In a general setting in which prior distributions that may take on the value $\infty$ are admitted, an inference based on a posterior for a prior, $\mu$, that is "minimally compatible" with the inference is shown to have a strong property of expectation consistency, that implies a corresponding property of coherence: A nonnegative expected payoff function for a gambler's strategy is necessary 0 almost everywhere $(\mu)$. In the converse direction, under appropriate regularity conditions involving continuity of the sampling distribution and of the inference, a weaker version of coherence implies that the inference is based on a posterior distribution.
"Fully Coherent Inference." Ann. Statist. 19 (2) 830 - 849, June, 1991. https://doi.org/10.1214/aos/1176348123