupation probability of each state. Specifically, the occupation probability is determined as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       γ 
                       
                         i 
                         , 
                         r 
                         , 
                         
                           s 
                           r 
                         
                       
                     
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           α 
                           t 
                         
                          
                         
                           ( 
                           i 
                           ) 
                         
                       
                        
                       
                         
                           β 
                           t 
                         
                          
                         
                           ( 
                           i 
                           ) 
                         
                       
                     
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         I 
                       
                        
                       
                         
                           α 
                           T 
                         
                          
                         
                           ( 
                           i 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   EQ 
                   . 
                   
                       
                   
                    
                   15 
                 
               
             
           
         
       
     
         [0046]    where α t (i) is a forward path score for state i at time point t, β t (i) is the backward path score for state i at time t, T is the total number of time points or frames for utterance r, and I is the total number of states at time point T. The forward score and the backward score can be determined recursively as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       α 
                       t 
                     
                      
                     
                       ( 
                       i 
                       ) 
                     
                   
                   = 
                   
                     
                       [ 
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             1 
                           
                           I 
                         
                          
                         
                           
                             
                               α 
                               
                                 t 
                                 - 
                                 1 
                               
                             
                              
                             
                               ( 
                               j 
                               ) 
                             
                           
                            
                           
                             a 
                             ji 
                           
                         
                       
                       ] 
                     
                      
                     
                       
                         b 
                         i 
                       
                        
                       
                         ( 
                         
                           x 
                           t 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   EQ 
                   . 
                   
                       
                   
                    
                   16 
                 
               
             
             
               
                 
                   
                     
                       β 
                       t 
                     
                      
                     
                       ( 
                       i 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         j 
                         = 
                         1 
                       
                       I 
                     
                      
                     
                       
                         
                           β 
                           
                             t 
                             + 
                             1 
                           
                         
                          
                         
                           ( 
                           j 
                           ) 
                         
                       
                        
                       
                         a 
                         ij 
                       
                        
                       
                         
                           b 
                           j 
                         
                          
                         
                           ( 
                           
                             x 
                             
                               t 
                               + 
                               1 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   EQ 
                   . 
                   
                       
                   
                    
                   17 
                 
               
             
           
         
       
     
         [0047]    where a ji  is the transition probability for transitioning from state j to state i, a ij  is the transition probability for transitioning from state i to state j, b i (x t ) is the observation probability determined from the model parameters for state i given feature vector x t  and b j (x t+1 ) is the observation probability determined from the model parameters for state j for feature vector x t+1 . In some embodiments, the lattice is constructed based on an alignment provided by decoder  102 , which aligns speech units and feature vectors of the utterance. As such, at certain time points in the lattice, only a limited set of states is available in the lattice. 
         [0048]    At step  310 , the process determines if there are more word sequences for the given utterance. If there are more word sequences, the next word sequence is selected by returning to step  304 . Steps  306  and  308  are then repeated for the new word sequence. When there are no more word sequences at step  310 , the process continues at step  312  where a probability of each word sequence given the utterance is determined. This probability is determined as: 
         [0000]    
       
         
           
             
               
                 
                   
                     p 
                      
                     
                       ( 
                       
                         
                           s 
                           r 
                         
                          
                         
                           X 
                           r 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         p 
                          
                         
                           ( 
                           
                             
                               X 
                               r 
                             
                              
                             
                               s 
                               r 
                             
                           
                           ) 
                         
                       
                        
                       
                         p 
                          
                         
                           ( 
                           
                             s 
                             r 
                           
                           ) 
                         
                       
                     
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         N 
                       
                        
                       
                         
                           p 
                            
                           
                             ( 
                             
                               
                                 X 
                                 r 
                               
                                
                               
                                 s 
                                 
                                   r 
                                   , 
                                   i 
                                 
                               
                             
                             ) 
                           
                         
                          
                         
                           p 
                            
                           
                             ( 
                             
                               s 
                               
                                 r 
                                 , 
                                 i 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   EQ 
                   . 
                   
                       
                   
                    
                   18 
                 
               
             
           
         
       
     
         [0049]    where p(s r |X r ) is the probability of the word sequence given the utterance, p(X r |s r ) is the probability of the utterance given the word sequence s r  which is provided by decoder  102 , p(s r ) is the language model probability of the word sequence, and s r,i  is the ith competitor word sequence of the N competitor word sequences for utterance X r . 
         [0050]    At step  314 , the process determines if there are more utterances. If there are more utterances, the process returns to step  300  to select the next utterance and steps  302  through  312  are repeated for the new utterance. 
         [0051]    When all of the utterances have been processed at step  314 , the method continues at step  316  where a product of weighted sums, Q(Λ′), is determined using EQ. 12 above. Note that in EQ. 12, a weighted sum is formed for each utterance by weighting each probability of a word sequence given an utterance by the weight for the word sequence. The weighted sums for the utterances are multiplied together to form Q(Λ′). 
         [0052]    At step  318 , for each combination of utterance r, state i, and time point t, a score is determined for each word sequence of utterance r using the portion of EQ. 10 within the outer summation. The scores for the word sequences are summed together for each utterance as indicated by EQ. 10 to form Δγ(i,r,t) at step  320  for each combination of utterance r, state i, and time point t. 
         [0053]    At step  322 , Q and γ i,r,s (t) and other terms are used in EQ. 11 to determine D i . At step  324 , Δγ(i,r,t) and D i  are used in EQS. 8 and 9 above to update the mean and variance for state i. Specifically, in EQ. 8, the feature vector x i  is multiplied by Δγ(i,r,t) and the previous mean is multiplied by D i . 
         [0054]      FIG. 4  provides a flow diagram for updating model parameters when decoder  102  provides a lattice of competitors  106  instead of an N-Best list. 
         [0055]      FIG. 5  provides an example word lattice  500  that could be used in the embodiment of  FIG. 4 . In word lattice  500 , point  502  represents the beginning of the utterance and point  520  represents the end of the utterance, with the time span of the utterance equally divided along the horizontal axis of the lattice. Boundaries between words are shown as dots such as word boundaries  504 ,  506  and  508 . Arcs between word boundaries represent a single word that spans the time points between the word boundaries. For example, the words on arcs  510 ,  512  and  514  all span the time points from beginning point  502  to word boundary  504 . Word arc  516  represents a word spanning the time points from beginning point  502  to word boundary  508 . Word sequences are formed by following a path from start point  502  to end point  520  and appending the words on each word arc along the path to form a word sequence. Note that each word arc can span multiple time points or frames of the utterance and that multiple word arcs can span the same time point or frame. 
         [0056]    At step  400  of  FIG. 4 , a recognition lattice, such as recognition lattice  500  of  FIG. 5 , is formed for each utterance by decoder  102  during decoding. At step  402 , one of the utterances is selected. At step  404 , the probability of the selected utterance is determined by determining the probability of all of the paths through the respective recognition lattice. This is performed by determining forward path scores. Specifically, the forward path score at the end of an arc c is computed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     a 
                      
                     
                       ( 
                       c 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         p 
                          
                         
                           ( 
                           
                             preceding 
                              
                             
                                 
                             
                              
                             c 
                           
                           ) 
                         
                       
                     
                      
                     
                       
                         
                           p 
                           
                             Λ 
                             ′ 
                           
                         
                          
                         
                           ( 
                           
                             c 
                              
                             p 
                           
                           ) 
                         
                       
                        
                       
                         
                           p 
                           
                             Λ 
                             ′ 
                           
                         
                          
                         
                           ( 
                           
                             
                               
                                 X 
                                 r 
                               
                                
                               
                                 ( 
                                 c 
                                 ) 
                               
                             
                              
                             c 
                           
                           ) 
                         
                       
                        
                       
                         α 
                          
                         
                           ( 
                           p 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   EQ 
                   . 
                   
                       
                   
                    
                   19 
                 
               
             
           
         
       
     
         [0057]    where p Λ′ (c|p) is the probability of the word on arc c given the preceding word p, which is determined using a bigram language model, p Λ′ (X r (c)|c) is the HMM probability of the utterance feature vectors of X r  associated with word arc c given the word on arc c, which is provided by decoder  102  as part of decoding, and α(p) is the forward path score for preceding word p. The summation in EQ. 19 is taken over all arcs that end at the beginning of arc c. 
         [0058]    The probability of the utterance is then determined as: 
         [0000]    
       
         
           
             
               
                 
                   
                     p 
                      
                     
                       ( 
                       
                         X 
                         r 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         c 
                         : 
                         
                           c 
                           ∈ 
                           
                             { 
                             
                               ending 
                                
                               
                                   
                               
                                
                               arcs 
                             
                             } 
                           
                         
                       
                     
                      
                     
                       α 
                        
                       
                         ( 
                         c 
                         ) 
                       
                     
                   
                 
               
               
                 
                   EQ 
                   . 
                   
                       
                   
                    
                   20 
                 
               
             
           
         
       
     
         [0059]    where the summation is taken over all of the arcs that end at the end of the utterance. 
         [0060]    At step  406 , the probability of the correct word sequence given the utterance is determined as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       p 
                       
                         Λ 
                         ′ 
                       
                     
                      
                     
                       ( 
                       
                         
                           S 
                           r 
                         
                          
                         
                           X 
                           r 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           p 
                           
                             Λ 
                             ′ 
                           
                         
                          
                         
                           ( 
                           
                             
                               X 
                               r 
                             
                              
                             
                               S 
                               r 
                             
                           
                           ) 
                         
                       
                        
                       
                         
                           p 
                           
                             Λ 
                             ′ 
                           
                         
                          
                         
                           ( 
                           
                             s 
                             r 
                           
                           ) 
                         
                       
                     
                     
                       
                         p 
                         
                           Λ 
                           ′ 
                         
                       
                        
                       
                         ( 
                         
                           X 
                           r 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   EQ 
                   . 
                   
                       
                   
                    
                   21 
                 
               
             
           
         
       
     
         [0061]    where p Λ′ (X r ) is determined using EQ. 20, p Λ′ (X r |S r ) is provided by the decoder, and p Λ′ (S r ) is provided by a language model. 
         [0062]    At step  408 , an arc in the recognition lattice is selected and at step  410 , for each state and time point in the selected arc, a probability of the state at that time point given the arc and the utterance is determined. Under one embodiment, this determination is made using a state lattice that begins at the beginning of the arc and ends at the end of the arc and is confined to the states associated with the phonemes of the word on the arc.  FIG. 6  provides an example of such a state lattice  601 . In  FIG. 6 , time is shown on the horizontal axis and HMM states are shown along the vertical axis. Time increases from left to right, with the arc beginning at state  600  and ending at state  602 . At each time point, such as time points  604  and  606 , only a limited number of states are available. For example, for time point  604  only states  610 ,  612 ,  614  and  616  are available. However, at time point  606 , states  618  and  620  are also available. Those skilled in the art will recognize that state lattice  601  is a simplified example of a state lattice and that an actual state lattice for a word arc will include more states and will extend over a greater time period. 
         [0063]    To determine the probability of a state at a time point given the arc and the utterance, the forward-backward recursion may be used. This forward-backward recursion to that described above for EQS. 15-17 with the recursions starting at the beginning and end of the arc instead of the entire word sequence. Thus, instead of EQ. 15, the following occupation probability is calculated at step  410 : 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       γ 
                       
                         i 
                         , 
                         r 
                         , 
                         c 
                       
                     
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           α 
                           t 
                         
                          
                         
                           ( 
                           i 
                           ) 
                         
                       
                        
                       
                         
                           β 
                           t 
                         
                          
                         
                           ( 
                           i 
                           ) 
                         
                       
                     
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         I 
                       
                        
                       
                         
                           α 
                           T 
                         
                          
                         
                           ( 
                           i 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   EQ 
                   . 
                   
                       
                   
                    
                   22 
                 
               
             
           
         
       
     
         [0064]    where T is the number of frames spanned by the arc, and I is the number of states at the end of the arc. 
         [0065]    At step  412 , the joint probability of the selected arc and the utterance is determined using paths in the recognition lattice, such as recognition lattice  500  of  FIG. 5 . In particular, the joint probability is computed as: 
         [0000]        p   Λ′ ( c,X   r )=α( c )β( c )  EQ. 23 
         [0066]    where α(c) is computed using EQ. 19 and β(c) is a backward path score which is calculated as: 
         [0000]    
       
         
           
             
               
                 
                   
                     β 
                      
                     
                       ( 
                       c 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         v 
                          
                         
                           ( 
                           
                             succeeding 
                              
                             
                                 
                             
                              
                             c 
                           
                           ) 
                         
                       
                     
                      
                     
                       
                         
                           p 
                           
                             Λ 
                             ′ 
                           
                         
                          
                         
                           ( 
                           
                             v 
                              
                             c 
                           
                           ) 
                         
                       
                        
                       
                         
                           p 
                           
                             Λ 
                             ′ 
                           
                         
                          
                         
                           ( 
                           
                             
                               
                                 X 
                                 r 
                               
                                
                               
                                 ( 
                                 v 
                                 ) 
                               
                             
                              
                             v 
                           
                           ) 
                         
                       
                        
                       
                         β 
                          
                         
                           ( 
                           v 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   EQ 
                   . 
                   
                       
                   
                    
                   24 
                 
               
             
           
         
       
     
         [0067]    where v is a word arc succeeding word arc c, p Λ′ (v|c) is the probability of succeeding word v given word c, which is computed using a bigram language model, p Λ′ (X r (v)|v) is the HMM probability of the feature vectors associated with word v given word v as provided by decoder  102  and β(v) is the backward path score for word arc v. 
         [0068]    In EQS. 19 and 24, α(c 0 ) are initialized to be equal to the prior probability of the word on arc c 0  times the HMM probability of the speech vectors associated with arc c 0  given arc c 0  as provided by decoder  102 . β(a) is initialized by setting β(c E ) equal to one where c E  are the ending arcs. 
         [0069]    At step  414 , the process determines if there are more arcs. If there are more arcs, the process returns to step  408  to select a new arc in the recognition lattice. Steps  410 ,  412  are then repeated for the new arc. When all of the arcs have been processed at step  414 , the method continues at step  416  where a term, Y i,r (t), used to compute Δγ(i,r,t) and D i  is computed. Specifically, in the embodiment of  FIG. 4 , the weights for the word sequences are all set to w(s r )=1 resulting in: 
         [0000]      Δγ( i,r,t )= p   Λ′ ( S   r   |X   r )γ i,r,s     r   ( t )− p   Λ′ ( S   r   |X   r ) Y   i,r ( t )  EQ. 25 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       D 
                       i 
                     
                     = 
                     
                       E 
                       · 
                       
                         
                           ∑ 
                           
                             r 
                             = 
                             1 
                           
                           R 
                         
                          
                         
                           
                             
                               p 
                               
                                 Λ 
                                 ′ 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   S 
                                   r 
                                 
                                  
                                 
                                   X 
                                   r 
                                 
                               
                               ) 
                             
                           
                            
                           
                             
                               ∑ 
                               t 
                             
                              
                             
                               
                                 Y 
                                 
                                   i 
                                   , 
                                   r 
                                 
                               
                                
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   where 
                 
               
               
                 
                   EQ 
                   . 
                   
                       
                   
                    
                   26 
                 
               
             
             
               
                 
                   
                     
                       Y 
                       
                         i 
                         , 
                         r 
                       
                     
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         s 
                         r 
                       
                     
                      
                     
                       
                         
                           p 
                           
                             Λ 
                             ′ 
                           
                         
                          
                         
                           ( 
                           
                             
                               s 
                               r 
                             
                              
                             
                               X 
                               r 
                             
                           
                           ) 
                         
                       
                        
                       
                         
                           γ 
                           
                             i 
                             , 
                             r 
                             , 
                             
                               s 
                               r 
                             
                           
                         
                          
                         
                           ( 
                           t 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   EQ 
                   . 
                   
                       
                   
                    
                   27 
                 
               
             
           
         
       
     
         [0070]    where γ i,r,S     r   (t) of EQ. 25 is computed according to EQ. 15. To efficiently compute EQ. 27 for a lattice representation of competitors s r , it is assumed that HMM state sequences are independent across arcs. In other words the occupancy probability γ i,r,s     r   (t) can be approximated as being dependent only upon the word arc that contains the time point t and not upon the entire word sequence. Based on this assumption, EQ. 27 becomes: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Y 
                       
                         i 
                         , 
                         r 
                       
                     
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         c 
                         : 
                         
                           t 
                           ∈ 
                           
                             [ 
                             
                               
                                 b 
                                 q 
                               
                               , 
                               
                                 e 
                                 q 
                               
                             
                             ] 
                           
                         
                       
                     
                      
                     
                       
                         
                           γ 
                           
                             i 
                             , 
                             r 
                             , 
                             c 
                           
                         
                          
                         
                           ( 
                           t 
                           ) 
                         
                       
                       · 
                       
                         
                           
                             p 
                             
                               Λ 
                               ′ 
                             
                           
                            
                           
                             ( 
                             
                               c 
                               , 
                               
                                 X 
                                 r 
                               
                             
                             ) 
                           
                         
                         
                           
                             p 
                             
                               Λ 
                               ′ 
                             
                           
                            
                           
                             ( 
                             
                               X 
                               r 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   EQ 
                   . 
                   
                       
                   
                    
                   28 
                 
               
             
           
         
       
     
         [0071]    where the summation is performed over all arcs that span time point t. 
         [0072]    At step  418 , EQ. 25 is used to compute Δγ(i,r,t) for each combination of time points t and states i for the current utterance r. At step  420 , the process determines if there are more utterances. If there are more utterances, the next utterance is selected at step  402  and the steps between step  402  and  420  are repeated for the new utterance. Once all of the utterances have been processed at step  420 , the method continues at step  422  where EQ. 26 is used to compute D i  for each state. 
         [0073]    At step  424 , Δγ(i,r,t) and D i  are used in EQS. 8 and 9 to update the model parameters for each state i. 
         [0074]      FIG. 7  illustrates an example of a suitable computing system environment  700  on which embodiments may be implemented. The computing system environment  700  is only one example of a suitable computing environment and is not intended to suggest any limitation as to the scope of use or functionality of the claimed subject matter. Neither should the computing environment  700  be interpreted as having any dependency or requirement relating to any one or combination of components illustrated in the exemplary operating environment  700 . 
         [0075]    Embodiments are operational with numerous other general purpose or special purpose computing system environments or configurations. Examples of well-known computing systems, environments, and/or configurations that may be suitable for use with various embodiments include, but are not limited to, personal computers, server computers, hand-held or laptop devices, multiprocessor systems, microprocessor-based systems, set top boxes, programmable consumer electronics, network PCs, minicomputers, mainframe computers, telephony systems, distributed computing environments that include any of the above systems or devices, and the like. 
         [0076]    Embodiments may be described in the general context of computer-executable instructions, such as program modules, being executed by a computer. Generally, program modules include routines, programs, objects, components, data structures, etc. that perform particular tasks or implement particular abstract data types. Some embodiments are designed to be practiced in distributed computing environments where tasks are performed by remote processing devices that are linked through a communications network. In a distributed computing environment, program modules are located in both local and remote computer storage media including memory storage devices. 
         [0077]    With reference to  FIG. 7 , an exemplary system for implementing some embodiments includes a general-purpose computing device in the form of a computer  710 . Components of computer  710  may include, but are not limited to, a processing unit  720 , a system memory  730 , and a system bus  721  that couples various system components including the system memory to the processing unit  720 . 
         [0078]    Computer  710  typically includes a variety of computer readable media. Computer readable media can be any available media that can be accessed by computer  710  and includes both volatile and nonvolatile media, removable and non-removable media. By way of example, and not limitation, computer readable media may comprise computer storage media and communication media. Computer storage media includes both volatile and nonvolatile, removable and non-removable media implemented in any method or technology for storage of information such as computer readable instructions, data structures, program modules or other data. Computer storage media includes, but is not limited to, RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, digital versatile disks (DVD) or other optical disk storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to store the desired information and which can be accessed by computer  710 . Communication media typically embodies computer readable instructions, data structures, program modules or other data in a modulated data signal such as a carrier wave or other transport mechanism and includes any information delivery media. The term “modulated data signal” means a signal that has one or more of its characteristics set or changed in such a manner as to encode information in the signal. By way of example, and not limitation, communication media includes wired media such as a wired network or direct-wired connection, and wireless media such as acoustic, RF, infrared and other wireless media. Combinations of any of the above should also be included within the scope of computer readable media. 
         [0079]    The system memory  730  includes computer storage media in the form of volatile and/or nonvolatile memory such as read only memory (ROM)  731  and random access memory (RAM)  732 . A basic input/output system  733  (BIOS), containing the basic routines that help to transfer information between elements within computer  710 , such as during start-up, is typically stored in ROM  731 . RAM  732  typically contains data and/or program modules that are immediately accessible to and/or presently being operated on by processing unit  720 . By way of example, and not limitation,  FIG. 7  illustrates operating system  734 , application programs  735 , other program modules  736 , and program data  737 . 
         [0080]    The computer  710  may also include other removable/non-removable volatile/nonvolatile computer storage media. By way of example only,  FIG. 7  illustrates a hard disk drive  741  that reads from or writes to non-removable, nonvolatile magnetic media, a magnetic disk drive  751  that reads from or writes to a removable, nonvolatile magnetic disk  752 , and an optical disk drive  755  that reads from or writes to a removable, nonvolatile optical disk  756  such as a CD ROM or other optical media. Other removable/non-removable, volatile/nonvolatile computer storage media that can be used in the exemplary operating environment include, but are not limited to, magnetic tape cassettes, flash memory cards, digital versatile disks, digital video tape, solid state RAM, solid state ROM, and the like. The hard disk drive  741  is typically connected to the system bus  721  through a non-removable memory interface such as interface  740 , and magnetic disk drive  751  and optical disk drive  755  are typically connected to the system bus  721  by a removable memory interface, such as interface  750 . 
         [0081]    The drives and their associated computer storage media discussed above and illustrated in  FIG. 7 , provide storage of computer readable instructions, data structures, program modules and other data for the computer  710 . In  FIG. 7 , for example, hard disk drive  741  is illustrated as storing operating system  744 , decoder  745 , discriminative trainer  746 , and program data  747 , such as training utterances  104 , baseline acoustic model  100 , true transcript  110 , and updated acoustic model  112  of  FIG. 1 . 
         [0082]    A user may enter commands and information into the computer  710  through input devices such as a keyboard  762 , a microphone  763 , and a pointing device  761 , such as a mouse, trackball or touch pad. These and other input devices are often connected to the processing unit  720  through a user input interface  760  that is coupled to the system bus, but may be connected by other interface and bus structures, such as a parallel port, game port or a universal serial bus (USB). A monitor  791  or other type of display device is also connected to the system bus  721  via an interface, such as a video interface  790 . 
         [0083]    The computer  710  is operated in a networked environment using logical connections to one or more remote computers, such as a remote computer  780 . The logical connections depicted in  FIG. 7  include a local area network (LAN)  771  and a wide area network (WAN)  773 , but may also include other networks. Such networking environments are commonplace in offices, enterprise-wide computer networks, intranets and the Internet. 
         [0084]    When used in a LAN networking environment, the computer  710  is connected to the LAN  771  through a network interface or adapter  770 . When used in a WAN networking environment, the computer  710  typically includes a modem  772  or other means for establishing communications over the WAN  773 , such as the Internet. The modem  772 , which may be internal or external, may be connected to the system bus  721  via the user input interface  760 , or other appropriate mechanism. In a networked environment, program modules depicted relative to the computer  710 , or portions thereof, may be stored in the remote memory storage device. By way of example, and not limitation,  FIG. 7  illustrates remote application programs  785  as residing on remote computer  780 . It will be appreciated that the network connections shown are exemplary and other means of establishing a communications link between the computers may be used. 
         [0085]    Although the subject matter has been described in language specific to structural features and/or methodological acts, it is to be understood that the subject matter defined in the appended claims is not necessarily limited to the specific features or acts described above. Rather, the specific features and acts described above are disclosed as example forms of implementing the claims.