Additive smoothing

In statistics, additive smoothing, also called Laplace smoothing (not to be confused with Laplacian smoothing), or Lidstone smoothing, is a technique used to smooth categorical data. Given an observation x = (x1, …, xd) from a multinomial distribution with N trials, a "smoothed" version of the data gives the estimator:

where the pseudocount α > 0 is the smoothing parameter (α = 0 corresponds to no smoothing). Additive smoothing is a type of shrinkage estimator, as the resulting estimate will be between the empirical estimate xi / N, and the uniform probability 1/d. Using Laplace's rule of succession, some authors have arguedthat α should be 1 (in which case the term add-one smoothing is also used), though in practice a smaller value is typically chosen.

From a Bayesian point of view, this corresponds to the expected value of the posterior distribution, using a symmetric Dirichlet distribution with parameter α as a prior. In the special case where the number of categories is 2, this is equivalent to using a Beta distribution as the conjugate prior for the parameters of Binomial distribution.