The probability/frequency of a word $w_i$ in an $N$-word corpus is given by

$$p(w_i) = \dfrac{c_i}{N}$$

where $c_i$ is the number of times the word $w_i$ appears in the corpus.

Suppose there are $V$ unique words in the corpus. Then add-one smoothing will leads to the new probability value of

$$p_{Laplace}(w_i) = \dfrac{c_i + 1}{N + V}$$

Now, how to understand the adjusted count $c_i^{*}$ given by

$$c_i^{*} = (c_i + 1) \dfrac{N}{N + V} $$

What does adjusted count it exactly tell us? How multiplying the new probability measure with $N$ manage to give us an adjusted count?


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