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}

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.

Add-K Smoothing

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.

University of Michigan - Ann Arbor

Smoothing (Discounting) is the modification that keeps a language model from assigning zero probability to unseen events, by shaving off a bit of probability mass from some more frequent events and give it to the events that were never seen.

Smoothing (Discounting)

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

Speech and Language Processing (3rd ed. draft) 

Absolute discounting is a method of discounting the counts for n-grams. In absolute discounting, a fixed (absolute) discount d is subtracted from each count. The intuition is that a small discount d will not significantly affect high count estimates, which are already well-estimated, but will modify the smaller counts in a generally positive way. 

Absolute Discounting

Laplace (Add-One) Smoothing

Backoff is the method of estimating probabilities using to a lower-order n-gram if we have zero evidence for a higher-order n-gram; if the n-gram we need has zero counts, we approximate it by backing off to the (n-1)-gram, and continue backing off until we reach a history that has some counts.

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