IBM Model 1

Model

IBM Model 1 defines the probability of a sentence $s_{1}^{J}$, with length $J$, being translated to a sentence $t_{1}^{I}$, with length $I$, with the alignment $a_{1}^{J}$ as:

$Pr(t,a|s)={\frac {\epsilon }{(J+1)^{I}}}\prod _{j=1}^{J}{tr(t_{j}|s_{a(j)})}$

Where the alignment $a_{1}^{J}$ is a function that maps each word $t_{j}$ to a word $s_{i}$, by their indexes. These alignments can be viewed as an object for indicating the corresponding words in a parallel text. We can see that the sentence translation probability $Pr(t,s)$, is decomposed into the product of the lexical translation probabilities $tr(t_{j}|s_{a(j)})$ of each word in the target $t_{1},...,t_{J}$ with the word that it is aligned to in the source $s_{a(j)}$. Additionally, target words that are not aligned with any source word are aligned with the null token, with the a lexical translation probability given by $tr(t_{j}|null)$. These are referred as null insertions. The normalizing factor ${\frac {\epsilon }{(I+1)^{J}}}$ ensures that $Pr(t,a|s)$ is a probability and is normalized over all possible alignments $a$ and all possible translations $t$.

One of the problems of the IBM Model 1 is that it is very weak to reordering, since $p(f,a|s)$ is calculated using only the lexical translation probabilities $tr(t|s)$. Because of this, if the model is presented with 2 translations candidates $t_{1}$ and $t_{2}$ with the same lexical translations, but with different reordering of the translated words, the model scores both translations with the same score.