  Likelihood ratios (LRs), which are commonly used for probabilistic data
processing, are often estimated based on the frequency counts of individual
elements obtained from samples. In natural language processing, an element can
be a continuous sequence of $N$ items, called an $N$-gram, in which each item
is a word, letter, etc. In this paper, we attempt to estimate LRs based on
$N$-gram frequency information. A naive estimation approach that uses only
$N$-gram frequencies is sensitive to low-frequency (rare) $N$-grams and not
applicable to zero-frequency (unobserved) $N$-grams; these are known as the
low- and zero-frequency problems, respectively. To address these problems, we
propose a method for decomposing $N$-grams into item units and then applying
their frequencies along with the original $N$-gram frequencies. Our method can
obtain the estimates of unobserved $N$-grams by using the unit frequencies.
Although using only unit frequencies ignores dependencies between items, our
method takes advantage of the fact that certain items often co-occur in
practice and therefore maintains their dependencies by using the relevant
$N$-gram frequencies. We also introduce a regularization to achieve robust
estimation for rare $N$-grams. Our experimental results demonstrate that our
method is effective at solving both problems and can effectively control
dependencies.
