## Comparing two populations using Bayesian Fourier series density estimation

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2017-04-12##### Author

Inacio, Marco Henrique de Almeida

http://lattes.cnpq.br/1931901020027887

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Given two samples from two populations, one could ask how similar the populations are, that is,
how close their probability distributions are. For absolutely continuous distributions, one way
to measure the proximity of such populations is to use a measure of distance (metric) between
the probability density functions (which are unknown given that only samples are observed). In
this work, we work with the integrated squared distance as metric. To measure the uncertainty
of the squared integrated distance, we first model the uncertainty of each of the probability
density functions using a nonparametric Bayesian method. The method consists of estimating the
probability density function f (or its logarithm) using Fourier series {f0;f1; :::;fI}. Assigning a
prior distribution to f is then equivalent to assigning a prior distribution to the coefficients of this
series. We used the prior suggested by Scricciolo (2006) (sieve prior), which not only places a
prior on such coefficients, but also on I itself, so that in reality we work with a Bayesian mixture
of finite dimensional models. To obtain posterior samples of such mixture, we marginalize out
the discrete model index parameter I and use a statistical software called Stan. We conclude
that the Bayesian Fourier series method has good performance when compared to kernel density
estimation, although both methods often have problems in the estimation of the probability
density function near the boundaries. Lastly, we showed how the methodology of Fourier series
can be used to access the uncertainty regarding the similarity of two samples. In particular, we
applied this method to dataset of patients with Alzheimer.