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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:

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

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 Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.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

According to Andrew Ng, Laplace came out with this smoothing technique when he tried to estimate the chance that Sun will rise tomorrow. His rational was that even given a large sample of days with rising Sun, we still can not be completely sure that Sun will rise also tomorrow.[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 :

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

As a consistency check, if the empirical estimator happens to equal the incidence rate, i.e. μi = xi / d , 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.

Chen & Goodman (1996) empirically compare additive smoothing to a variety of other techniques, using both α fixed at one and a more general value.