Bayesian Nonparametrics

We provide a new proof of the fact that the only Species Sampling Models where the probability of observing a new value does depend just on the sample size n and the number of clusters h are the Gibbs-type Processes.

We introduce Tail-Free processes, with particular reference to Pòlya Trees. And we develop some closed formulas for the mean posterior density of particular Pòlya Trees. We also provide some graphical illustrations.

The aim of this paper is to investigate the relationship between continuous and discrete dynamical systems in a unified approach. In order to do so, we briefly present the recent theory of time scales. In the main part, we find a general solution to homogeneous, constant-coefficients equations in both continuous and discrete time; we focus on the convergence between the two solutions. Finally, we apply time scales to other fields, we describe a unified Laplace-Z transform ad we derive a generalised Euler-Lagrange equation.

We introduce Classical Kernel Density Estimation, we find optimal bandwidth and Kernel wrt a square loss criterion and we implement an algorithm on Python. We introduce the Bayesian framework and we prove two formulas by Lo and we apply them to Exponential and Normal Kernels and we develop a Python algorithm for the computation of the optimal Bayesian Density estimator with square loss. We finally report some results in Analysis linked with the choice of the Kernel in the Bayesian setting.