Penalised Complexity-priors: A new tool to define default priors and robustify Bayesian models

Bayesian statistics aims firstly to accurately describe a priori beliefs about a phenomenon based on expert knowledge and previously performed studies, and then to update these beliefs based on the evidence in observed data. This results in robust methods that accurately describe the uncertainty not only arising from the model, but also from the parameters that describe the model. The prior beliefs are expressed through a probability distribution called the prior distribution. However, eliciting prior distributions are challenging, and, in practice, prior distributions are elicited for each parameter separately. In this project, we construct intuitive joint prior distributions for the parameters that respect the model structure and allow a more accurate description of the a priori beliefs. In the case of an additive model, we decompose the total variance hierarchically along a tree structure to the individual parts of the model. This provides an intuitive and proper joint prior distribution for the variance parameters, and the clear graphical representation encourages increased transparency within the scientific community. We have seen that this new way of defining prior distributions is useful in many scientific areas. One example is genomic modelling. Here we can take advantage of the expert knowledge existing about the relative magnitude of the sources of phenotypic variation in an intuitive way, which is easy to communicate to applied scientists and leads to robust statistical inference. We provide a new R-package called "makemyprior", which offers an intuitive graphical user interface to facilitate construction and assessment of different choices of priors through visualization of the tree and joint prior. After completing the prior specification, inference can be carried out directly with the popular R packages INLA (www.r-inla.org) and rstan (www.mc-stan.org), or pre-computed priors can be fed into other Bayesian software. The package aims to expand the toolbox of applied researchers and make priors an active component in their Bayesian workflow.

Providing a well-motivated, transparent and intuitive framework for formulating priors is relevant for researchers in applied statistics and sciences. We developed a general framework for constructing joint priors for variance parameters in a Bayesian hierarchical model. Prior knowledge, obtained from previous experiments or comparable investigations, and expert knowledge can be intuitively included to make the model more robust while the model structure is acknowledged. Up til now, there has been lack of software that intuitively allows the proper inclusion of such knowledge and simple visualisation of chosen priors, and with our new framework implemented in the R-package "makemyprior" we close this gap.

A long lasting problem within Bayesian statistics, is the choice of prior distributions. Although various approaches have been suggested to approach this issue, the current practice among applied statisticians is not good. In this project we will develop a recent proof of concept idea of Penalised Complexity (PC) priors, which is a principled approach to construct priors. This approach constructs priors that are invariant to reparameterisations, are designed to support Occam's razor and seem to have excellent robustness properties. We will develop this idea further and define default priors suitable for routine applied use. The priors and with them model robustness against possible model deviations can be controlled by the user in a transparent, consistent and intuitive way. Instead of considering each model component separately, the user needs only to provide an intuition about the linear predictor as a whole, which is then like a chain reaction transferred to the single model components. The framework will be integrated into the program-system R-INLA for doing Bayesian inference in latent Gaussian models.