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There are many statements I feel more confident in than others. For example, I would wager that almost everyone would feel more confident in the statement "The sun will rise tomorrow" than "a murder will happen in my neighborhood tomorrow". Now, if I were to ask you to assign a probability to either of those statements, you would obviously not be able to. However, you would be able to say that the first is more likely.

With examples like the above, a difference in confidence of belief seems intuitive. But what about when considering cases that both equally seem to have no evidence for them?

For example, imagine two scenarios. Assume that in both of these scenarios, there is no cheating done by humans as a matter of fact. This is important for later.

Scenario A : There are a million games being played in private. With each game, there is a 1/million chance of winning a prize. One person in one of these games wins.

Scenario B: 999,999 games are played in private. The last one is played in public in front of a massive audience. The person in the last game wins. Assume that person is a Christian.

Assume you are now tasked with figuring out whether or not the Christian God intervened in either of those scenarios. Most would find it unreasonable to think this.

However, now suppose you are asked about what your comparative degree of belief is? As in, is your confidence in (God intervening in Scenario A) greater than or less than (Go d intervening in Scenario B)? Even if both confidences are infinitesimal, many might say that scenario B should bring greater confidence given that it's possible that the Christian God might want to send some sort of sign down in public in front of an audience.

On the face of it, part of this seems ridiculous to me. After all, there is no evidence of the Christian God, much less any prior evidence of the Christian God intervening in lotteries. Given that there is no prior evidence of the Christian God, what would be the most rational state of mind?

A) Have more confidence in (God intervened in Scenario B) compared to Scenario A)

B) Have more confidence in (God intervened in Scenario A) compared to Scenario B)

or

C) Have equal confidence in that neither of them occurred by a Christian God

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  • It's easy to assign probabilities to both your initial statements: for the run rising tomorrow: 0.999999. For the crime in my neighborhood: 0.1. Done.
    – Frank
    Jan 21 at 0:25
  • If there is no evidence, the most rational thing to do is to assign a uniform probability, 0.5 if there are two alternatives, since this is the most symmetric, and there is no evidence to perturb that symmetry. Done.
    – Frank
    Jan 21 at 0:26
  • Clearly I meant assigning it but also justifying it even if I didn't explicitly state the latter. Would you be able to derive exactly how you got the 0.1? No. As for the second, why is that the most rational thing to do? Jan 21 at 0:32
  • 0.1 of course is arbitrary, I could have said 0.11, or 0.9, or 0.25. The point is, it's less than 0.5 and more than 0. The rest is subjective, and you can't get around that, especially if you don't have other information, that's inescapable. So if you want to act on this and not be paralyzed by indecision, just assign e.g. 0.2 and carry on. Another thing you could is look at average crime stats and use that as a prior, if you want to add information. But otherwise, subjective is subjective and will not have any further grip on finding the "right" value.
    – Frank
    Jan 21 at 0:36
  • It is most rational because you have no reason to deviate from equivalence of the outcomes. There is no reason whatsoever to prefer A over B, no information to tilt the balance any way, so you are stuck with equal probability among the outcomes. Do you have something more rational in mind?
    – Frank
    Jan 21 at 0:37

3 Answers 3

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The concept of having a degree of belief is a rational one. It is a central feature of decision theory. The use of probabilities to represent degrees of belief can be justified using Dutch book arguments, as shown by Bruno De Finetti.

It is also fairly common to use Bayesian conditioning as a belief updating mechanism. This involves taking a prior probability, and updating it against some evidence E by replacing the prior P(H) with P(H|E). One of the main problems arises from deciding where the priors come from.

There are two contrasting schools of thought here. Objective Bayesians maintain that there are ways to determine a rational assignment of prior probabilities. Subjective Bayesians hold that the priors are simply whatever you happen to believe, and they should not be set or determined by any procedure. Subjective Bayesians often point out that the whole point of the Bayesian approach to belief revision is that Bayesian conditioning represents the distinctive rational way to update one's beliefs, and erasing one's existing priors and replacing them with objective ones is a violation of Bayesian conditioning.

A fundamental problem with setting priors objectively is that it is often difficult or even impossible to find a satisfactory distribution. In very simple cases one can appeal to the principle of indifference, or use a uniform distribution over a given variable. But in even moderately complex examples, there may be no obvious choice of variable, and different choices may lead to completely different priors. One group of problematic cases is the Bertrand paradoxes. In some cases, it is possible to find priors that are uninformative and scale invariant. Then there is the problem of dependencies between the priors. In some cases these can be handled using the principle of maximum information entropy, or by minimising the Kullback-Leibler divergence between the priors and posteriors.

Applying Bayesian conditioning to issues such as whether God exists is really an imponderable problem. For some people, the prior degree of belief that God exists is very high and for others it is very low. In both cases, they would no doubt think that it just seems obvious. If the priors are sufficiently high or low, then even a great deal of evidence will not shift the posteriors much. And trying to assign objective priors just seems impossible. What would be a prior for God's existence? 0.5 for God existing and 0.5 for God not existing? Or 1/n for each of the n gods that some person or other believes in? Or something else? And what even counts as evidence for the existence of God anyway?

In practice, Bayesian methods are really only useful when we have a firm handle on the nature of the hypotheses, the appropriate values for the priors and the relation between the hypotheses and the evidence.

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If you want to use a probabilistic framework to decide the question, and if there is no evidence or prior to tilt the balance when you are faced with N possible alternatives, the most rational thing to do is to assign the probability that results from a uniform distribution over the N possible outcomes, i.e. 1/N to each outcome. That is the definition of the uniform distribution: it is the distribution we use in probability theory when there is no information to differentiate the probabilities of the outcomes.

What else could be done within a probabilistic framework? If something else could be chosen i.e. favor some outcome with a higher probability, it would have to have some justification to count as rational, but if the problem does not provide any further information about the likelihood of the outcomes, the possibility of any justification has been eliminated. No other rational choice is possible, because none will be justifiable. Nothing can be said to differentiate the outcomes, so no justification will be possible. So there cannot be anything more rational to choose than complete ignorance/equal probabilities.

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If the aim is to deliver a message, the more spectacularly done, the more likely is it going to be taken seriously. After all the objective is to display power & wisdom. I'd say the probability of a god intervening in scenario B is more likely than scenario A.

However, I can imagine a god with little interest in the affairs of people, all except one; the so-called chosen one is a well-worn trope in fiction. In that case, divine intervention is more likely in scenario A.

On the larger point of degrees of truth, there's fuzzy logic which has truth values ranging from 0 (false) to 1 (true).

Then there's Bayes' theorem, which applied to the OP's question would look like this:

P(I) = the probability of god intervening in a scenario

P(S) = the probability of a scenario (A or B)

P(I/S) = the probability that god intervenes given a particular scenario

P(S/I) = the probability of a scenario given god intervenes

P(I/S) = [P(I) × P(S/I)]/P(S)

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  • But Bayes only provides a way to calculate with probabilities if you know probabilities for the priors, which is really the difficulty - you never really know the prior probabilities.
    – Frank
    Jan 21 at 2:11
  • @Frank, indeed. There's a subjective component to Bayes' theorem, but it does allow one to calculate posterior probability from relative if not absolute probabilities. For example, Is P(I) greater/less than P(S/I) and how does that affect the probability given P(S) is comparatively greater/less than P(I) and P(S/I)? Jan 21 at 2:35
  • Sure you can update the ratio of the priors to get the ratio of the posteriors. If the ratio of the conditionals is > 1, the ratio of the posteriors will be in the same direction as for the priors. May or may not be useful.
    – Frank
    Jan 21 at 2:59
  • The subjective component is not in the theorem itself - the mechanics of the theorem - but in the value of the priors, and possibly in the values of the conditionals that enter the formula. The formula itself it not subjective at all.
    – Frank
    Jan 21 at 3:01
  • @Frank, true. The prior probabilities need to be estimated and then plugged into the equation. Jan 21 at 3:32

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