Under uncertainty, precise probability cannot be assigned, see my other question: How valid is assignment of probabilities when evidence is totally lacking, as in Pascal's Wager? In this case, either probability cannot be assigned, or a range of probabilities is assigned (in the case of complete uncertainty, [0, 1]).

How does this change when there is some evidence, but not full evidence? For example, many cosmologists posit the existence of an infinite universe under the assumption that the universe is flat. This does not directly follow, yet many physicists say that it is "likely."

In general, how can evidence slightly favoring one hypothesis over another be used in support of that hypothesis, if the prior probability of the hypothesis being true is not known? Indeed, if we have a probability ranging over [0, 1], and any value can rationally be chosen from that, we could choose 1. This makes evidence useless in support or refutation of that hypothesis, according to Bayes' Theorem.

Question: In what sense can we use evidence to change our probabilities of a given proposition if the prior probability of that proposition cannot be known (or range over an interval)?

  • Given that probabilities are in the range [0,1], we can apply some math to them, we can often compute maximum likelihoods. In particular, enough repetition can make the original distribution and probabilities highly unlikely to matter. We really can measure the maximum likelihood they do matter due to the Law of Large numbers. The whole mechanism used by most scientists is modern Normal Theory statistical techniques, which rely upon the normality of measures on large enough data sets to give answers that are less than a fixed probability of lying outside a given range.
    – user9166
    Aug 22 '19 at 19:56
  • @jobermark First, my question isn't really aimed at cases where evidence is abundant, but more where evidence is few and far between. Second, I'm not sure that it really matters how much evidence you have if the probability is either 0 or 1 (Bayes' theorem). But yes, I think the probability does converge in (0, 1), but not [0, 1]. Perhaps (0, 1) should be used in scientific questions?
    – Josh
    Aug 22 '19 at 20:15
  • It is not that prior probabilities can not be known, it is that there is nothing to know. In many cases, Bayesian priors are an artifice assigned more or less based on technical convenience (Wikipedia lists some schemes). It is how they are updated that really matters.
    – Conifold
    Aug 22 '19 at 20:44
  • @Conifold So the ability to incorporate evidence, and how much pull that evidence has, is dependent on the prior you choose? Is there, perhaps, another way to evaluate evidence besides having to update your prior? Besides, of course, frequentist probability.
    – Josh
    Aug 22 '19 at 22:20
  • 1
    @Josh But then the answer is obviously 'It can't that is why we verify theories with more than one test.' People don't judge theories on little bits of data, they judge it on subjective criteria, or adequate data.
    – user9166
    Aug 22 '19 at 22:53

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