40 PRIOR-FREE PROBABILISTIC INFERENCE
posterior probabilities, except for possibly in a large-sample limit, and now Propo-
sition 2.1 provides some explanation of what has gone wrong. In other words, the
Bayes theorem is a perfectly valid tool for combining beliefs coming from different
sources, but the belief of ignorance about θ, which we think is a typical one in ap-
plications, cannot be described by a (proper or improper) prior, and if one or both of
the beliefs to be combined are not trustworthy, then neither is the output.
2.4.2 Additivity and evidence in data
Probability is predictive in nature, that is, it is designed to describe our uncertainties
about a yet-to-be-observed outcome. In our particular statistical inference problem,
however