Acoustic Model and Language Model

nao113 · Apr 25, 2023

Question:

$Slide with text defining a vowel V, feature vector O, acoustic model P_AM(O|V), language model P_LM(V), the argmax formula for V-hat over {a,i,u,e,o}, and a table of log-likelihoods for each vowel.$

My Answer:

Handwritten derivation: y-hat = argmax P(V|O) y, rewritten via Bayes' rule as argmax P(O|V)P(V)/P(O), then argmax P(O|V)(P(V)), with log-probability sums (-13.4 + -18.5 + -30.1 + -10.2 + -17.0) times (-1.61 + -2.30 + -1.61 + -1.39 + -1.38), giving argmax (-69.1) and argmax (138.639), with some steps crossed out.

Is it correct? Thank you

Mark44 · Apr 25, 2023

nao113 said:

Homework Statement: Suppose 𝑉 is a vowel and 𝑂 is a feature vector.
Suppose that 𝑃 AM (𝑂|𝑉) is an acoustic model and 𝑃 𝐿M (𝑉) is a language model. Obtain a vowel 𝑉 that maximizes 𝑃(𝑉|𝑂) when the acoustic and language model log likelihoods are given in the following table.
Relevant Equations: W: a vowel v (v ∊ {a,i,u,e,o})
O: a feature vector

Question:
View attachment 325473

My Answer:
View attachment 325474

Is it correct? Thank you

No idea without some more context.
Is P(V|O) a conditional probability?
What does argmax mean?
How did you go from ##P(V|O)## to ##\frac{P(O|V)P(V)}{P(O)}## in the 2nd line of your work and similar for the 3rd line?
What role do the numbers in the log table play?

nao113 · Apr 26, 2023

This is the reference that I got, I don t know about what argmax mean here, so I assumed it has the same meaning as log e (P(V|O)).

Slide titled Acoustic Model and Language Model showing W-hat = argmax over W of P(W|O), then Bayes decomposition argmax of P(O|W)P(W)/P(O), then argmax of P(O|W)P(W); labels mark P(O|W) as acoustic model (AM), P(W) as language model (LM), and the whole as speech recognizer.

Slide titled Frame-wise Vowel Recognition: W is a vowel v in {a,i,u,e,o} and O is a feature vector o. A box labeled o points to a blue Pattern recognition block, which points to a box labeled v; below, a small orange bar sits under o and the label "a" under v.

Slide titled Recognition Using AM and LM with equations for argmax of Gaussian acoustic model and categorical language model, plus a plot of Gaussian contour ellipses labelled with Japanese kana v=い, え, あ, う, お.

Mark44 · Apr 26, 2023

What you've posted so far doesn't give any definition of "argmax". In your work that you showed in post #1, you added the numbers in the first row of the table to get one sum, and then added the numbers in the second row to get another sum. You then multiplied the two sums.

Given that I know nothing more about this than what you posted, I think your work is incorrect. My guess, and this is only a guess, is that to maximize ##P(O|W)P(O)## what you need to do is to look at the five separate products of the numbers in the five columns, and pick whichever one is the largest. You might get better advice by contacting your instructor.

Acoustic Model and Language Model

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