I think that there were some difficulties in communicating our opinions to each other because of the different ways in which probability can be used, and after some consideration I think I've found the core problem of the most common argument made against miracles and the like. Stopping short of appealing to philosophical naturalism, the argument basically states that since the poster has seen no "confirmed" examples of paranormal events, the probability of such events must be considered to be zero - and therefore any more normal explanations will necessarily be more plausible.
The problem with the argument is that zero "confirmed" paranormal events may not be significantly different from one or two or even a dozen confirmed events, at least at any high level of significance: And by making that argument, in asserting their 'zero' figure to be significant, proponents are implicitly assuming answers to the relevant questions beforehand. Obviously if it's not a valid probabilistic argument, it then amounts to little more than an appeal to ignorance and personal incredulity of the various reported miracle observations we've all heard of. Further details are below to avoid clutter.
Is this a fair analysis of the situation, and does it invalidate arguments of that type?
If not, how can those arguments be refined to ensure their validity? What further premises or qualifications are required?
Or if they are not valid, what is a more appropriate way to view or evaluate the probability of miracles (or indeed any events which are rare and non-repeatable)?
Mithrae wrote: As is my wont I edited and then re-edited my last post a few times, but figured I should do some quick Wiki-ing before trying to explain what I wanted to get at.
It seems that the position which Rikuoamero (and I believe Justin also) is advancing is similar to the theory of frequentist probability:The position which Liamconner is advancing seems to be more akin to propensity probability:
- Frequentist probability or frequentism is an interpretation of probability; it defines an event's probability as the limit of its relative frequency in a large number of trials. This interpretation supports the statistical needs of experimental scientists and pollsters; probabilities can be found (in principle) by a repeatable objective process (and are thus ideally devoid of opinion). It does not support all needs; gamblers typically require estimates of the odds without experiments.
As I understand it (again, from only some brief reading) these both belong to the category dubbed 'physical' or 'objectivist' probabilities, in that they attempt to assess a real likelihood of a given event occurring. I think that usually I would be inclined towards viewing the latter, propensity probability, as being more appropriate even for most mundane purposes, because repeating large numbers of trials to establish a frequentist probability is so often impractical, impossible, unnecessary or if there aren't enough trials even misleading (though in cases where no valid information on propensities can be obtained, frequencies may serve an important purpose, not least in trying to discover the underlying causes).
- The propensity theory of probability is one interpretation of the concept of probability. Theorists who adopt this interpretation think of probability as a physical propensity, or disposition, or tendency of a given type of physical situation to yield an outcome of a certain kind, or to yield a long run relative frequency of such an outcome.[1] Propensities are not relative frequencies, but purported causes of the observed stable relative frequencies. Propensities are invoked to explain why repeating a certain kind of experiment will generate a given outcome type at a persistent rate.
In the case of exceptionally rare or singular events, frequentist probability seems to be all but useless or fundamentally fallacious. This has nothing to do with the 'supernatural': Frequentist probability presumably would have implied a zero probability of black holes before they were discovered for example, or a 100% probability of there being life on Earth-like planets until we find one without. So if we pretend to be discussing real likelihoods, propensity probability would be more appropriate. But that soon runs into the problem (as Liam has suggested) that we'd be trying to answer questions like "is there a God," "what would the creator of the universe do" or even "is it a deity responsible for 'supernatural' events at all" before considering the physical evidence for or against a miracle claim.
Far more reasonable in my view is an approach which explicitly quantifies our evaluation of likelihood, fully recognising that our evaluation may not perfectly match the real likelihood (which is obviously true of objectivist probabilities too), though we'll hopefully come close. This is what I have been (and generally do) talk about, and it seems that it is pretty much along the lines of Bayesian probability:To my delight, Bayesian probability even incorporates those terms 'prior probability' and 'posterior probability' that I've been using: How we evaluate the likelihood of a result before we have access to some or all of the relevant data on it (ie, before an event has happened in our case), and how we evaluate the likelihood of it being the case after we have access to all the information that we can (the claim's plausibility in our case, since we would now be talking about past events).
- Bayesian probability is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation[1] representing a state of knowledge[2] or as quantification of a personal belief.[3] The Bayesian interpretation of probability can be seen as an extension of propositional logic that enables reasoning with hypotheses, i.e., the propositions whose truth or falsity is uncertain.
Perhaps this will help clear up some confusion or miscommunication going forwards.
Edit: To further explain why frequentist probability is inappropriate here, I've been known to play the odd RPG game at times, and sometimes delve into the mechanics such as loot drop rates or critical hit proc rates. And in doing so, I have always held that to get a valid estimate on a low drop or proc chance, I'd need at least three and preferably five or more positive results (to allow some room for a +/-2 estimate). The reason is that any one or even two drops, or the absence of them, could easily be sheer coincidence rather than being statistically representative.
Suppose a given event had a real likelihood of 2 in 1000, or 0.002. That would mean that while you might expect 2 positive results in 1000 trials, you would still have a 13.5% chance of getting zero positives (0.998^1000). Even in 2000 trials, you'd still have a 1.8% chance of getting zero positives. So zero results is only different from two positive results at an 86.5% level of significance from a thousand trials (which really isn't significant) or at a 98.2% level of significance from two thousand.
Justin and Rikuo argue that there have been zero "confirmed" positive results for the supernatural; but depending both on the real likelihood of the supernatural and (given their qualification) the likelihood of such events being confirmed to their satisfaction, zero positive results may not be significantly different from two positive results - or even a dozen positive results, for that matter! Obviously if there were in fact some "confirmed" positive results they wouldn't be making that argument, so in making it - in suggesting that their zero figure is a significant one - they are unintentionally implying or presupposing those answers already.
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It can hardly be anything else. 
