A Note on Inferential Decisions, Errors and Path-Dependency
Consider sequential binary testing under model uncertainty or misspecification in an otherwise 'ideal' setting: the a posteriori belief process and its objective conditional probability counterpart may differ but converge to the same correct outcome. We show that under common conditions (defined) unless the two are 'essentially identical', differing only by a priori factors, time-homogeneous continuous decisions based on one must fail to be path-independent with respect to state-variables based on the other or any non-essentially-identical process. The difference between them, inferential error, decomposes uniquely into two independent components: a path-independent systematic bias and a path-dependent, not necessarily systematic, error.
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