The IBM alignment models are a sequence of increasingly complex models used in statistical machine translation to train a translation model and an alignment model, starting with lexical translation probabilities and moving to reordering and word duplication. They underpinned the majority of statistical machine translation systems for almost twenty years starting in the early 1990s, until neural machine translation began to dominate. These models offer principled probabilistic formulation and (mostly) tractable inference. The IBM alignment models were published in parts in 1988 and 1990, and the entire series is published in 1993. Every author of the 1993 paper subsequently went to the hedge fund Renaissance Technologies. The original work on statistical machine translation at IBM proposed five models, and a model 6 was proposed later. The sequence of the six models can be summarized as: Model 1: lexical translation Model 2: additional absolute alignment model Model 3: extra fertility model Model 4: added relative alignment model Model 5: fixed deficiency problem. Model 6: Model 4 combined with a HMM alignment model in a log linear way == Mathematical setup == The IBM alignment models translation as a conditional probability model. For each source-language ("foreign") sentence f {\displaystyle f} , we generate both a target-language ("English") sentence e {\displaystyle e} and an alignment a {\displaystyle a} . The problem then is to find a good statistical model for p ( e , a | f ) {\displaystyle p(e,a|f)} , the probability that we would generate English language sentence e {\displaystyle e} and an alignment a {\displaystyle a} given a foreign sentence f {\displaystyle f} . The meaning of an alignment grows increasingly complicated as the model version number grew. See Model 1 for the most simple and understandable version. == Model 1 == === Word alignment === Given any foreign-English sentence pair ( e , f ) {\displaystyle (e,f)} , an alignment for the sentence pair is a function of type { 1 , . , . . . , l e } → { 0 , 1 , . , . . . , l f } {\displaystyle \{1,.,...,l_{e}\}\to \{0,1,.,...,l_{f}\}} . That is, we assume that the English word at location i {\displaystyle i} is "explained" by the foreign word at location a ( i ) {\displaystyle a(i)} . For example, consider the following pair of sentences It will surely rain tomorrow -- 明日 は きっと 雨 だWe can align some English words to corresponding Japanese words, but not everyone:it -> ? will -> ? surely -> きっと rain -> 雨 tomorrow -> 明日This in general happens due to the different grammar and conventions of speech in different languages. English sentences require a subject, and when there is no subject available, it uses a dummy pronoun it. Japanese verbs do not have different forms for future and present tense, and the future tense is implied by the noun 明日 (tomorrow). Conversely, the topic-marker は and the grammar word だ (roughly "to be") do not correspond to any word in the English sentence. So, we can write the alignment as 1-> 0; 2 -> 0; 3 -> 3; 4 -> 4; 5 -> 1where 0 means that there is no corresponding alignment. Thus, we see that the alignment function is in general a function of type { 1 , . , . . . , l e } → { 0 , 1 , . , . . . , l f } {\displaystyle \{1,.,...,l_{e}\}\to \{0,1,.,...,l_{f}\}} . Future models will allow one English world to be aligned with multiple foreign words. === Statistical model === Given the above definition of alignment, we can define the statistical model used by Model 1: Start with a "dictionary". Its entries are of form t ( e i | f j ) {\displaystyle t(e_{i}|f_{j})} , which can be interpreted as saying "the foreign word f j {\displaystyle f_{j}} is translated to the English word e i {\displaystyle e_{i}} with probability t ( e i | f j ) {\displaystyle t(e_{i}|f_{j})} ". After being given a foreign sentence f {\displaystyle f} with length l f {\displaystyle l_{f}} , we first generate an English sentence length l e {\displaystyle l_{e}} uniformly in a range U n i f o r m [ 1 , 2 , . . . , N ] {\displaystyle Uniform[1,2,...,N]} . In particular, it does not depend on f {\displaystyle f} or l f {\displaystyle l_{f}} . Then, we generate an alignment uniformly in the set of all possible alignment functions { 1 , . , . . . , l e } → { 0 , 1 , . , . . . , l f } {\displaystyle \{1,.,...,l_{e}\}\to \{0,1,.,...,l_{f}\}} . Finally, for each English word e 1 , e 2 , . . . e l e {\displaystyle e_{1},e_{2},...e_{l_{e}}} , generate each one independently of every other English word. For the word e i {\displaystyle e_{i}} , generate it according to t ( e i | f a ( i ) ) {\displaystyle t(e_{i}|f_{a(i)})} . Together, we have the probability p ( e , a | f ) = 1 / N ( 1 + l f ) l e ∏ i = 1 l e t ( e i | f a ( i ) ) {\displaystyle p(e,a|f)={\frac {1/N}{(1+l_{f})^{l_{e}}}}\prod _{i=1}^{l_{e}}t(e_{i}|f_{a(i)})} IBM Model 1 uses very simplistic assumptions on the statistical model, in order to allow the following algorithm to have closed-form solution. === Learning from a corpus === If a dictionary is not provided at the start, but we have a corpus of English-foreign language pairs { ( e ( k ) , f ( k ) ) } k {\displaystyle \{(e^{(k)},f^{(k)})\}_{k}} (without alignment information), then the model can be cast into the following form: fixed parameters: the foreign sentences { f ( k ) } k {\displaystyle \{f^{(k)}\}_{k}} . learnable parameters: the entries of the dictionary t ( e i | f j ) {\displaystyle t(e_{i}|f_{j})} . observable variables: the English sentences { e ( k ) } k {\displaystyle \{e^{(k)}\}_{k}} . latent variables: the alignments { a ( k ) } k {\displaystyle \{a^{(k)}\}_{k}} In this form, this is exactly the kind of problem solved by expectation–maximization algorithm. Due to the simplistic assumptions, the algorithm has a closed-form, efficiently computable solution, which is the solution to the following equations: { max t ′ ∑ k ∑ i ∑ a ( k ) t ( a ( k ) | e ( k ) , f ( k ) ) ln t ( e i ( k ) | f a ( k ) ( i ) ( k ) ) ∑ x t ′ ( e x | f y ) = 1 ∀ y {\displaystyle {\begin{cases}\max _{t'}\sum _{k}\sum _{i}\sum _{a^{(k)}}t(a^{(k)}|e^{(k)},f^{(k)})\ln t(e_{i}^{(k)}|f_{a^{(k)}(i)}^{(k)})\\\sum _{x}t'(e_{x}|f_{y})=1\quad \forall y\end{cases}}} This can be solved by Lagrangian multipliers, then simplified. For a detailed derivation of the algorithm, see chapter 4 and. In short, the EM algorithm goes as follows:INPUT. a corpus of English-foreign sentence pairs { ( e ( k ) , f ( k ) ) } k {\displaystyle \{(e^{(k)},f^{(k)})\}_{k}} INITIALIZE. matrix of translations probabilities t ( e x | f y ) {\displaystyle t(e_{x}|f_{y})} .This could either be uniform or random. It is only required that every entry is positive, and for each y {\displaystyle y} , the probability sums to one: ∑ x t ( e x | f y ) = 1 {\displaystyle \sum _{x}t(e_{x}|f_{y})=1} . LOOP. until t ( e x | f y ) {\displaystyle t(e_{x}|f_{y})} converges: t ( e x | f y ) ← t ( e x | f y ) λ y ∑ k , i , j δ ( e x , e i ( k ) ) δ ( f y , f j ( k ) ) ∑ j ′ t ( e i ( k ) | f j ′ ( k ) ) {\displaystyle t(e_{x}|f_{y})\leftarrow {\frac {t(e_{x}|f_{y})}{\lambda _{y}}}\sum _{k,i,j}{\frac {\delta (e_{x},e_{i}^{(k)})\delta (f_{y},f_{j}^{(k)})}{\sum _{j'}t(e_{i}^{(k)}|f_{j'}^{(k)})}}} where each λ y {\displaystyle \lambda _{y}} is a normalization constant that makes sure each ∑ x t ( e x | f y ) = 1 {\displaystyle \sum _{x}t(e_{x}|f_{y})=1} .RETURN. t ( e x | f y ) {\displaystyle t(e_{x}|f_{y})} .In the above formula, δ {\displaystyle \delta } is the Dirac delta function -- it equals 1 if the two entries are equal, and 0 otherwise. The index notation is as follows: k {\displaystyle k} ranges over English-foreign sentence pairs in corpus; i {\displaystyle i} ranges over words in English sentences; j {\displaystyle j} ranges over words in foreign language sentences; x {\displaystyle x} ranges over the entire vocabulary of English words in the corpus; y {\displaystyle y} ranges over the entire vocabulary of foreign words in the corpus. === Limitations === There are several limitations to the IBM model 1. No fluency: Given any sentence pair ( e , f ) {\displaystyle (e,f)} , any permutation of the English sentence is equally likely: p ( e | f ) = p ( e ′ | f ) {\displaystyle p(e|f)=p(e'|f)} for any permutation of the English sentence e {\displaystyle e} into e ′ {\displaystyle e'} . No length preference: The probability of each length of translation is equal: ∑ e has length l p ( e | f ) = 1 N {\displaystyle \sum _{e{\text{ has length }}l}p(e|f)={\frac {1}{N}}} for any l ∈ { 1 , 2 , . . . , N } {\displaystyle l\in \{1,2,...,N\}} . Does not explicitly model fertility: some foreign words tend to produce a fixed number of English words. For example, for German-to-English translation, ja is usually omitted, and zum is usually translated to one of to the, for the, to a, for a. == Model 2 == Model 2 allows alignment to be conditional on sentence lengths. That is, we have a probability distribution p a ( j | i , l e , l f ) {\displaystyle
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