Laplace smoothing applied to unigram probability of word wi with count $c_i$, which is normalized by total number of work tokens $N$ and number of words in the vocabulary $V$:
$$P_{Laplace}(w_i) = \frac{c_i+1}{N+V}$$
It is more convenient to define an adjusted count $c_i^*$, and turn it into a probability $P_i^*$ by normalizing by $N$:
$$c_i^* = (c_i + 1)\frac{N}{N+V}$$
The add-one smoothed bigram probability and adjust count are given by
$$P^*_{Laplace}(w_n|w_{n-1}) = \frac{C(w_{n-1}w_n) + 1}{\sum_w (C(w_{n-1}w_n) + 1)} = \frac{C(w_{n-1}w_n) + 1}{C(w_{n-1}) + V}$$
$$c^*(w_{n-1}w_n) = \frac{(C(w_{n-1}w_n) + 1) \times C(w_{n-1})}{C(w_{n-1}) + V}

University of Michigan - Ann Arbor

Laplace (Add-One) Smoothing is a smoothing algorithm that adds one to all the bigram counts, before normalizing them into probabilities, hence all counts that were zero are now one, and so on. This algorithm provides a useful baseline, but does not perform well enough to be used in modern n-gram models.

Laplace (Add-One) Smoothing

An on-going but a helpful book resource about NLP
https://web.stanford.edu/~jurafsky/slp3/

Speech and Language Processing (3rd ed. draft) 

Add-One Smoothing Representation

Add-K Smoothing adds a fractional count k to each count instead of adding one, it moves less probability mass from the seen to the unseen events than add-one smoothing. Add-k smoothing requires a method for choosing k; this can be done, for example, by optimizing on a development set.

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