  Probabilistic (Bayesian) modeling has experienced a surge of applications in
almost all quantitative sciences and industrial areas. This development is
driven by a combination of several factors, including better probabilistic
estimation algorithms, flexible software, increased computing power, and a
growing awareness of the benefits of probabilistic learning. However, a
principled Bayesian model building workflow is far from complete and many
challenges remain. To aid future research and applications of a principled
Bayesian workflow, we ask and provide answers for what we perceive as two
fundamental questions of Bayesian modeling, namely (a) "What actually is a
Bayesian model?" and (b) "What makes a good Bayesian model?". As an answer to
the first question, we propose the PAD model taxonomy that defines four basic
kinds of Bayesian models, each representing some combination of the assumed
joint distribution of all (known or unknown) variables (P), a posterior
approximator (A), and training data (D). As an answer to the second question,
we propose ten utility dimensions according to which we can evaluate Bayesian
models holistically, namely, (1) causal consistency, (2) parameter
recoverability, (3) predictive performance, (4) fairness, (5) structural
faithfulness, (6) parsimony, (7) interpretability, (8) convergence, (9)
estimation speed, and (10) robustness. Further, we propose two example utility
decision trees that describe hierarchies and trade-offs between utilities
depending on the inferential goals that drive model building and testing.
