Katz's back-off model

Katz back-off is a generative n-gram language model that estimates the conditional probability of a word given its history in the n-gram. It accomplishes this estimation by "backing-off" to models with smaller histories under certain conditions. By doing so, the model with the most reliable information about a given history is used to provide the better results.

The method

The equation for Katz's back-off model is deceptively simple:

$P_{bo}(w_{i}|w_{i-n+1}\cdots w_{i-1})={\begin{cases}d_{w_{i-n+1}\cdots w_{i-1}}{\frac {C(w_{i-n+1}...w_{i-1}w_{i})}{C(w_{i-n+1}\cdots w_{i-1})}}{\mbox{ if }}C(w_{i-n+1}\cdots w_{i})>k\\\alpha _{w_{i-n+1}\cdots w_{i-1}}P_{bo}(w_{i}|w_{i-n+2}\cdots w_{i-1}){\mbox{ otherwise}}\end{cases}}$

where,

$C(x)$ = number of times x appears in training

$w_{i}$ = ith word in the given context

Essentially, this means that if the n-gram has been seen k or more times in training, the conditional probability of a word given its history is proportional to the maximum likelihood estimate of that n-gram. Otherwise, the conditional probability is equal to the back-off conditional probability of the "(n-1)-gram".

The more difficult part is determining the values for k, d and α.

Computing the parameters

k is the least important of the parameters. It is usually chosen to be 0. However, empirical testing may find better values for k.

d is typically the amount of discounting found by Good-Turing estimation. In other words, if Good-Turing estimates $C$ as $C^{*}$, then $d={\frac {C^{*}}{C}}$

To compute α, it is useful to first define a quantity β, which is the left-over probability mass for the (n-1)-gram:

$\beta _{w_{i-n+1}\cdots w_{i-1}}=1-\sum _{\{w_{i}:C(w_{i-n+1}\cdots w_{i})>0\}}d_{w_{i-n+1}\cdots w_{i-1}}{\frac {C(w_{i-n+1}...w_{i})}{C(w_{i-n+1}\cdots w_{i-1})}}$

Then the back-off weight, α, is computed as follows:

$\alpha _{w_{i-n+1}\cdots w_{i-1}}={\frac {\beta _{w_{i-n+1}\cdots w_{i-1}}}{\sum _{\{w_{i}:C(w_{i-n+1}\cdots w_{i})=0\}}P_{bo}(w_{i}|w_{i-n+2}\cdots w_{i-1})}}$

Discussion

This model generally works well in practice, but fails in some circumstances. For example, suppose that the bigram "a b" and the unigram "c" are very common, but the trigram "a b c" is rarely seen (less than k times). Instead of assigning a more appropriate value of 0, the method will back off to the bigram and estimate P(c | b).