train.tex

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\[
    \alpha_j(t) = \left[ \sum_{i=2}^{N-1} \alpha_i(t-1) a_{ij} \right]
                     b_j(\bm{o}_t)
\]
with initial conditions given by 
\[
    \alpha_1(1) = 1
\]
\[
    \alpha_j(1) = a_{1j} b_j(\bm{o}_1)
\]
for $1<j<N$ and final condition given by
\[
    \alpha_N(T) = \sum_{i=2}^{N-1} \alpha_i(T) a_{iN}
\]

The backward probability $\beta_i(t)$ for $1<i<N$ and $T>t \geq 1$ is 
calculated by the backward recursion
\[
   \beta_i(t) = \sum_{j=2}^{N-1} a_{ij} b_j(\bm{o}_{t+1}) \beta_j(t+1)
\]
with initial conditions given by
\[
   \beta_i(T) = a_{iN}
\]
for $1<i<N$ and final condition given by
\[
   \beta_1(1) = \sum_{j=2}^{N-1} a_{1j} b_j(\bm{o}_1) \beta_j(1)
\]

In the case of embedded training where the HMM spanning the observations
is a composite constructed by concatenating $Q$ subword models, it is
assumed that at time $t$, the $\alpha$ and $\beta$
values corresponding to the entry state and exit states of a HMM
represent the forward and backward probabilities at time $t-\Delta t$
and $t+\Delta t$, respectively, where $\Delta t$ is small.  The equations
for calculating $\alpha$ and $\beta$ are then as follows.

For the forward probability, the initial conditions are established at
time $t=1$ as follows
\[
   \alpha^{(q)}_{1}(1) = 
        \left\{ \begin{array}{cl}
                              1 & \mbox{if $q=1$} \\
                   \alpha^{(q-1)}_1(1)  a^{(q-1)}_{1N_{q-1}} & \mbox{otherwise}
                \end{array}
        \right.
\]
\[
   \alpha^{(q)}_{j}(1) = a^{(q)}_{1j} b^{(q)}_j(\bm{o}_1)
\]
\[
   \alpha^{(q)}_{N_q}(1) = 
        \sum_{i=2}^{N_q-1} \alpha^{(q)}_{i}(1) a^{(q)}_{iN_q}
\]
where the superscript in parentheses refers to the index of the model in 
the sequence of concatenated models.  All unspecified values of $\alpha$
are zero.  For time $t > 1$, 
\[
   \alpha^{(q)}_{1}(t) = 
        \left\{ \begin{array}{cl}
                              0 & \mbox{if $q=1$} \\
                   \alpha^{(q-1)}_{N_{q-1}}(t-1) + 
                   \alpha^{(q-1)}_1(t)  a^{(q-1)}_{1N_{q-1}}& \mbox{otherwise}
                \end{array}
        \right.
\]
\[
    \alpha^{(q)}_j(t) = 
          \left[ 
               \alpha^{(q)}_1(t) a^{(q)}_{1j} + 
                \sum_{i=2}^{N_q-1} \alpha^{(q)}_{i}(t-1) a^{(q)}_{ij}
          \right]
          b^{(q)}_j(\bm{o}_t)
\]
\[
   \alpha^{(q)}_{N_q}(t) = 
        \sum_{i=2}^{N_q-1} \alpha^{(q)}_{i}(t) a^{(q)}_{iN_q}
\]
For the backward probability, the initial conditions are set at time
$t=T$ as follows
\[
   \beta^{(q)}_{N_q}(T) = 
        \left\{ \begin{array}{cl}
                              1 & \mbox{if $q=Q$} \\
                   \beta^{(q+1)}_{N_{q+1}}(T) a^{(q+1)}_{1N_{q+1}} & \mbox{otherwise}
                \end{array}
        \right.
\]
\[
   \beta^{(q)}_i(T) = a^{(q)}_{iN_q} \beta^{(q)}_{N_q}(T)
\]
\[
   \beta^{(q)}_1(T) = \sum^{N_q - 1}_{j=2} 
                  a^{(q)}_{1j} b^{(q)}_j(\bm{o}_T) \beta^{(q)}_j(T)
\]
where once again, all unspecified $\beta$ values are zero.  For
time $t<T$,
\[
   \beta^{(q)}_{N_q}(t) = 
        \left\{ \begin{array}{cl}
                              0 & \mbox{if $q=Q$} \\
                   \beta^{(q+1)}_1(t+1)+ \beta^{(q+1)}_{N_{q+1}} (t) 
                               a^{(q+1)}_{1N_{q+1}} & \mbox{otherwise}
                \end{array}
        \right.
\]
\[
    \beta^{(q)}_i(t) = 
                a^{(q)}_{iN_q} \beta^{(q)}_{N_q}(t) +
               \sum_{j=2}^{N_q-1} a^{(q)}_{ij} 
                      b^{(q)}_j(\bm{o}_{t+1}) \beta^{(q)}_{j}(t+1) 
\]
\[
   \beta^{(q)}_{1}(t) = 
        \sum_{j=2}^{N_q-1} a^{(q)}_{1j} b^{(q)}_j(\bm{o}_t) 
                          \beta^{(q)}_{j}(t) 
\]




The total probability $P = \mbox{prob}(\bm{O} | \lambda)$ can be computed
from either the forward or backward probabilities
\[
P = \alpha_N(T) = \beta_1(1)
\]

\subsection{Single Model Reestimation(\htool{HRest})}

\index{model training!isolated unit formulae}
In this style of model training, a set of training observations
$\bm{O}^r, \;\; 1 \leq r \leq R$ is used to estimate the 
parameters of a single HMM. The
basic formula for the reestimation of the transition probabilities is
\newcommand{\albe}[1]{
                  \sum_{r=1}^R \frac{1}{P_r}
                  \sum_{t=1}^{T_r}
                  \alpha^r_#1(t)\beta^r_#1(t)
}
\[
   \hat{a}_{ij} = \frac{
                  \sum_{r=1}^R \frac{1}{P_r}
                  \sum_{t=1}^{T_r-1}
                  \alpha^r_i(t)a_{ij}b_j(\bm{o}^r_{t+1})\beta^r_j(t+1)
                    }{\albe{i}}
\]
where $1<i<N$ and $1<j<N$ and $P_r$ is the total probability
$P = \mbox{prob}(\bm{O}^r | \lambda)$ of the $r$'th observation.  
The transitions from the non-emitting entry state are reestimated by
\[
   \hat{a}_{1j} = \frac{1}{R} 
                  \sum_{r=1}^R \frac{1}{P_r}
                  \alpha^r_j(1) \beta^r_j(1)
\]
where $1<j<N$ and the transitions from the emitting states to the final
non-emitting exit state are reestimated by
\[   
   \hat{a}_{iN} = \frac{
                  \sum_{r=1}^R \frac{1}{P_r}
                  \alpha^r_i(T)\beta^r_i(T)
                    }{
                  \sum_{r=1}^R \frac{1}{P_r}
                  \sum_{t=1}^{T_r}
                  \alpha^r_i(t)\beta^r_i(t)
                    }
\]
where $1<i<N$.

For a HMM with $M_s$ mixture components in stream $s$, the means, covariances
and mixture weights for that stream are reestimated as follows.
Firstly, the probability of occupying the $m$'th mixture component in stream
$s$ at time $t$ for the $r$'th observation is
\[
  L^r_{jsm}(t) = \frac{1}{P_r} U^r_j(t) c_{jsm} b_{jsm}(\bm{o}^r_{st})
                  \beta^r_j(t) b^*_{js}(\bm{o}^r_t)
\]
where
\hequation{
  U^r_j(t) = \left\{ \begin{array}{cl}
                              a_{1j}             & \mbox{if $t=1$} \\
                   \sum^{N-1}_{i=2} \alpha^r_i(t-1) 
                       a_{ij}         & \mbox{otherwise}
                \end{array}
        \right.
}{urjt}
and
\[
     b^*_{js}(\bm{o}^r_t) = \prod_{k \neq s} b_{jk}(\bm{o}^r_{kt})
\]
For single Gaussian streams, the probability of mixture component occupancy is
equal to the probability of state occupancy and hence it is more efficient
in this case to use
\[
        L^r_{jsm}(t) =  L^r_{j}(t) = \frac{1}{P_r} \alpha_j(t) \beta_j(t)
\]

Given the above definitions, the re-estimation formulae may 
now be expressed in terms of $L^r_{jsm}(t)$ as 
follows.

\newcommand{\liksum}[1]{
                  \sum_{r=1}^R  \sum_{t=1}^{T_r} L^r_{#1}(t)
}

\[
   \hat{\bm{\mu}}_{jsm} = \frac{
                \liksum{jsm}\bm{o}^r_{st}}{\liksum{jsm}}
\]

\hequation{
   \hat{\bm{\Sigma}}_{jsm} = \frac{
        \liksum{jsm}(\bm{o}^r_{st} - \hat{\bm{\mu}}_{jsm})
                                        (\bm{o}^r_{st} - \hat{\bm{\mu}}_{jsm})'
                }{\liksum{jsm}}
}{sigjsm}

\[
   \bm{c}_{jsm} = \frac{\liksum{jsm}}{\liksum{j}}
\]

\subsection{Embedded Model Reestimation(\htool{HERest})}

\index{model training!embedded subword formulae}
The re-estimation formulae for the embedded model case have
to be modified to take account of the fact that the
entry states can be occupied at any time as a result
of transitions out of the previous model.  The basic
formulae for the re-estimation of the transition
probabilities is
\newcommand{\albeq}[1]{
                  \sum_{r=1}^R \frac{1}{P_r}
                  \sum_{t=1}^{T_r}
                  \alpha^{(q)r}_#1(t)\beta^{(q)r}_#1(t)
}
\[
   \hat{a}^{(q)}_{ij} = \frac{
                  \sum_{r=1}^R \frac{1}{P_r}
                  \sum_{t=1}^{T_r-1}
      \alpha^{(q)r}_i(t) a^{(q)}_{ij}b^{(q)}_j(\bm{o}^r_{t+1})
         \beta^{(q)r}_j(t+1)
                    }{\albeq{i}}
\]
The transitions from the non-emitting entry states into the HMM are re-estimated by
\[
   \hat{a}^{(q)}_{1j} = \frac{
                  \sum_{r=1}^R \frac{1}{P_r}
                  \sum_{t=1}^{T_r-1}
      \alpha^{(q)r}_1(t) a^{(q)}_{1j}b^{(q)}_j(\bm{o}^r_{t})
         \beta^{(q)r}_j(t)
                    }{\albeq{1} + \alpha^{(q)r}_{1}(t)a^{(q)}_{1N_q}\beta^{(q+1)r}_1(t)}
\]
and the transitions out of the HMM into the non-emitting exit states are re-estimated by
\[
   \hat{a}^{(q)}_{iN_q} = \frac{
                  \sum_{r=1}^R \frac{1}{P_r}
                  \sum_{t=1}^{T_r-1}
      \alpha^{(q)r}_i(t) a^{(q)}_{iN_q} \beta^{(q)r}_{N_q}(t)
                    }{\albeq{i}}
\]
Finally, the direct transitions from non-emitting entry to 
non-emitting exit states are
re-estimated by
\[
   \hat{a}^{(q)}_{1N_q} = \frac{
                  \sum_{r=1}^R \frac{1}{P_r}
                  \sum_{t=1}^{T_r-1}
      \alpha^{(q)r}_1(t) a^{(q)}_{1N_q}
         \beta^{(q+1)r}_1(t)
                    }{\albeq{i} + \alpha^{(q)r}_{1}(t)a^{(q)}_{1N_q}\beta^{(q+1)r}_1(t)}
\]



The re-estimation formulae for the output distributions are the
same as for the single model case except 
for the obvious additional subscript for $q$.  However, the
probability calculations must now allow for transitions from the
entry states by changing $U^r_j(t)$ in equation~\ref{e:urjt} to
\[
  U^{(q)r}_j(t) = \left\{ \begin{array}{cl}
                              \alpha^{(q)r}_1(t) a^{(q)}_{1j}   & \mbox{if $t=1$} \\
                   \alpha^{(q)r}_1(t) a^{(q)}_{1j} + 
                   \sum^{N_q-1}_{i=2} \alpha^{(q)r}_i(t-1) 
                       a^{(q)}_{ij}         & \mbox{otherwise}
                \end{array}
        \right.
\]


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