Abstract

Symbolic data analysis (SDA) aggregates large individual-level datasets into a small number of distributional summaries, such as intervals or histograms. Inference is carried out using these summaries in place of the original dataset, resulting in computational gains at the loss of some information. In likelihood-based SDA, the likelihood function is characterised by an integral with a large exponent and in some circumstances the likelihood function is known to produce biased parameter estimates. To remedy for this Yang and coauthors (Yang et al., 2024) have developed a Bayesian framework for SDA methods in these settings using pseudo-marginal Markov-chain Monte Carlo methods based on path sampling, block-Poisson estimator, and a much faster but approximate method based on Taylor expansion. In our article we extend their approach to the study of Bayesian likelihood approaches for Irpino-Verde two-component regression for histogram data.

We propose a calibrated Bayesian composite likelihood approach which reduces the need for a fully specified likelihood function and allows for complex within-study correlations. To enable robust posterior inference we develop a hybrid Gibbs sampler algorithm with Open-Faced Sandwich post-sampling adjustment (see e.g. Wang et al., 2025). We derive a Bernstein-von Mises theorem and study contraction rates for the Bayesian estimator proposed. Through simulation studies we demonstrate that the approach yields unbiased estimates and maintains coverage probabilities close to the nominal levels. Our approach upgrades the previous approach of Yang and coauthors with allowing for a more complex and adequate SDA study design and shows a solid computational performance and improved asymptotic efficiency.

In conclusion we discuss connections between Bayesian probability and statistics and symbolic data analysis. Both areas share some very interesting common features, in particular as related to Bayesian nonparametrics and concept of random measure. This promises highly interesting future pathways of study in SDA.

Our analysis is AI-assisted using ChatGPT 5.6 and Lean 4.