This paper is devoted to the problem of characterizing coherent inferences in the countably additive setting. The concept of coherence distinguishes inference procedures with certain types of inconsistency. The main idea is to ask a statistician to accept bets on subsets of a parameter space at prices based on his own inference. The inference is coherent if we cannot beat the statistician.
There are different interpretations of the concept of coherence. We approach it as follows. Some reference measure is selected to fix null sets of the parameter space. Depending on an observation, a statistician chooses certain subsets of this space and posts his odds according to his inference method. Then a gambler is allowed to place bets on these subsets. We ask whether the inference permits the existence of betting strategies with a positive expected payoff function for all $\theta$ not in a null set. Actually, we extend our definition of coherence to consider certain limits of sequences of betting strategies. Our main result says that an inference function is coherent if and only if it arises from a proper countably additive prior equivalent to the selected reference measure.
We provide several examples to demonstrate that some classical procedures produce incoherent inferences. In some cases incoherence can be shown only by considering the limiting behavior of sequences of betting strategies.
Coherence in betting has a clear financial interpretation; the relationship of our results to the arbitrage theorems of mathematical finance is particularly relevant since we use similar techniques. The existence of a prior for an inference function can be seen as an analog for existence of a martingale measure.