Additive smoothing

In statistics, additive smoothing, also called Laplace smoothing^[1] or Lidstone smoothing, is a technique used to smooth count data, eliminating issues caused by certain values having 0 occurrences. Given a set of observation counts ${\displaystyle \mathbf {x} =\langle x_{1},x_{2},\ldots ,x_{d}\rangle }$ from a ${\displaystyle d}$-dimensional multinomial distribution with ${\displaystyle N}$ trials, a "smoothed" version of the counts gives the estimator

${\displaystyle {\hat {\theta }}_{i}={\frac {x_{i}+\alpha }{N+\alpha d}}\qquad (i=1,\ldots ,d),}$

where the smoothed count ${\displaystyle {\hat {x}}_{i}=N{\hat {\theta }}_{i}}$, and the "pseudocount" α > 0 is a smoothing parameter, with α = 0 corresponding to no smoothing (this parameter is explained in § Pseudocount below). Additive smoothing is a type of shrinkage estimator, as the resulting estimate will be between the empirical probability (relative frequency) ${\displaystyle x_{i}/N}$ and the uniform probability ${\displaystyle 1/d.}$ Common choices for α are 0 (no smoothing), +1⁄2 (the Jeffreys prior), or 1 (Laplace's rule of succession),^[2][3] but the parameter may also be set empirically based on the observed data.

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 distribution. 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 the binomial distribution.