This question uses a toy problem to test your knowledge of HMM$-$based ASR.

In this question we will consider a small vocabulary ASR task. Here are the

words and their pronunciations.

uh ah

yeah y eh

no $,n$ ow

Each phone is modelled with a single$-$state context$-$independent HMM.

We will work with costs rather than probabilities, with a lower cost indicating a

higher probability. A particular utterance has four observations, and the obser$-$

vation costs for each HMM state at each time $t$ are given as follows.

An empty cell in the table indicates an infinite cost $($equivalent to a probability

of zero$).$

$($a$)$ Draw the HMM structure that would allow you to recognise any sequence of

words from the vocabulary, clearly labelling the HMM states. You may use

the conventional visualisation of an HMM, or use the WFST representation.

$($b$)$ Find the best $($lowest cost$)$ word sequence for the four$-$frame utterance de$-$

scribed in the table above in each of the following conditions. You may find

it helpful to draw a state$-$time trellis.

i$.$ All HMM transition costs are zero, but there are no self loops

ii$.$ As $($i$)$ above, but each state has a self loop with a cost of zero

iii. As $($ii$)$ above, but there is a transition cost of two between words If

there is more than one best$-$scoring word sequence in any of the condi$-$

tions, you should clearly list each of these in your answer.

$($c$)$ Comment briefly on the differences between your answers above $-$ which

HMM structure might you chose to use in a real ASR scenario?

$($d$)$ You decide to switch to an context$-$dependent HMM system modelling cross$-$

word left$-$biphone units. Draw the new HMM structure that would allow

you to recognise any sequence of words, clearly labelling the states.

$[2$ marks$]$

$[4$ marks$]$

$[4$ marks$]$

$[4$ marks$]$

$[2$ marks$]$

$[5$ marks$]$

$($e$)$ Describe one scheme that would allow a left$-$biphone HMM system to be

trained on a data set with limited examples of each left$-$biphone unit.

$[4$ marks$]$