Additive smoothing

In statistics, additive smoothing, also called Laplace smoothing[1] or Lidstone smoothing, is a technique used to smooth categorical data. Given a set of observation counts ${\textstyle \textstyle {\mathbf {x} \ =\ \left\langle x_{1},\,x_{2},\,\ldots ,\,x_{d}\right\rangle }}$ from a ${\textstyle \textstyle {d}}$-dimensional multinomial distribution with ${\textstyle \textstyle {N}}$ 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 ${\textstyle \textstyle {{\hat {x}}_{i}=N{\hat {\theta }}_{i}}}$ and the "pseudocount" α > 0 is a smoothing parameter. α = 0 corresponds 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) ${\textstyle \textstyle {x_{i}/N}}$, and the uniform probability ${\textstyle \textstyle {1/d}}$. Invoking Laplace's rule of succession, some authors have argued^{[citation needed]} that α should be 1 (in which case the term add-one smoothing[2][3] is also used)^{[further explanation needed]}, 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 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.