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What are the main differences between a language model and a machine translation model?

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The simple language model will give you the probability of a sequence of tokens(sentence) for that language. So lets say if you have trained a model for English language your model can give you the probability for any random english sentence.

Consider some sentence

$X$ $=$ "the quick brown fox jumps over the lazy dog" $=$ $x_1 \ x_2 \ x_3 \ ... \ x_n$

model will give you $P(X)$

Moreover if the model has been trained properly, in the following scenario, where

$X$ $=$ "the quick brown fox jumps over the lazy dog" $=$ $x_1 \ x_2 \ x_3 \ ... \ x_n$

$Y$ $=$ "dog brown quick fox over the jumps lazy the" $=$ $y_1 \ y_2 \ y_3 \ ... \ y_m$

model will always give $P(X) > P(Y)$, as it has learnt the structure of language.

On the other hand Machine translation model gives you the conditional probability of the next token given your source sentence and partial-target sentence. So if

$X = $ "I am a student" and $Y = $ "je suis" $=y_1, \ y_2$

model will give you $P(y_3 | X,y1_,y_2)$ .

where $X$ is a source sentence and $Y= y_1, y_2$ is a partial target sentence. The probability of word/token "étudiant" would be maximum among all words of vocabulary.

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