On Bayesian inference in pulsar timing arrays: identifiability and choice of the priors

Pulsar Timing Arrays (PTAs) represent one of the most active and rapidly evolving research areas in contemporary astrophysics.PTAs use pulsar observations to search for gravitational waves, relying on accurate statistical models of pulsar timing data. In this work, we revisit the bias problem reported in both Bayesian and frequentist analyses and identify the structural features of the statistical models that give rise to these effects. We show that it originates from an underlying identifiability issue in commonly used linear Gaussian models. We outline and discuss different possible remedies to the non-identifiability. For instance, one possible approach is the use of hierarchical priors, which can alleviate non-identifiability but at the cost of making the inference strongly prior-dependent: the choice of the prior effectively selects one specific set of parameters among infinitely many equivalent solutions. Hence, in these non-identifiable models, the resulting inference outputs along the non-identifiable directions are effectively determined by the choice of the prior, rather than by the data themselves (becoming completely prior-driven).Moreover, we also highlight the limitations of the use of diffuse and improper priors in Bayesian model selection, as they can distort Bayes factors and compromise the robustness of gravitational-wave detection analyses.Finally, we give an overview of best practice for prior choice in model selection and suggest possible remedies to the limitations of linear PTA models reported in literature.