Additive smoothing

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, …, x_d) from a multinomial distribution with N trials and parameter vector θ = (θ₁, …, θ_d), 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 the pseudocount 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 x_i / N, and the uniform probability 1/d. Using Laplace's rule of succession, some authors have argued 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. 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 Binomial distribution.

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, …, μ_d). In this case the uniform probability 1/d should be replaced by the known incidence rate of the control population μ_i 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. μ_i = x_i / 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. Recent studies have proven that additive smoothing is more effective than other probability smoothing methods in several retrieval tasks such as language-model-based pseudo-relevance feedback and recommender systems.^{[5] [6]}

See also

• Bayesian average
• Pseudocount

References

1. C.D. Manning, P. Raghavan and M. Schütze (2008). Introduction to Information Retrieval. Cambridge University Press, p. 260.
2. Jurafsky, Daniel; Martin, James H. (June 2008). Speech and Language Processing (2nd ed.). Prentice Hall. p. 132. ISBN 978-0-13-187321-6.
3. Russell, Stuart; Norvig, Peter (2010). Artificial Intelligence: A Modern Approach (2nd ed.). Pearson Education, Inc. p. 863.
4. Lecture 5 | Machine Learning (Stanford) at 1h10m into the lecture
5. Hazimeh, Hussein; Zhai, ChengXiang. "Axiomatic Analysis of Smoothing Methods in Language Models for Pseudo-Relevance Feedback". ICTIR '15 Proceedings of the 2015 International Conference on The Theory of Information Retrieval.
6. Valcarce, Daniel; Parapar, Javier; Barreiro, Álvaro. "Additive Smoothing for Relevance-Based Language Modelling of Recommender Systems". CERI '16 Proceedings of the 4th Spanish Conference on Information Retrieval.

External links

• 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.