Additive Smoothing

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An Additive Smoothing is an image processing technique for smoothing categorical data



[math]\hat\theta_i= \frac{x_i + \alpha}{N + \alpha d} \qquad (i=1,\ldots,d),[/math]
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 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 smoothingis 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.