Tutor: Tommaso Rigon

Politecnico di Milano - Mathematical Engineering - Bayesian Statistics

Author
Affiliation

Tommaso Rigon

Università degli Studi di Milano-Bicocca

Published

September 26, 2025

Short Bio

  • My name is Tommaso Rigon and I am a Senior Assistant Professor (RTD-B).

  • University of Milano-Bicocca, Department of Economics, Management and Statistics (DEMS), Milan, Italy.

    • Senior Assistant Professor (RTD-B), 2023 - Present
    • Junior Assistant Professor (RTD-A), 2020 - 2023

  • Duke University, Department of Statistical Science, Durham (NC), U.S.A.

    • Postdoctoral Associate, 2020 - 2020
    • Research Associate, 2019 - 2020
Education

Hyperbolic secant regression

Background

Let Y_1,\dots,Y_n be independent random variable distributed according to a hyperbolic secant distribution, whose density is f(y_i; \theta_i) = \frac{\exp\{\theta_i y_i - \log{\cos(\theta_i)} \}}{2 \cosh(\pi y_i / 2)}, \qquad y_i \in \mathbb{R}, \; \theta_i \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right). This distribution is discussed in Morris (1982) and is an instance of highly tractable exponential family having a quadratic variance function.

  • This distribution can be employed to build a novel generalized linear model (GLM), which automatically incorporates heteroskedasticity and exhibits heavier tails than the Gaussian law.

  • There are multiple application areas for such a regression technique, including (but not limited to) financial data.

Objectives and expectations

Research gap

  • A systematic Bayesian investigation of such a GLM is entirely lacking. In principle, this GLM framework can be combined with modern Bayesian tools for regression, such as:

    • Variable selection and regularization, e.g., shrinkage priors or spike-and-slab priors.
    • Random effects (parametric and nonparametric).
    • Generalized Additive Models (GAMs), e.g., Gaussian Processes.

  • Overdispersion may require novel generalized Bayes techniques (Agnoletto et al. 2025).

  • A Polya-gamma data augmentation scheme may be applicable by adapting the results in Polson et al. (2013). If not, Hamiltonian Monte Carlo (HMC) remains a feasible option.

Expected outcome

The group is expected to implement and develop 2–3 of the research gaps outlined above.

I expect the group to be proficient in R programming (knowledge of C++ is a strong asset, though probably not essential). Most of the work will focus on the practical implementation of these ideas.

References

Agnoletto, D., Rigon, T., and Dunson, D. B. (2025), “Bayesian inference for generalized linear models via quasi-posteriors,” Biometrika, 112.
Morris, C. N. (1982), “Natural exponential families with quadratic variance functions,” Annals of Statistics, 10, 65–80.
Polson, N. G., Scott, J. G., and Windle, J. (2013), “Bayesian inference for logistic models using polya-gamma latent variables,” Journal of the American Statistical Association, Taylor {&} Francis, 108, 1–42.