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For example,
In model 1, P(a|e,f) defines the alignment probability. Where, e and f are the English and French words respectively, if the translation is initiated between French-->English.
== Model 2 ==
The IBM Model 2 has an additional model for alignment that is not present in Model 1. For example, using only IBM Model 1 the translation probabilities for these translations would be the same:
[[File:IBM models 01.jpg|none]]
The IBM Model 2 addressed this issue by modeling the translation of a foreign input word in position <math>i</math> to a native language word in position <math>j</math> using an alignment probability distribution defined as:
:<math>a(i\lor j,l_e,l_f)</math>
In the above equation, the length of the input sentence f is denoted as l<sub>f</sub>, and the length of the translated sentence e as l<sub>e</sub>. The translation done by IBM Model 2 can be presented as a process divided into two steps (lexical translation and alignment).
[[File:IBM models 02.jpg|none]]
Assuming <math>t(e\mid f)</math> is the translation probability and <math>a(i\lor j,l_e,l_f)</math> is the alignment probability, IBM Model 2 can be defined as:
:<math>p(e,a\mid f)=\in \prod_{j=1}^{l_e} t(e_{j}\lor f_{a\mid j})a(a(j)\lor j,l_e,l_f)</math>
In this equation, the alignment function <math>a</math> maps each output word <math>j</math> to a foreign input position <math>a(j)</math>.<ref>{{cite journal | last1 = Och | first1 = Franz Josef | last2 = Ney | first2 = Hermann | title = A systematic comparison of various statistical alignment models | journal = Computational Linguistics | volume = 29 | year = 2003 | issue = 29 | pages = 19–51 | doi = 10.1162/089120103321337421 | doi-access = free }}</ref>
== Model 3 ==
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