In statistics, additive smoothing, also called Laplace smoothing[1] (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 and parameter vector θ = (θ1, …, θd), a "smoothed" version of the data gives the estimator:

$\hat\theta_i= \frac{x_i + \alpha}{N + \alpha d} \qquad (i=1,\ldots,d),$

where 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 argued[citation needed]that α should be 1 (in which case the term add-one smoothing[2][3] 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.

## History

Laplace came up with this smoothing technique when he tried to estimate the chance that the sun will rise tomorrow. His rationale was that even given a large sample of days with the rising sun, we still can not be completely sure that the sun will still rise tomorrow (known as the sunrise problem).[4]

## Generalized to the case of known incidence rates

Often you are testing the bias of an unknown trial population against a control population with known parameters (incidence rates) μ = (μ1, …, μd). In this case the uniform probability 1/d should be replaced by the known incidence rate of the control population μi to calculate the smoothed estimator :

$\hat\theta_i= \frac{x_i + \mu_i \alpha d }{N + \alpha d } \qquad (i=1,\ldots,d),$

As a consistency check, if the empirical estimator happens to equal the incidence rate, i.e. μi = xi / N, the smoothed estimator is independent of α and also equals the incidence rate.

## Applications

### Classification

Additive smoothing is commonly a component of naive Bayes classifiers.

### Statistical language modelling

In a bag of words model of natural language processing and information retrieval, the data consists of the number of occurrences of each word in a document. Additive smoothing allows the assignment of non-zero probabilities to words which do not occur in the sample.