%/* ----------------------------------------------------------- */
%/*                                                             */
%/*                          ___                                */
%/*                       |_| | |_/   SPEECH                    */
%/*                       | | | | \   RECOGNITION               */
%/*                       =========   SOFTWARE                  */ 
%/*                                                             */
%/*                                                             */
%/* ----------------------------------------------------------- */
%/*         Copyright: Microsoft Corporation                    */
%/*          1995-2000 Redmond, Washington USA                  */
%/*                    http://www.microsoft.com                */
%/*                                                             */
%/*   Use of this software is governed by a License Agreement   */
%/*    ** See the file License for the Conditions of Use  **    */
%/*    **     This banner notice must not be removed      **    */
%/*                                                             */
%/* ----------------------------------------------------------- */
%
% HTKBook - Steve Young 1/12/97
%

\mychap{The Fundamentals of \HTK}{fundaments}

\sidepic{overview}{40}{}
\HTK\ is a toolkit for building Hidden Markov Models (HMMs). 
HMMs can be used to model any time series and the core of 
\HTK\ is similarly general-purpose. However, \HTK\ is primarily
designed for building HMM-based speech processing tools,  in
particular recognisers.  Thus, much of the infrastructure
support in \HTK\ is dedicated to this task.  As shown in the picture
above, there are two major processing stages involved.  Firstly,
the \HTK\ training tools are used to estimate the parameters of a set
of HMMs using 
training utterances and their associated transcriptions. Secondly,
unknown utterances are transcribed using the \HTK\ recognition tools.


The main body of this book is mostly concerned with the mechanics of these
two processes. However, before launching into detail it is necessary to
understand some of the basic principles of HMMs.  It is also helpful to have
an overview of the toolkit and to have some appreciation of 
how training and recognition in \HTK\ is organised. 

This first part of the book attempts to provide this information. In this
chapter, the basic ideas of HMMs and their use in speech recognition are
introduced.  The following chapter then presents a brief overview
of \HTK\ and, for users of older versions, it highlights the main
differences in version 2.0 and later.  Finally in this tutorial part of the book, chapter 3
describes how a HMM-based speech
recogniser can be built using
\HTK.  It does this by describing the construction of a simple small vocabulary
continuous speech recogniser.

The second part of the book then revisits the topics skimmed over here
and discusses each in detail. This can be 
read in conjunction with the third and final part
of the book which provides a reference manual for \HTK.  This includes
a description of each tool,  summaries of the various
parameters used to configure \HTK\ and a list of the error messages that
it generates when things go wrong.

Finally, note that this book is concerned only with \HTK\ as a tool-kit.
It does not provide  information for using the \HTK\ libraries as a programming
environment.

\mysect{General Principles of HMMs}{genpHMM}

\sidefig{messencode}{50}{Message Encoding/Decoding}{-4}{}
Speech recognition systems generally assume that the speech signal is
a  realisation of some message encoded as a sequence of one or more
symbols (see Fig.~\href{f:messencode}). To effect the reverse
operation of recognising the underlying symbol sequence given a spoken
utterance, the continuous speech waveform is first converted to a
sequence of equally spaced discrete parameter vectors. This sequence
of parameter vectors is assumed to form an exact representation of
the speech waveform on the basis that for the duration covered by a
single vector (typically 10ms or so), the speech waveform can be
regarded as being stationary.  Although this is not strictly true, it
is  a reasonable approximation.  Typical parametric representations in
common use are smoothed spectra or linear prediction coefficients plus
various other representations derived from these.

The r\^{o}le of the recogniser is to effect a mapping between  sequences
of speech vectors and the wanted underlying symbol sequences.  Two
problems make this very difficult.  Firstly, the mapping from symbols
to speech is not one-to-one since different underlying symbols can
give rise to similar speech sounds.  Furthermore, there are large
variations in the realised speech waveform due to speaker variability,
mood, environment, etc.  Secondly, the boundaries between symbols
cannot be identified explicitly from the speech waveform.  Hence, it
is not possible to treat the speech waveform as a sequence of
concatenated static patterns. 

The second problem of not knowing the word boundary locations
can be avoided by restricting the task to
isolated word recognition.  As shown in Fig.~\href{f:isoprob},
this implies that the speech waveform corresponds to a single
underlying symbol (e.g. word) chosen from a fixed vocabulary.
Despite the fact that this
simpler problem is somewhat artificial, it nevertheless has 
a wide range of practical applications.
Furthermore, it serves as a good basis for 
introducing the basic ideas of HMM-based
recognition before dealing with the more complex continuous speech
case.  Hence, isolated word recognition using HMMs will be dealt
with first.

\mysect{Isolated Word Recognition}{isowrdrec}

Let each spoken word be represented by a sequence of speech vectors or {\it
observations} $\bm{O}$,  defined as
\begin{equation}
\bm{O} = \bm{o}_1, \bm{o}_2, \ldots, \bm{o}_T
\end{equation}
where $\bm{o}_t$ is the speech vector observed at time $t$.  The
isolated word recognition problem can then be regarded as that of
computing
\begin{equation}  \label{e:2}
 \arg\max_i \left\{ P(w_i | \bm{O}) \right\}
\end{equation}
where $w_i$ is the $i$'th vocabulary word.  This probability
is not computable directly but using Bayes' Rule\index{Bayes' Rule} gives
\begin{equation} \label{e:3}
  P(w_i | \bm{O}) = \frac{P(\bm{O}|w_i) P(w_i)}{P(\bm{O})}
\end{equation}
Thus, for a given set of prior probabilities $P(w_i)$, the most
probable spoken word depends only on the likelihood $P(\bm{O}|w_i) $.
Given the dimensionality of the observation sequence $\bm{O}$, the
direct estimation of the joint conditional probability 
$P(\bm{o}_1,\bm{o}_2,\ldots | w_i)$ from examples of spoken words
is not practicable. However, if a parametric model of word production
such as a Markov model
is assumed, then estimation from data is possible since the problem
of estimating the class conditional observation densities $P(\bm{O}|w_i)$
is replaced by the much simpler problem of estimating the Markov
model parameters.

\sidefig{isoprob}{50}{Isolated Word Problem}{-4}
In HMM based speech recognition, it is assumed that the sequence of
observed speech vectors corresponding to each word is generated
by a Markov model\index{HMM!definitions} as shown in Fig.~\href{f:markovgen}.
A Markov model is a finite state machine which changes state
once every time unit and each time $t$ that a state $j$ is entered, a
speech vector $\bm{o}_t$ is generated from the probability density
$b_j(\bm{o}_t)$.  Furthermore, the transition from state $i$ to state $j$
is also probabilistic and is governed by the discrete probability $a_{ij}$.
Fig.~\href{f:markovgen} shows an example of this process where the six state
model moves through the state sequence $X=1,2,2,3,4,4,5,6$ in
order to generate the sequence $\bm{o}_1$ to $\bm{o}_6$. Notice that
in \HTK, the entry and exit states of a HMM are non-emitting.  This
is to facilitate the construction of composite models as explained in
more detail later.


The joint probability that $\bm{O}$ is generated by the model $M$ moving
through the state sequence
$X$ is calculated simply as the product of the transition
probabilities and the output probabilities.  So for the state sequence $X$ in
Fig.~\href{f:markovgen}
\begin{equation} \label{e:4}
P(\bm{O},X|M) = a_{12} b_2(\bm{o}_1) a_{22} b_2(\bm{o}_2) a_{23} 
           b_3(\bm{o}_3) \ldots
\end{equation}
However, in practice, only the observation sequence 
$\bm{O}$ is known and the
underlying state sequence $X$ is hidden.  This is why it is
called a {\it Hidden Markov Model}.  

\centrefig{markovgen}{85}{The Markov Generation Model}

Given that $X$ is unknown, the
required likelihood\index{likelihood computation} is computed 
by summing over all possible state
sequences $X = x(1), x(2), x(3), \ldots, x(T)$, that is
\begin{equation}  \label{e:5}
P(\bm{O}|M) = \sum_X a_{x(0)x(1)} \prod_{t=1}^T b_{x(t)}(\bm{o}_t)
a_{x(t)x(t+1)} \end{equation}
where $x(0)$ is constrained to be the model entry state and $x(T+1)$
is constrained to be the model exit state.

As an alternative to equation~\ref{e:5}, the likelihood can be
approximated by only considering the most likely state
sequence, that is
\begin{equation} \label{e:6}
\hat{P}(\bm{O}|M) = \max_X \left\{ 
        a_{x(0)x(1)} \prod_{t=1}^T b_{x(t)}(\bm{o}_t) a_{x(t)x(t+1)}
        \right\}
\end{equation}

Although the direct computation of equations \ref{e:5} and \ref{e:6}
is not tractable, simple recursive procedures exist which allow
both quantities to be calculated very efficiently.
Before going any further, however, notice that if equation~\ref{e:2} is
computable then the recognition problem is solved.  Given a set of models
$M_i$ corresponding to words $w_i$, equation~\ref{e:2} is
solved by using \ref{e:3} and assuming that
\begin{equation} \label{e:7}
P(\bm{O}|w_i) = P(\bm{O}|M_i).
\end{equation}

All this, of course, assumes that the parameters $\{a_{ij}\}$ and
$\{b_{j}(\bm{o}_t)\}$ are known for each model $M_i$.  Herein lies the
elegance and power of the HMM framework.  Given a set of training examples
corresponding to a particular model, the parameters of that model can be
determined automatically by a robust and efficient re-estimation 
procedure.  Thus, provided that a sufficient number of representative
examples of each word can be collected then a HMM can be constructed
which implicitly models all of the many sources of variability inherent
in real speech.  Fig.~\href{f:useforiso} summarises the use of HMMs
for isolated word recognition.  Firstly, a
HMM is trained for each vocabulary word using a number of examples
of that word.  In this case, the vocabulary consists of
just three words: ``one'', ``two'' and ``three''.
Secondly, to recognise some unknown word, the likelihood of 
each model generating that word is calculated and the most likely
model identifies the word.

\centrefig{useforiso}{84}{Using HMMs for Isolated Word Recognition}

\mysect{Output Probability Specification}{outprobspec}

Before\index{output probability!continuous case} the problem of parameter estimation can be discussed in more
detail, the form of the output distributions $\{b_{j}(\bm{o}_t)\}$
needs to be made explicit. \HTK\ is designed primarily for modelling
continuous parameters using continuous density multivariate output
distributions.  It can also handle observation sequences
consisting of discrete symbols in which case, the output distributions
are discrete probabilities.  For simplicity, however, the presentation 
in this chapter will assume that continuous density 
distributions are being used. The minor differences that the use
of discrete probabilities entail are noted in chapter~\ref{c:HMMDefs}
and discussed in more detail in chapter~\ref{c:discmods}.

In common with most other\index{Gaussian mixture}\index{streams}\index{codebooks}
continuous density HMM systems, \HTK\ represents output distributions
by Gaussian Mixture Densities.  
In \HTK, however, a further
generalisation is made.  \HTK\ allows each observation vector at time $t$
to be split into a number of $S$ independent data streams $o_{st}$.  The
formula for computing $b_{j}(\bm{o}_t)$ is then
\begin{equation} \label{e:8}
  b_{j}(\bm{o}_t) = \prod_{s=1}^S \left[
     \sum_{m=1}^{M_s} c_{jsm} {\cal N}(\bm{o}_{st};
                    \bm{\mu}_{jsm}, \bm{\Sigma}_{jsm})
  \right]^{\gamma_s}
\end{equation}
where $M_s$ is the number of mixture components in stream $s$, $c_{jsm}$
is the weight of the $m$'th component and 
${\cal N}(\cdot; \bm{\mu}, \bm{\Sigma})$ is a multivariate Gaussian with
mean vector $\bm{\mu}$ and covariance matrix $\bm{\Sigma}$,
that is\index{stream weight}\index{codebook exponent}
\begin{equation}
{\cal N}(\bm{o}; \bm{\mu}, \bm{\Sigma}) =
       \frac{1}{\sqrt{(2 \pi)^n | \bm{\Sigma} |}} 
       e^{- \frac{1}{2}(\bm{o}-\bm{\mu})' \bm{\Sigma}^{-1}(\bm{o}-\bm{\mu})}
\end{equation}
where $n$ is the dimensionality of $\bm{o}$.

The exponent $\gamma_s$ is a stream weight\footnote{often 
referred to as a codebook exponent.}.  It 
can be used to give a particular stream more emphasis, however,
it can only be set manually.  No current \HTK\ training tools 
can estimate values for it. 

Multiple data streams are used to
enable separate modelling of multiple information sources.  In
\HTK, the processing of streams is completely general.  However,
the speech input modules assume that the 
source data is split into at most 4 streams.  Chapter~\ref{c:speechio}
discusses this in more detail but for now it is sufficient to
remark that the default streams are the
basic parameter vector, first (delta) and second (acceleration)
difference coefficients and log energy.

\mysect{Baum-Welch Re-Estimation}{bwrest}

To determine the parameters of a HMM it is first necessary to make
a rough guess at what they might be.  Once this is done, more
accurate (in the maximum likelihood sense) parameters
can be found by applying the so-called 
Baum-Welch re-estimation\index{Baum-Welch re-estimation}
formulae.  

\sidefig{subsmixrep}{60}{Representing a Mixture}{-4}{
Chapter~\ref{c:Training} gives the formulae used 
in \HTK\ in full detail.  
Here the basis
of the formulae will be presented in a very informal way.
Firstly, it should be noted that the inclusion of multiple data
streams does not alter matters significantly since each stream
is considered to be statistically independent.  Furthermore,
mixture components can be considered to be a special form of 
sub-state in which the transition probabilities are the mixture
weights (see Fig.~\href{f:subsmixrep}).
}

Thus, the essential problem is to estimate the means and 
variances of a HMM in which each state output distribution is a single
component Gaussian, that is
\begin{equation} \label{e:10}
b_j(\bm{o}_t) = \frac{1}{\sqrt{(2 \pi)^n | \bm{\Sigma_j} |}} 
      e^{- \frac{1}{2}(\bm{o}_t - \bm{\mu}_j)'\bm{\Sigma}_j^{-1}(\bm{o}_t - \bm{\mu}_j)}
\end{equation}
If there was just one state $j$ in the HMM, this parameter
estimation would be easy.  The maximum likelihood estimates of 
$\bm{\mu}_j$ and $\bm{\Sigma}_j$ would be just the simple averages, 
that is
\begin{equation} \label{e:11}
   \hat{\bm{\mu}}_j = \frac{1}{T} \sum_{t=1}^{T} \bm{o}_t
\end{equation}
and
\begin{equation} \label{e:12}
   \hat{\bm{\Sigma}}_j = \frac{1}{T} \sum_{t=1}^{T} 
        (\bm{o}_t - \bm{\mu}_j) (\bm{o}_t - \bm{\mu}_j)'
\end{equation}
In practice, of course, there are multiple states and there is no
direct assignment of observation vectors to individual states 
because the underlying state sequence is unknown.  Note, however,
that if some approximate assignment of vectors to states could be made then
equations \ref{e:11} and \ref{e:12} could be used to give the
required initial values for the parameters.  Indeed, this is exactly
what is done in the \HTK\ tool called \htool{HInit}\index{hinit@\htool{HInit}}.  
\htool{HInit} first divides the
training observation vectors equally amongst the model states and then
uses equations \ref{e:11} and \ref{e:12} to give initial values for
the mean and variance of each state.  It then finds the maximum
likelihood state sequence using the Viterbi\index{Viterbi training} 
algorithm described  below,
reassigns the observation vectors to states and then uses
equations \ref{e:11} and \ref{e:12} again to get better initial 
values.  This process is repeated until the estimates
do not change.

Since the full likelihood of each observation sequence
is based on the summation of all possible state sequences,
each observation vector $\bm{o}_t$ contributes to the computation
of the maximum likelihood parameter values for each state $j$.
In other words, instead of assigning each observation vector
to a specific state as in the above approximation, each
observation is assigned to every state in proportion to
the probability of the model being in that state when the
vector was observed.  Thus, if $L_j(t)$ denotes the probability
of being in state $j$ at time $t$ then the 
equations \ref{e:11} and \ref{e:12} given above become the
following weighted averages
\begin{equation} \label{e:13}
   \hat{\bm{\mu}}_j = \frac{ \sum_{t=1}^{T} L_j(t) \bm{o}_t}
                          {\sum_{t=1}^{T} L_j(t)}
\end{equation}
and
\begin{equation} \label{e:14}
   \hat{\bm{\Sigma}}_j = \frac{ \sum_{t=1}^{T} L_j(t)
        (\bm{o}_t - \bm{\mu}_j) (\bm{o}_t - \bm{\mu}_j)' }
                      {\sum_{t=1}^{T} L_j(t)}
\end{equation}
where the summations in the denominators are included to give
the required normalisation.

Equations \ref{e:13} and \ref{e:14} are the 
Baum-Welch re-estimation\index{Baum-Welch re-estimation}
formulae for the means and covariances of a HMM.  A similar but
slightly more complex formula can be derived for the transition
probabilities (see chapter~\ref{c:Training}).

Of course, to apply equations \ref{e:13} and \ref{e:14}, the
probability of state occupation $L_j(t)$ must be calculated.
This is done efficiently using the so-called {\it Forward-Backward}
\index{Forward-Backward algorithm}
algorithm.  Let the forward probability\footnote{
Since the output distributions are densities, these are not
really probabilities but it is a convenient fiction.
}
\index{forward probability} $\alpha_j(t)$ for some model
$M$ with $N$ states be defined as 
\begin{equation} \label{e:15}
   \alpha_j(t) = P(\bm{o}_1,\ldots,\bm{o}_t, x(t)=j | M).
\end{equation}
That is, $\alpha_j(t)$ is the joint probability of observing the
first $t$ speech vectors and being in state $j$ at time $t$.  This
forward probability can be efficiently calculated by the following
recursion
\begin{equation} \label{e:16}
    \alpha_j(t) = \left[ \sum_{i=2}^{N-1} \alpha_i(t-1) a_{ij} \right]
                     b_j(\bm{o}_t).
\end{equation}
This recursion depends on the fact that the probability
of being in state $j$ at time $t$ and seeing observation $\bm{o}_t$
can be deduced by summing the forward probabilities for all
possible predecessor states $i$ weighted by the transition
probability $a_{ij}$.  The slightly odd limits are caused by
the fact that states $1$ and $N$ are non-emitting\footnote{
To understand equations involving a non-emitting state at time $t$, the time
should be thought of as being $t-\delta t$ if it is an entry state, and $t+\delta t$
if it is an exit state.  This becomes important when HMMs are connected together
in sequence so that transitions across non-emitting states take place
\textit{between frames}.
}.   The
initial conditions for the above recursion are
\begin{equation}
    \alpha_1(1) = 1
\end{equation}
\begin{equation}
    \alpha_j(1) = a_{1j} b_j(\bm{o}_1)
\end{equation}
for $1<j<N$ and the final condition is given by
\begin{equation}
    \alpha_N(T) = \sum_{i=2}^{N-1} \alpha_i(T) a_{iN}.
\end{equation}
Notice here that from the definition of $\alpha_j(t)$, 
\begin{equation}
   P(\bm{O}|M) = \alpha_N(T).
\end{equation}
Hence, the calculation of the forward probability also
yields the total likelihood\index{total likelihood} $P(\bm{O}|M)$.

The backward probability\index{backward probability} $\beta_j(t)$ is defined as 
\begin{equation} \label{e:21}
   \beta_j(t) = P(\bm{o}_{t+1},\ldots,\bm{o}_T | x(t)=j , M).
\end{equation}
As in the forward case, this backward probability can be 
computed efficiently using the following recursion
\begin{equation}
   \beta_i(t) = \sum_{j=2}^{N-1} a_{ij} b_j(\bm{o}_{t+1}) \beta_j(t+1)
\end{equation}
with initial condition given by
\begin{equation}
   \beta_i(T) = a_{iN}
\end{equation}
for $1<i<N$ and final condition given by
\begin{equation}
   \beta_1(1) = \sum_{j=2}^{N-1} a_{1j} b_j(\bm{o}_1) \beta_j(1).
\end{equation}

Notice that in the definitions above, the forward
probability is a joint probability whereas the backward
probability is a conditional probability.  This somewhat
asymmetric definition is deliberate since it allows the
probability of state occupation to be determined by taking the
product of the two probabilities.  From the definitions,
\begin{equation} \label{e:25}
 \alpha_j(t) \beta_j(t) = P(\bm{O},x(t)=j | M).
\end{equation}
Hence, 
\begin{eqnarray}
  L_j(t) & = & P(x(t)=j|\bm{O},M) \\ \nonumber
         & = & \frac{P(\bm{O},x(t)=j | M)}{P(\bm{O}|M)} \\ \nonumber
                & = & \frac{1}{P} \alpha_j(t) \beta_j(t)
\end{eqnarray}
where $P=P(\bm{O}|M)$.

All of the information needed to perform HMM
parameter re-estimation using the Baum-Welch algorithm\index{Baum-Welch algorithm}
is now in place.
The steps in this algorithm may be summarised as follows
\begin{enumerate}
\item For every parameter vector/matrix requiring 
      re-estimation, allocate storage for the numerator
      and denominator summations of the form illustrated
      by equations~\ref{e:13} and \ref{e:14}.  These
      storage locations are referred to as\index{accumulators}
      {\it accumulators}\footnote{
      Note that normally the summations in
      the denominators of the re-estimation formulae are identical
      across the parameter sets of a given state and therefore
      only a single common storage location for the denominators
      is required and it need only be calculated once. However,
      \HTK\ supports a generalised parameter tying mechanism
      which can result in the denominator summations being
      different.  Hence, in \HTK\, the denominator summations
      are always stored and calculated individually 
      for each distinct parameter vector or matrix.}.
\item Calculate the forward and backward probabilities
      for all states $j$ and times $t$.
\item For each state $j$ and time $t$, use the probability
      $L_j(t)$ and the current observation vector $\bm{o}_t$
      to update the accumulators for that state.
\item Use the final accumulator values to calculate new
      parameter values.
\item If the value of $P=P(\bm{O}|M)$ for this iteration is not
      higher than the value at the previous
      iteration then stop, otherwise repeat the above steps
      using the new re-estimated parameter values.
\end{enumerate}

All of the above assumes that the parameters for a HMM are
re-estimated from a single observation sequence, that is a
single example of the spoken word.  In practice, many
examples are needed to get good parameter estimates.  However,
the use of multiple observation sequences adds no additional complexity
to the algorithm. Steps 2 and 3 above are simply repeated for
each distinct training sequence.

One final point that should be mentioned is that the computation
of the forward and backward probabilities involves taking the
product of a large number of probabilities.  In practice, this
means that the actual numbers involved become very small.  Hence,
to avoid numerical problems, the forward-backward computation
is computed in \HTK\ using log arithmetic\index{log arithmetic}.

The \HTK\ program which implements the above 
algorithm is called \htool{HRest}\index{hrest@\htool{HRest}}.  
In combination with the tool \htool{HInit}\index{hinit@\htool{HInit}} for
estimating initial values mentioned earlier, \htool{HRest} allows isolated
word HMMs to be constructed from a set of training examples
using Baum-Welch re-estimation.

\mysect{Recognition and Viterbi Decoding}{recandvit}

The previous section has described the basic ideas underlying
HMM parameter re-estima\-tion using the Baum-Welch algorithm.
In passing, it was noted that the efficient recursive
algorithm for computing the forward probability also yielded
as a by-product the total likelihood\index{total likelihood} 
$P(\bm{O}|M)$.  Thus, this
algorithm could also be used to find the model which yields
the maximum value of $P(\bm{O}|M_i)$, and hence, it could be used
for recognition.

In practice, however, it is preferable to base recognition
on the maximum likelihood state sequence since this generalises
easily to the continuous speech case whereas the use
of the total probability does not.  This likelihood is 
computed using essentially the same algorithm as the forward
probability calculation except that the summation is replaced
by a maximum operation.  For a given model $M$, 
let $\phi_j(t)$ represent the 
maximum likelihood of observing speech vectors $\bm{o}_1$ to
$\bm{o}_t$ and being in state $j$ at time $t$.
This partial likelihood can be computed efficiently using
the following recursion (cf. equation~\ref{e:16})
\begin{equation} \label{e:27}
    \phi_j(t) = \max_i \left\{ \phi_i(t-1) a_{ij} \right\}
                     b_j(\bm{o}_t).
\end{equation}
where
\begin{equation}
    \phi_1(1) = 1
\end{equation}
\begin{equation}
    \phi_j(1) = a_{1j} b_j(\bm{o}_1)
\end{equation}
for $1<j<N$.  The maximum likelihood $\hat{P}(O|M)$ is then given by
\begin{equation}
    \phi_N(T) = \max_i \left\{ \phi_i(T) a_{iN} \right\}
\end{equation}

As for the re-estimation case, the direct computation of likelihoods
leads to underflow, hence, log likelihoods are used instead.  The
recursion of equation~\ref{e:27} then becomes
\begin{equation} \label{e:30}
   \psi_j(t) = \max_i \left\{ \psi_i(t-1) + log(a_{ij}) \right\}
                       + log(b_j(\bm{o}_t)).
\end{equation}
This recursion forms the basis of the so-called Viterbi algorithm.
As shown in Fig.~\href{f:vtrellis}, this algorithm can be visualised
as finding the best path through a matrix where the vertical
dimension represents the states of the HMM and
the horizontal dimension represents the frames of speech (i.e. time).
Each large dot in the picture represents the log probability
of observing that frame at that time and each arc between dots
corresponds to a log transition probability.  The log probability
of any path is computed simply by summing the log transition probabilities
and the log output probabilities along that path.  The paths are
grown from left-to-right column-by-column.  At time $t$, each
partial path\index{path@partial} $\psi_i(t-1)$ is known for all states $i$, hence
equation~\ref{e:30} can be used to compute $\psi_j(t)$
thereby extending the partial paths by one time frame.

\centrefig{vtrellis}{100}{The Viterbi Algorithm for Isolated
Word Recognition}

This concept of
a path\index{path} is extremely important and it is generalised below to deal
with the continuous speech case.

This completes the discussion of isolated word recognition using
HMMs.  There is no \HTK\ tool which implements the above Viterbi
algorithm directly.  Instead, a tool 
called \htool{HVite}\index{hvite@\htool{HVite}} is provided which
along with its supporting libraries, \htool{HNet} and \htool{HRec}, 
is designed to handle continuous speech.  Since this recogniser is syntax
directed, it can also perform isolated word recognition as a special case.
This is discussed in more detail below.

\mysect{Continuous Speech Recognition}{consprec}

Returning now to the conceptual model of speech production and
recognition exemplified by Fig.~\href{f:messencode}, it should be
clear that the extension to continuous speech simply involves
connecting HMMs together in sequence.  Each model in the sequence
corresponds directly to the assumed underlying symbol.  These
could be either whole words for so-called 
{\it connected speech recognition} or sub-words such as phonemes
for {\it continuous speech recognition}.  The reason for including
the non-emitting entry and exit 
states\index{non-emitting states} should now be evident, these
states provide the {\it glue} needed to join models together.

There are, however, some
practical difficulties to overcome.  The training
data for continuous speech must consist of continuous utterances and,
in general, the boundaries dividing the segments of speech
corresponding to each underlying sub-word model in the sequence will not
be known.  In practice, it is usually feasible to mark the boundaries
of a small amount of data by hand.  All of the segments corresponding
to a given model can then be extracted and the {\it isolated word}
style of training described above can be used.  However, the 
amount of data obtainable in this way is usually very limited and
the resultant models will be poor estimates.  Furthermore, even
if there was a large amount of data, the boundaries imposed by
hand-marking may not be optimal as far as the HMMs are concerned.
Hence, in \HTK\ the use of \htool{HInit} and \htool{HRest}
for initialising sub-word
models is regarded as a {\it bootstrap}\index{bootstrapping} operation\footnote{
They can even be avoided altogether by using a \textit{flat start}
as described in section~\ref{s:flatstart}.}.
The main
training phase involves the use of a tool called 
\htool{HERest}\index{herest@\htool{HERest}} which does
{\it embedded training}.

Embedded training\index{embedded training} uses the same 
Baum-Welch procedure as for the 
isolated case but rather than training each model individually
all models are trained in parallel.  It works in the following 
steps:
\begin{enumerate}
\item Allocate and zero accumulators for all parameters of all HMMs.
\item Get the next training utterance.
\item Construct a composite HMM by joining in sequence  the
      HMMs corresponding to the symbol transcription of the
      training utterance. 
\item Calculate the forward and backward probabilities for the
      composite HMM.  The inclusion of
      intermediate non-emitting states in the composite model 
      requires some changes to the computation of the forward
      and backward probabilities but these are only minor.  The
      details are given in chapter~\ref{c:Training}.
\item Use the forward and backward probabilities to compute 
      the probabilities of state occupation at each time frame
      and update the accumulators in the usual way.
\item Repeat from 2 until all training utterances have been
      processed.
\item Use the accumulators to calculate new parameter estimates
      for all of the HMMs.
\end{enumerate}
These steps can then all be repeated as many times as is necessary
to achieve the required convergence.  Notice that although the
location of symbol boundaries in the training data is not 
required (or wanted) for this procedure, the symbolic transcription
of each training utterance is needed.

Whereas the extensions needed to the Baum-Welch procedure for
training sub-word models are relatively minor\footnote{
In practice, a good deal of extra work is needed to achieve
efficient operation on large training databases.  For example,
the \htool{HERest} tool includes facilities for
pruning on both the forward and backward passes and 
parallel operation on a network of machines.}, the corresponding
extensions to the Viterbi algorithm are more substantial.

In \HTK, an alternative formulation of the Viterbi algorithm is
used\index{Token Passing Model} called the {\it Token Passing Model} \footnote{
See ``Token Passing: a Conceptual Model for Connected Speech
Recognition Systems'', SJ Young, NH Russell and JHS Thornton,
CUED Technical Report F\_INFENG/TR38, Cambridge University, 1989.
Available by anonymous ftp from \texttt{svr-ftp.eng.cam.ac.uk}.}.  
In brief,
the token\index{path!as a token} passing model makes the concept of a state alignment
path explicit.  Imagine each state $j$ of a HMM at time $t$ holds a single
moveable token which contains, amongst other information,
the partial log probability $\psi_j(t)$.  This token then represents
a partial match between the observation sequence $\bm{o}_1$ to
$\bm{o}_t$ and the model subject to the constraint that the model
is in state $j$ at time $t$.  The path extension algorithm represented
by the recursion of equation~\ref{e:30} is then replaced by the
equivalent {\it token passing algorithm} which is
executed at each time frame $t$.  The key steps in this algorithm
are as follows
\begin{enumerate}
\item Pass a copy of every token in state $i$ 
      to all connecting states $j$, incrementing the log probability
      of the copy by $log[a_{ij}]+log[b_j(\bm{o}(t)]$.
\item Examine the tokens in every state and discard all but
      the token with the highest probability.
\end{enumerate}
In practice, some modifications are needed to deal with the non-emitting
states but these are straightforward  if the tokens in entry
states are assumed to represent paths extended to time $t-\delta t$
and tokens in exit
states are assumed to represent paths extended to time $t+\delta t$.

The point of using the Token Passing Model is that it extends
very simply to the continuous speech case.  Suppose that the
allowed sequence of HMMs is defined by a finite state network.
For example, Fig.~\href{f:netforcsr} shows a simple network in
which each word is defined as a sequence of phoneme-based HMMs
and all of the words are placed in a loop. 
In this network, the oval boxes denote HMM 
instances\index{HMM!instance of} and the square
boxes denote \textit{word-end} nodes\index{word-end nodes}. This composite
network is essentially just a single large HMM and the above
Token Passing algorithm applies.  The only difference now is that
more information is needed beyond the log probability of the best
token.  When the best token reaches the end of the speech,
the route it took through the network must be known in order 
to recover the recognised sequence of models.

\centrefig{netforcsr}{65}{Recognition Network for Continuously
Spoken Word Recognition}

The history of a token's route through the network may be 
recorded efficiently as follows.  Every token carries a pointer
called a {\it word end link}.  When a token is propagated from the 
exit state of a word (indicated by passing through a 
word-end node\index{word-end nodes})
to the entry state of another, that transition
represents a potential word boundary.  Hence a record called 
a {\it Word Link Record} is generated\index{WLR}\index{Word Link Record}
in which is stored the identity of the word from which the token
has just emerged and the current value of the token's link.  The
token's actual link is then replaced by a pointer to the newly
created WLR.  Fig.~\href{f:wlroper} illustrates this process.

Once all of the unknown speech has been processed, the WLRs
attached to the link of the best matching token
(i.e. the token with the highest log probability)
can be traced back to give the best matching sequence of words.
At the same time the positions of the word boundaries can also
be extracted if required.  

\centrefig{wlroper}{100}{Recording Word Boundary Decisions}

The token passing algorithm for continuous speech has been described
in terms of recording the word sequence only.  If required, the same
principle can be used to record decisions at the model and state level.
Also, more than just the best token at each word boundary can be saved.
This gives the potential for generating a lattice of hypotheses rather
than just the single best hypothesis.  Algorithms based on this idea
are called \textit{lattice N-best}\index{lattice!N-best}.   
They are suboptimal because the
use of a single token per state limits the number of different token
histories that can be maintained.  This limitation can be avoided by
allowing each model state to hold multiple-tokens and regarding
tokens as distinct if they come from different preceding words.  This
gives a class of algorithm called  \textit{word N-best} which has been
shown empirically to be comparable in performance to an optimal N-best
algorithm. \index{N-best}\index{word N-best}

The above outlines the main idea of Token Passing as it is
implemented within \HTK. The algorithms are embedded in the
library modules \htool{HNet}\index{hnet@\htool{HNet}} and 
\htool{HRec}\index{hrec@\htool{HRec}} and they may be
invoked using the recogniser tool called \htool{HVite}\index{hvite@\htool{HVite}}.
They provide single and multiple-token\index{multiple-tokens} 
passing recognition, single-best
output, lattice output, N-best lists, support for cross-word context-dependency,
lattice rescoring\index{lattice!rescoring} and forced alignment\index{forced alignment}.

\mysect{Speaker Adaptation}{spk_adapt}
\index{adaptation}
Although the training and recognition techniques described previously can
produce high performance recognition systems, these systems 
can be improved upon by customising the HMMs to the characteristics of a 
particular speaker. \HTK\ provides the tools 
\htool{HERest}\index{headapt@\htool{HEAdapt}} and 
\htool{HVite}\index{hvite@\htool{HVite}} to perform adaptation 
using a small amount of enrollment or 
adaptation data. The two tools differ in that \htool{HERest} performs
offline supervised adaptation while \htool{HVite} recognises the
adaptation data and uses the generated transcriptions to 
perform the adaptation. 
Generally, more robust adaptation is performed in a supervised mode,
as provided by \htool{HERest}, but 
given an initial well trained model set, \htool{HVite} can still achieve 
noticeable improvements in performance. 
Full details of adaptation and how it is used in \HTK\ can be found in 
Chapter~\ref{c:Adapt}.



%%% Local Variables: 
%%% mode: latex
%%% TeX-master: "htkbook"
%%% End: 
