Tens of millions of snippets are harvested and cleaned up in this way, yet expecting even this large set to provide decent coverage over the test data is still unrealistic. We calculated the proba­bility of an example in the test data to appear in KB at less than 1%. To overcome the coverage is­sue, a loosely coupled analysis and representation of compounds is employed. Despite the fact that both modifier and head can influence the ranking of a paraphrase, we believe that either the modifier or the head is the dominating factor in most cases. This assumption has been shown to be plausible by earlier work (Butnariu and Veale, 2008). Thus, instead of storing complete triples in our KB, we divide each complete triple into two partial triples as shown below:

We can also retrieve these partial triples directly from the n-grams corpus using partial patterns like "Head Para" and "Para Mod". However, just as shorter incomplete patterns can produce a larger KB, they also accept much more noise. For in­stance, single-verb paraphrases are very common among the test data. In these cases, the partial pat­tern approach would need to harvest snippets with the form "NN W" or "W NN" from 2-grams, which are too common to be reliable.

In the probabilistic framework, we define the B-score as the conditional probability of a para­phrase, Para, being suggested for a given com­pound Comp:

Using the KB, we can estimate this conditional probability by applying the Bayes theorem:

P(Comp\Para)P(Para) P(Comp) P(Para\Comp)

The loose-coupling assumption (Sect. 2.2) allows us to estimate P(Comp) as:

P(Comp) = P(Mod V Head).

Meanwhile, a priori probabilities such as P(Para) can be easily inferred from the KB.

After applying the loose-coupling technique de­scribed in Section 2.2, the coverage of the KB rises to 31.78% (see Figure 1). To further in­crease this coverage, an inference layer is added to the system. This layer aims to stretch the con­tents of the KB via semantic slippage to the KB, as guided by the maximization of a fitness function. A WordNet-based similarity matrix is employed (Seco et al., 2004) to provide a similarity measure between nouns (so sim(x, x) is 1). Then, a su­perset of Head or Mod (denoted as H and M re­spectively) can be extracted by including all nouns with similarity greater than 0 to any of them in the test data. Formally, for Head we have:

H = {h\sim(h, Head) > 0, Head in dataset}.

The definition of M is analogous to that of H.

A system of equations is defined to produce al­ternatives for Head and Mod and their smoothed corpus frequencies (we show only the functions for head here):

Similar to various types of data studied by social scientists, the distribution of strings in our corpus tends to obey Zipf's law (Zipf, 1936). The same Zipfian trend was also observed in the compound-paraphrase dataset: more than 190 out of 250 com­pounds in the training data have 60% of their para­phrases in an undiscriminating tail, while 245 of 250 have 50% of their paraphrases in the tail. We thus assume the existence of a long tail in the para­phrase list for each compound.

The tail of each paraphrase list can be a valuable heuristic for modeling human paraphrase prefer­ences. We refer to this model as the tail-based preference model. We assume that an occurrence of a paraphrase is deemed to occur in the tail iff il is mentioned by the human raters only once. Thus, the tail preference is defined as the probability that a paraphrase appears in the non-tail part of the list for all compounds in the training data. Formally, it can be expressed as:

where C is the set of all compounds in the training data and f(c,p) is the frequency of paraphrase p on compound c as given by the human raters. The 6(c, p) is a filter coefficient as shown below:

1, f(c,p) > 1, 0, f(c,p) = l.

The tail-based preference model is simple but effective when used in conjunction with seman­tic ranking via the KB acquired from n-grams. However, an important drawback is that the tail model assigns a static preference to paraphrase (i.e., tail preferences are assumed to be context-independent). More than that, this preference does not take information from non-tail paraphrases into consideration. Due to these downsides, we use pairwise preferences described below.

To fully utilize the training data, we employ an­other preference mining approach called pairwise preference modeling. This approach applies the principle of pairwise comparison (David, 1988) to determine the rank of a paraphrase inside a list.

We build a pairwise comparison matrix II for paraphrases using the values of Equation 10 (here we have assumed that each of the paraphrases has been mapped into numeric values):

otherwise.

where n(pi,pj) is the relative preferability of pi to pj. To illustrate the logic behind n(x,y), we imagine a scenario with three compounds shown in Table 1 :

lln this example, abor. prob, stands for abortion problem, abor. vote stands for abortion vote, and arti. desc. stands for artifact description

The relative preferability is given by the number of times that the frequency of Pi from human raters is greater than that of pj. Observing that 1 out of 3 times involve is ranked higher than concern, we can calculate their relative preferability as:

n(involve, concern) = 1 n(concern, involve) = 2

Once the matrix is built, the preference score for a paraphrase i is calculated as:

where Vc is the list of paraphrases for a given com­pound c in the test data. The pairwise preference puts a paraphrase in the context of its company, so that the opinions of human raters can be approxi­mated more precisely.