In this part, you will be implementing the interesting parts
of a simple HMM *decoder*, *i.e.*, the program that computes
the most likely word sequence given an utterance.
We have pre-trained the transition and observation probabilities
of an HMM on data consisting of isolated digits, and this
is the model you will be decoding with.

As discussed in the slides for Lecture 5 (10/6), we can use an HMM to model each word in a vocabulary. For the lab, we use the same HMM topology and parameterization as in the slides. That is, for each word, we compute how many phonemes its canonical pronunciation has, and use three times that many states plus a final state. We line up the states linearly, and each (non-final) state has an arc to itself and to the next state. We place output distributions on arcs (as in the slides) rather than states (as in the readings), and each output distribution is modeled using a GMM. For each state, the GMM on each of its outgoing arcs is taken to be the same GMM.

For example, consider the word `TWO` whose canonical
pronunciation can be written as the two phonemes `T UW`.
Its HMM contains `2*3=6` states plus a final
state. Intuitively, the GMM attached to the outgoing
arcs of the first state can be thought of as modeling the
feature vectors associated with the first third of the
phoneme `T`, the second state's GMM models the second
third of a `T`, etc. The topology of the HMM can be
thought of as accommodating one or more feature vectors
representing the first third of a `T`, followed by
one or more feature vectors
representing the second third of a `T`, etc.

Now, we can use word HMM's to perform isolated word recognition
in the same way that recognition is performed with DTW. Instead
of a training example (or *template*) for each word,
we have a word HMM. Instead of computing an ad hoc distance
between each word and the test example, we compute the
probability that each word HMM assigns the test example,
and select the word that assigns the highest probability. We
can compute the total probability an HMM assigns to a sequence
of feature vectors efficiently using dynamic programming.

However, this approach doesn't scale well, and we can do better using an alternate approach. Instead of scoring each word HMM separately, we can build one big HMM consisting of each word HMM glued together in parallel, and keep track of which word each part of the big HMM belongs to. Then, we can use the Viterbi algorithm on this big HMM, and do backtracing to recover the highest scoring path. By seeing which word HMM the best path belongs to, we know what the best word is. In theory, doing dynamic programming on this big HMM takes about the same amount of time as the corresponding computation on all of the individual word HMM's, but this approach lends itself better to pruning, which can vastly accelerate computation. Furthermore, this approach can easily be extended beyond isolated word recognition.

For example, consider the case of wanting to recognize continuous digit strings rather than single digits. We can do this by taking the big HMM we would use for single digits, and simply adding an arc from the final state back to the start state. The HMM can now accept digit strings consisting of multiple digits, and we can use the Viterbi algorithm to find the best word sequence in the same way as we did for isolated digits. In this case, the best path may loop through the machine several times, producing several words of output. We use the “one big HMM” framework in this lab.

For this part of the lab, we supply much of a decoder for you, and you have to fill in a couple parts. Here's an outline of what the decoder does:

Load in the big HMM graph to use for decoding.

For each acoustic signal to decode:

Use the front end from Lab 1 to produce feature vectors.

Perform Viterbi on the big HMM with the feature vectors (*).

Recover the best word sequence via backtracing.

`get_arc_log_prob()`

.
However, this function calls the function `LprObservR()`

that evaluates the likelihood that a Gaussian assigns to
a feature vector, which you must also fill in.
We take care of handling the static arc probabilities
and shunting around the acoustic feature vectors to where
they need to go, so you don't need to deal with this.
We store the big HMM in a structure of type `GraphType`

.
We assume states are numbered starting from 1. To find
out the number of states in a graph `graph`, you can call
`graph.get_state_count()`. We assume that HMM's have
a single “start” state, which can be found by calling
`graph.get_start()`. To find the outgoing arcs for
a state, you can call `graph.get_out_arcs()`. (There
is no easy way to find the incoming arcs for a state, but
you don't need to do this for the lab.) Finally, you
can call `graph.get_arc_log_prob()` to find the
total (log) probability for an arc at a frame, *i.e.*, the product of the
static arc probability (usually denoted `a _{ij}`)
and the observation probability (usually denoted

Arcs in the HMM are of type `ArcType`

. To find
the destination state of an arc `arc`, call
`arc.get_dest_state()`. Under the covers, each
arc also has a transition probability index that lets us
look up the
static arc probability, an index specifying
which GMM is attached to it, and possibly a word label.
We attach a word label to the final arc in each word HMM,
so that when we do backtracing, we can figure out which
word HMM's we pass through.
However, you don't need to directly access any of these other
fields for the lab.

Aside: in addition to having a start state (the state that all legal paths much start in), an HMM can also have final states, states that legal paths must end in. In our implementation, a graph can have multiple final states, and each final state can have a “final probability” that is multiplied in when computing the probability of a path ending there. However, you don't have to worry about this, because we supply all of the code dealing with final states.

When doing dynamic programming as in any of the three main HMM
tasks (likelihood computation, Viterbi, Forward-Backward),
you need to fill in a matrix of values, *e.g.*, forward probabilities
or backtrace pointers or the such. This matrix of values is
sometimes referred to as a *dynamic programming chart*,
and the tuple of values you need to compute at each location
in the chart is sometimes called a *cell* of the chart.
Appropriately, we define a structure named `ChartCell`

that can store the values you might possibly need in a cell,
and allocate a matrix of cells for you in a variable named
`dpChart` for you to fill in for the Viterbi computation.

For Viterbi, you should fill in the member `forwLogProbM`
(this is a slight misnomer, since a forward probability is different
than a Viterbi probability) and the backtrace pointer `backPtrM`
for each cell in the chart.

One thing to note is that
instead of storing probabilities or likelihoods directly, we
will be storing log probabilities (base `e`). This is because
if we store probabilities directly, we may cause numerical
underflow. For more information, read Section 9.12 (p. 153) in
Holmes!!! *We're not kidding: if you don't understand the concepts in
section 9.12 before starting, you're going to be in a world of pain!!*
(This can be found on the web site.) For example,
we would initialize `forwLogProbM` for the start state at
frame 0 to the logprob value 0, since `ln 1 = 0`.
If we want to set a value to correspond to the probability 0,
we would set it to the logprob `-(infinity)`,
or to the constant `zeroLogProbS` that we have provided
which is pretty close. Hint: you may be tempted to convert
log probabilities into regular probabilities to make things
clearer in your mind, but resist the temptation!
The reason we use log probabilities is because converting to
a regular probability may result in an underflow.

To prepare for the exercise, create the relevant subdirectory and copy over the needed files:

mkdir -p ~/e6884/lab2/ cd ~/e6884/lab2/ cp ~stanchen/e6884/lab2/Lab2_AM.C . cp ~stanchen/e6884/lab2/Lab2_DP.C . cp ~stanchen/e6884/lab2/.mk_chain . |

`ViterbiLab2G`

in `LprObservR`

in
Your code will be compiled into the program **DcdLab2**.
To compile this program with your code, type

smk DcdLab2 |

To run this decoder on some utterances representing isolated digits, run

lab2p1a.sh |

This script takes some HMM's and GMM's that we have trained for
you, and runs the decoder using your Viterbi implementation
and Gaussian evaluator on ten utterances, each containing a single digit.
The big HMM, or *decoding graph*, used in this run consists
of an (optional) silence HMM followed by each digit HMM in parallel
followed by another (optional) silence HMM.

The “correct” output of this script can be found in the
file `p1a.out` in `~stanchen/e6884/lab2/`. It's OK
if your output doesn't match exactly, but it should be very, very
close. In particular, look to see if the logprob for each utterance
matches (*i.e.*, the basic Viterbi algorithm and Gaussian eval are correct) and
the output word sequence matches (*i.e.*, the back pointers are correct).

To help with debugging, we have provided the file `lab2p1a.debug`
which contains portions of the DP chart for the first utterance
when running `lab2p1a.sh`. You should try to match
these `forwLogProbM` values. Another hint: for frame 0 in
the first utterance, the Gaussian associated with the outgoing
arcs of state 1 should return a log prob of 20.273.

The instructions in `lab2.txt` will ask you to run the
script `lab2p1b.sh`, which does the same thing as
`lab2p1a.sh` except on a different test set.