Beyond Symmetry in Bayesian Posterior Approximations
Bayesian statistics plays a central role in modern science, medicine, economics, and artificial intelligence. It allows researchers to modify, in a principled way, their prior beliefs about unknown quantities of interest as new data become available. The result of this update is contained in a key object: the posterior distribution.
Unfortunately, although elegant and theoretically sound, the calculations required to obtain the exact posterior distribution are often too complex to carry out. For this reason, to make Bayesian methods applicable in practice, statisticians typically rely on tractable symmetric approximations of these posteriors. Many widely used techniques, such as the Laplace method and variational Bayes, yield approximations that are accurate when the posterior displays a symmetric bell-shaped form.
There’s just one issue: real posterior distributions are often not symmetric.
In many applications, posterior distributions are skewed, with more mass on one side than the other. Replacing them with symmetric approximations can therefore introduce bias and reduce accuracy, particularly when data are limited or when the model is complex and high-dimensional. Although in recent years researchers have developed more flexible skewed approximations, these methods are often computationally demanding, model-specific, or lack broad theoretical guarantees.
In their forthcoming paper in the Journal of the Royal Statistical Society: Series B, Francesco Pozza, Daniele Durante, and Botond Szabó (all from Bocconi’s Department of Decision Sciences and the Bocconi Institute for Data Science and Analytics research center) propose a simple and broadly applicable solution. Instead of replacing existing symmetric approximations with more elaborate (and computationally expensive) alternatives, the authors show how to systematically “perturb” them to incorporate skewness.
Starting from any symmetric approximation, the method produces a skewed version that remains mathematically tractable and requires no additional optimization. Importantly, the approach is supported by rigorous theory: the resulting skewed approximation is provably optimal within its class and improves accuracy both in finite samples and in large-data settings. In numerical studies and real-world applications, applying the method to state-of-the-art Gaussian approximations consistently enhanced performance in characterizing complex posteriors.
Rather than discarding established tools, this work offers a general upgrade to them. By embracing asymmetry in a principled and efficient way, it brings Bayesian approximations closer to the complex realities they are meant to represent.
