Causal Interpretations of Probability

scientific theories or hypotheses. Since the range of alternatives is not known in these cases, it seems implausible to construct a collective and relatedly the measure remains undetermined. If one requires probabilities to be predictive, the range of hypotheses to which probabilities should be ascribed is thus rather restricted. 44 We are therefore in the position to assess the plausibility of the various Bayesian programs from the perspective of causal probability. Sometimes, the hypothesis space and the measure are objectively determined by the causal set-up. Consider for example the following experiment with three urns, each containing both black and white balls but in different ratios, e.g. 1:2, 1:1, 2:1, corresponding to three hypotheses. Now, one of these urns is randomly chosen and then balls are drawn with replacement. Given a certain sequence of draws as evidence, e.g. w-w-b, a probability for each of the three hypotheses can be calculated. In this specific situation, an objective Bayesian approach is feasible because all relevant elements are determined by the physical set-up: the hypothesis space, the initial probability measure over the hypothesis space, and the probability of evidence given a certain hypothesis is true. In other circumstances, we might not be so lucky. We may for example be confronted with limited information about a single urn, e.g. that the colors of the balls are only black and white and that there are no more than five balls in the urn. In this case, the hypothesis space is determined by the set-up but there is flexibility in the choice of measure since the actual process with which the urn was prepared is unknown. In analogy to the discussion in point iii) of Section 6b, the Bayesian can now construct in her mind a collective to which the urn is attributed, e.g. an ensemble in which every ratio of balls has equal prior probability. With respect to such a collective, the posterior probabilities of the various hypotheses can then be calculated taking into account additional evidence. However, the Bayesian might just as well have chosen a different measure over the hypothesis space and would have come up with a different result for the posterior probabilities. There is no contradiction, since strictly speaking the probabilities only hold relative to the respective collective and measure. In cases, where the measure is underdetermined by given knowledge and somewhat arbitrarily construed with respect to an imagined collective, we may plausibly speak of subjective Bayesianism. Of course, much more should be said how Bayesianism is to be integrated into the framework of causal probability. But the brief discussion above already suggests how the notion of causal probability allows determining the limits of a Bayesian approach.