Symmetry of evidence without evidence of symmetry

The de Finetti Theorem is a cornerstone of the Bayesian approach. Bernardo (1996) writes that its "message is very clear: if a sequence of observations is judged to be exchangeable, then any subset of them must be regarded as a random sample from some model, and there exists a prior distribution on the parameter of such model, hence requiring a Bayesian approach." We argue that while exchangeability, interpreted as symmetry of evidence, is a weak assumption, when combined with subjective expected utility theory, it implies also complete confidence that experiments are identical. When evidence is sparse, and there is little evidence of symmetry, this implication of de Finetti's hypotheses is not intuitive. This motivates our adoption of multiple-priors utility as the benchmark model of preference. We provide two alternative generalizations of the de Finetti Theorem for this framework. A model of updating is also provided.