When dealing with n-gram models, smoothing refers to the practice of adjusting empirical probability estimates to account for insufficient data.
In the descriptions below, we use the notation w^{j}_{i}, i < j, to denote the (j - i)-gram (w_{i}, w_{i+1}, \ldots, w_{j}).
Laplace smoothing is the assumption that each n-gram in a corpus occurs exactly one more time than it actually does.
p(w_i \mid w^{i-1}_{i-n+1}) = \frac{1 + c(w^{i}_{i-n+1})}{|V| \sum_{w_i} c(w^{i}_{i-n+1})}
where c(a) denotes the empirical count of the n-gram a in the corpus, and |V| corresponds to the number of unique n-grams in the corpus.
Models
Additive/Lidstone smoothing is a generalization of Laplace smoothing, where we assume that each n-gram in a corpus occurs k more times than it actually does (where k can be any non-negative value, but typically ranges between [0, 1]):
p(w_i \mid w^{i-1}_{i-n+1}) = \frac{k + c(w^{i}_{i-n+1})}{k |V| \sum_{w_i} c(w^{i}_{i-n+1})}
where c(a) denotes the empirical count of the n-gram a in the corpus, and |V| corresponds to the number of unique n-grams in the corpus.
Models
Good-Turing smoothing is a more sophisticated technique which takes into account the identity of the particular n-gram when deciding the amount of smoothing to apply. It proceeds by allocating a portion of the probability space occupied by n-grams which occur with count r+1 and dividing it among the n-grams which occur with rate r.
r^* = (r + 1) \frac{g(r + 1)}{g(r)} \\
p(w^{i}_{i-n+1} \mid c(w^{i}_{i-n+1}) = r) = \frac{r^*}{N}
where r^* is the adjusted count for an n-gram which occurs r times, g(x) is the number of n-grams in the corpus which occur x times, and N is the total number of n-grams in the corpus.
Models
References
| [1] | Chen & Goodman (1998). "An empirical study of smoothing techniques for language modeling". Harvard Computer Science Group Technical Report TR-10-98. |
| [2] | Gale & Sampson (1995). "Good-Turing frequency estimation without tears". Journal of Quantitative Linguistics, 2(3), 217-237. |
.. toctree:: :maxdepth: 3 :hidden: numpy_ml.ngram.mle numpy_ml.ngram.additive numpy_ml.ngram.goodturing