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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 and parameter vector θ = (θ1, …, θd), a "smoothed" version of the data gives the estimator:
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 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 conjugagate prior for the parameters of Binomial distribution.
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).
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 :
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.
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.
- C.D. Manning, P. Raghavan and M. Schütze (2008). Introduction to Information Retrieval. Cambridge University Press, p. 260.
- Jurafsky, Daniel; Martin, James H. (June 2008). Speech and Language Processing (2nd ed.). Prentice Hall. p. 132. ISBN 978-0-13-187321-6.
- Russell, Stuart; Norvig, Peter (2010). Artificial Intelligence: A Modern Approach (2nd ed.). Pearson Education, Inc. p. 863.
- Lecture 5 | Machine Learning (Stanford) at 1h10m into the lecture
- SF Chen, J Goodman (1996). "An empirical study of smoothing techniques for language modeling". Proceedings of the 34th annual meeting on Association for Computational Linguistics.