**Organizer: **Francisco Javier Rubio, King’s College London, UK

**Speakers:**

Title:

Posterior distribution error control in the Bayesian approach to UQ in inverse problems

Abstract:

We generalize the results of previous results on expected Bayes factors (BF) to control the numerical error in the posterior distribution to an infinite dimensional setting when considering Banach functional spaces and now in a prior setting. The main result is a bound on the absolute global error to be tolerated by the Forward Map numerical solver, to keep the BF of the numerical vs. the theoretical model near to 1, now in this more general setting, possibly including a truncated, finite dimensional approximate prior measure. In so doing we found a far more general setting to define and prove existence of the infinite dimensional posterior distribution than what is commonly known in this area. Discretization consistency and rates of convergence are also investigated in this general setting for the Bayesian inverse problem.

Title:

Families of multivariate distributions derived from conjugate exponential models

Abstract:

Predictive distributions derived from exponential family models with conjugate priors are not necessarily exponential family models themselves, but may contain the generating exponential family as a limiting case. Observations that are i.i.d. under the generating exponential family are only exchangeable under the corresponding predictive distributions, and hence are identically distributed but not necessarily independent. We use a related idea to derived families of multivariate “predictive” distributions whose marginal distributions belong to the same family but are not equal.

Title:

A hierarchical Bayesian unit level model for small area estimation with log skew-normal random errors

Abstract:

Small Area Estimation (SAE) of population parameters is straight forward task when the hierarchical Bayesian formulation of the Unit Level Model (ULM) when the normality assumption is used. However, there are response variables which are positive and its empirical distribution exhibits skewness and outliers, then a probability distribution able to model skewness and a heavy left tail is needed. Many heavy tailed and skewed probability distributions have been used to estimate poverty indicators based on income data; nevertheless, under such models inclusion of the auxiliary covariates to build synthetic estimators on small areas is an active research field. This work proposes a hierarchical Bayesian model to estimate the average income in small areas by using a MCMC scheme based on the hidden truncation genesis of the multivariate skew normal distribution. This formulation allows us to use the standard results and interpretations of the ULM but assuming a log skew normal distribution of the errors. The application of the proposed model is illustrated with data from National Survey of Household Income and Expenditure in Mexico (2016) to estimate the average income in unobserved municipalities of the State of Mexico.

Keywords: Small Area Estimation, Unit Level Model, skewness, log skew normal distribution, hidden truncation.