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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)?

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  • 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
    Commented Aug 22, 2019 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
    Commented Aug 22, 2019 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
    Commented Aug 22, 2019 at 20:44
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    @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
    Commented Aug 22, 2019 at 22:53
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    A major issue here is that according to subjectivist Bayesianism there is no such thing as a true unknown prior probability/distribution, rather this is chosen from the point of view of the person to which the uncertainty refers. So the issue is not whether you "know" or "don't know" the prior, rather you have to construct it, optimally expressing your own uncertainty. Having done that, you can update your priors using Bayes' rule when new evidence comes in, as explained in existing answers. Commented Sep 5, 2023 at 18:20

2 Answers 2

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Bayesians wouldn't assign a specific numeric probability to such an event. Instead they would adopt a prior that was a probability distribution over the probability of the proposition being true. An objectivist Bayesian would probably choose a distribution than encoded only the fact that we don't know what that probability is, and use something like a Beta(1,1) prior which is uniform on the interval [0,1], i.e. every value of the probability the proposition is true is equally likely.

If any evidence came in, they could use Bayes rule to update their prior to produce a posterior, which would also be a probability distribution over the probability that the proposition is true.

Either way, if you want to decide what course of action to take, you work out a loss function, which tells you how much you will lose or gain under each strategy depending on whether the proposition is true or not in reality. We then work out the "expected loss" of each course of action by marginalising over our uncertainty of whether the proposition was true or not (in this case it would involve a sum of the losses, weighted by their posterior probabilities - I'm not going to spell it out on a SE without LateX). We then rationally choose the course of action with the lowest expected loss.

In the case of Pascals wager, the losses consist of:

(i) the cost of behaving like God exists if it does exist (ii) the cost of behaving like God exists if it does not exist (iii) the cost of behaving like God does not exist when it does (iv) the cost of behaving like God does not exist when it doesn't

So the Bayesian scheme doesn't tell you what to do, that depends on the losses, which are going to be your judgement and your prior (which may be an objective uninformative prior). It does however provide a rational means of going from a prior belief and a set of losses (and perhaps some evidence) to the course of action that is most likely to minimise your losses.

For details, the standard reference is Berger "Statistical Decision Theory and Bayesian Analysis", which is published by Springer.

Just a few quotes from the previous question that the OP mentions, first from Keynes:

About these matters there is no scientific basis on which to form any calculable probability whatever. We simply do not know."

The uniform uninformative prior encodes the knowledge that we simply do not know. It expresses no preference for any probability that God exists (in the case of Pascal's wager). Note it is only an interval in the sense that it covers the entire interval on which probabilities are defined. I suspect "interval" in the previous discussion is likely to mean a subset of [0,1].

And from Taleb:

It eliminates the need for us to understand the probabilities of a rare event (there are fundamental limits to our knowledge of these); rather, we can focus on the payoff and benefits of an event if it takes place.

If we reason from an uninformative prior, and have little or no evidence, the optimal course of action is essentially determined by the "payoff and benefits" (or in this case losses - Bayesians are a pessimistic lot! ;o). The marginalisation over the probability that the proposition is true means that it doesn't affect the outcome greatly simply because it is so vague and uninformative. So it all comes down to the costs. The real problem with Pascal's wager is that the cost of an eternity in Hell is essentially infinite, to all intents and purposes, so that dominates the decision. The difficulty is entirely in justifying the losses.

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    Choosing a supposedly uninformative prior in most real situations is hard. There is dependence on the parametrisation, and the principle of indifference can often be interpreted in various ways. In multivariate parameter spaces supposedly uninformative priors can implicitly impose far stronger information than what most people believe. See for example here tandfonline.com/doi/abs/10.1080/01621459.1996.10477003 Commented Sep 5, 2023 at 17:18
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    To add on to that, the idea of an “uninformative” prior makes no sense. If one is truly uninformative, one can’t assign a probability to it.
    – user62907
    Commented Sep 5, 2023 at 18:00
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    @ChristianHennig I completely agree, and if this was the statistics SE I would have mentioned it, but the discussion is currently at a far lower level than that so I thought it inappropriate here. As it happens, I wouldn't view a Jeffrey's prior as completely uninformative in that it encodes the knowledge that an invariance to parameterization is required (however that can be objectively justified). There is also the point that if you use an uninformative prior in situations where you do have solid knowledge (e.g. from physics) it can give obviously wrong [given that knowledge] conclusions. Commented Sep 5, 2023 at 18:10
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    " the principle of indifference may not produce definite, well-defined results for probabilities if it is applied uncritically when the domain of possibilities is infinite." is whether Pascal's deity exists or not an infinite domain? No, “Either Pascal's deity exists or it doesn't.”. Quit quote mining. Commented Sep 5, 2023 at 18:56
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    "We just map it into an infinite domain." why would we do that? You are just making stuff up now. Commented Sep 5, 2023 at 19:08
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P(H|E) = [P(H)*(P(E/H)]/P(E) is Bayes' Theorem.

Hypothesis (H): All Swans are White

The probability that all swans are white = P(H) = 50%

The probability that there's evidence (there are white swans) given all swans are white = P(E|H) > 50% (say P(E|H) = 70%)

The probability that there's evidence (there are white swans) = P(E) = 50%

P(H|E) = [50% * 70%]/50% = 70%.

So, the probability that all swans are white went up from 50% to 70%, after we saw some white swans (after we found some evidence).

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