Bayesian inference is my main field of research.
I am working on an alternative approach to Bayesian inference where parameters are not estimated in the usual "likelihood-prior to posterior" way. The idea is instead to specify a predictive model for the data, impute the missing part of the population and estimate parameters by taking summary statistics of the reconstructed population (i.e. observed sample and imputed values).
In the literature, this approach is called Predictive resampling.
Choosing a predictive model entails interesting probabilistic and statistical challenges which are deeply interconnected.
On the one hand, it is crucial that the sequence of predictive distributions converges to some random measure. This is to ensure that resampled observations come from the same population. Then, to ensure that the reconstructed population is close as possible to the true one, predictive distributions have to be chosen in such a way that they have a good fit on the observed sample (since this contains all the available information on the true population).
My goal is to introduce new classes of predictive distributions that satisfy the above two requirements and apply them to different contexts, such as parameter estimation on univariate/multivariate data, regression and classification.
I have recently started working on Bayesian non-parametrics and in particular on the use of birth-death processes for the construction of non-parametric prior distributions. I am studying conditions for the difference of Poisson processes (P_1 - P_2, where P_1 is the birth process and P_2 is the death process) to be positive, with the goal of obtaining probability measures suitable to tempering of information.
