Last modified: 2018-06-17

#### Abstract

Entropy is a measure of heterogeneity widely used in applied sciences, where often spatial data are present. Recently, many approaches have been proposed to include spatial information in entropy, in order to synthesize the observed data in a single, interpretable number. In other studies the objective is, rather, entropy estimation; at this regard several proposals appear in the literature, which basically are corrections of the plug-in estimator, where proportions take the place of the required probabilities. In these estimation procedures, usually independence is assumed and spatial correlation is not considered.

We propose a new perspective for spatial entropy estimation: instead of intervening on the entropy estimator, we focus on improving the estimation of its components, i.e. the probabilities. Once probabilities are suitably evaluated, estimating entropy is straightforward since it is a deterministic function of the probability distribution. Therefore, we estimate the probabilities of a multinomial distribution for categorical variables, accounting for spatial correlation following a Bayesian approach. We obtain a posterior distribution for the parameters, hence a posterior distribution for entropy, which can be summarized as wished.