Specialized Session 2: Bayesian Data Science

The replication crisis is affecting scientific research, leading to a increased focus on methods for conducting replication studies. We introduce a Bayesian method based on mixture priors to asses replication success. This involves combining an informative prior distribution based on the original study with a non-informative prior. Then, the mixture weight specifies how the data from both studies are integrated. In this context, it is shown how Bayes factors allow for formal hypothesis testing concerning effects. We illustrate our approach with data from three replication studies, showing that mixture priors present a practical and intuitive alternative to other Bayesian techniques, such as hierarchical models and power priors.

We introduce a general strategy for defining distributions over the space of sparse symmetric positive definite matrices by leveraging its Cholesky factorization. In particular, we propose a modification of the Bartlett decomposition which we name S-Bartlett that recovers the typical Wishart distribution as a particular case. Notably, the proposed hierarchical form does not require the numerical evaluation of normalization constants. By incorporating such hierarchical specification with a Spike-and-Slab prior, our approach enforces sparsity and enables efficient inference via a tailored MCMC routine.

Partially exchangeable data can be described with a random probability measure for each group of observations. The joint de Finetti prior of the vector of random probability measures governs group dependence, affecting the sharing of information. For two groups, the most common index in the literature is the set-wise correlation, which in some situations is unstable with respect to the selection of the set. To address this, we propose a novel RKHS-based correlation index using kernel mean embeddings of random probability measures. This index provides a robust quantification of dependence, generalizing the set-wise correlation for an appropriate choice of the kernel and addressing its limitations under additional assumptions on the kernel. We illustrate some theoretical advantages with examples. This work is based on some recent results in [5] by the same authors.