Probability and rare or paranormal events

[Mithrae]

In the recent thread Assuming the supernatural is possible I advanced the opinion that so-called 'supernatural' events should be treated in a similar way to claims of a winning lottery ticket: They have a prior probability of essentially zero, and hence we should quite rightly maintain a higher level of scepticism than we do for more normal claims, whilst recognizing that in assessing the actual evidence we may well find a much higher probability in hindsight that the claim is true.

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)?

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:

• 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.

The position which Liamconner is advancing seems to be more akin to propensity probability:

• 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.

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).

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:

• 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.

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).

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.

[Mithrae]

The same could be said of any event: Either it happened (100% 'probability') or it didn't (0%). So how do you ever tell if anything ever happened or not, if that is the whole extent of your approach? Or are you suggesting that rare and non-repeatable events are somehow in a unique position here that I'm just not seeing?

You're talking about individual events. I'm talking about the entire category or type of events.

The probability that supernatural events can occur in general needs to be 100% before any individual supernatural event could even occur.

If the probability of supernatural events in general is zero (i.e. they are impossible) then the probability of any individual supernatural ever occurring must also be zero.

The fact that there has never been a verified supernatural event ever recorded certainly suggests that the probability that supernatural events are possible is more likely to be zero than 100%. So there is a very high probability that supernatural events never occur.

And in the early 20th century, presumably you would have argued...

• The fact that there has never been a verified black hole ever recorded certainly suggests that the probability that black holes are possible is more likely to be zero than 100%. So there is a very high probability that black holes never occur.

The problem with the argument, as I pointed out in the OP, is that in this case and in all cases of rare and elusive or non-repeatable phenomena, there is no significant difference between zero or several verified records of it. If you had solid grounds in the early 20th century for believing that if black holes existed there should have been a very high probability that they would have been confirmed, it would be a valid argument. But if there were a very low probability that they would have been confirmed, that absence of confirmation is a correspondingly weak argument.

Now to be fair, some alleged paranormal phenomena (eg. sun standing still in the bible, ghosts, ESP and the like) arguably should be easily confirmable since they should leave conclusive evidence or be subject to repeatable observations. In those cases, the absence of strong evidence becomes strong evidence of their absence, to greater or lesser extents. But in other cases (eg. less dramatic miracles, alien visitation) the events are not subject to repeatable observation and often are not of a sort which we'd expect to leave conclusive evidence. The argument from silence fails in those cases, just as it would have for black holes in the early 20th century.

And in fact it's even weaker than an argument from silence: It's an argument from "I think all the noise should be louder than that"

However the situation isn't that we have zero evidence: Going by the standard of observational reports by multiple not-obviously-unreliable witnesses, paranormal events have been independently reported
- in all regions of the world,
- in most if not all cultures,
- in all periods of history through to the enlightened 20th and 21st centuries (still in all regions and cultures).

We also know that these not-obviously-unreliable witnesses have reported conflicting accounts of paranormal events. Thus we can certainly concluded that many of these reports are clearly undependable. And there's really no reason to think that any of them are. After all the overwhelming majority of them would need to be false reports.

Once again, you're conflating an individual event with a specific interpretation of an entire philosophical or theological corpus of the culture in which it occurred. That never has been and never will be a valid assumption to make.

[Mithrae]

[Replying to post 1 by Mithrae]

I advanced the opinion that so-called 'supernatural' events should be treated in a similar way to claims of a winning lottery ticket: They have a prior probability of essentially zero,

In what way are you looking at the chances of a lottery ticket? The odds of one specific person getting the winning numbers, or the odds of anyone at all out of a player-pool of potentially millions or tens of millions?

The prior odds that a particular person's ticket will win, and the posterior plausibility of their claim to hold a winning ticket.

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.

It's not only that I have never seen/experienced anything paranormal/supernatural/miracle, it's that very frequently these things are described as violating the known laws of physics/chemistry/biology.

Not always. Perhaps rarely or even never, unless we include assumptions about specifically how it happened. But even assuming that they do: Then we would be talking about observational reports which are anomalies or exceptions to the patterns we see from observing perhaps 1 or 2% of what happens in the visible spectrum on the populated land surfaces of our planet.

What conclusions do you draw from patterns observed in 1 or 2% of what happens in that limited scope?

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

Oh? So how is 0 not different to 1, or 2, or 12?

Perhaps the way I phrased it in my response to DI above will be easier to understand: If there were a very low probability of conclusively confirming some particular phenomenon even if it were real, the absence of conclusive confirmation is a correspondingly weak argument that it's not real. That's not quite the same thing as the frequentist probability argument, but it's a similar concept as to how significant (or not) that zero figure is.

And by making that argument, in asserting their 'zero' figure to be significant, proponents are implicitly assuming answers to the relevant questions beforehand.

Which I have a very high confidence for. If someone wants to get their claim of a resurrected Jew across, they'll need some actual evidence.

It doesn't matter how confident you are in them, if you're assuming answers to the relevant questions beforehand - ie, how probable are paranormal events? - it is a circular argument and therefore not valid.

[Divine Insight]

And in the early 20th century, presumably you would have argued...

• The fact that there has never been a verified black hole ever recorded certainly suggests that the probability that black holes are possible is more likely to be zero than 100%. So there is a very high probability that black holes never occur.

Why would you think that?

I am not arguing that a probability can even be assigned to supernatural events. I simply pointed out that they are either possible, (100% probability) or they cannot occur (zero probability).

The possible existence of black holes is an entirely different question. To begin with the very idea of black holes never even came up until the question of whether gravity could ever be strong enough to create one. At that point we actually have a mechanism for the potential creation of a black hole. Because of this we can assign probabilities to whether the laws of gravity and physics might permit such a thing to exist.

Of course, even then we wouldn't actually speak in terms of probabilities. Instead we would just do the math using the known equations of the laws of physics and see what those laws have to say. As it turns out the laws not only allow for black holes to exist, but provide nothing to prevent them from existing. At that point the probability that a black hole exists becomes extremely high. And of course once we have discovered one the probability that they exist instantly becomes 100%. So today we now know that the probability that black holes exist in our universe is indeed 100%.

We can't approach the question of supernatural events in this way because we have no laws associated with the creation of supernatural events. If we did, then we could start talking about probabilities. But we don't so we can't. We would actually need to record a verified supernatural event. But that would be extremely difficult because even if we observed an event that we can't yet explain, that doesn't automatically mean that it is "supernatural". All it would mean is that we don't yet have a natural explanation for it.

[Mithrae]

[Replying to post 1 by Mithrae]

I continue to question the validity of assigning probability to any supernatural claim; in terms of Christianity, it seems the question would have to run, "How likely is it that there is a power behind the universe whose plan it is to redeem a messed up world by suspending the normal behavior of Nature?"

I can only reply, I don't know, we'd have to ask that power; in the same way we'd have to as a bachelor how probable it is that he will get married. If he has no intention of getting married, it is zero; regardless of the fact that there are more couples than their are confirmed bachelors.

Does not a similar situation face cosmologists? What was the antecedent probability of the big bang occurring, by which time and space erupted? But time and space are the necessary conditions for measuring probability.

Yes, in most cases we don't know the real likelihood of an event occurring with any great precision, either in fore- or hindsight; people's actions, deities' actions, political and economic outcomes, weather events... Many of the things which concern us most, in other words.

Applying frequentist probability to rare events is an attempt to fudge that "I don't know" into an answer, one way or another, and it is not a valid approach. It does not progress our understanding of those events in the slightest, and in fact impedes understanding by burdening us with presuppositions. But the propensity approach to probability which you're using doesn't progress our understanding either. Sometimes we do know the real likelihood of a thing, and in those cases one of these 'objectivist' approaches to probability makes sense. But in cases where we don't and probably can't know the real likelihood of an event occurring, rather than continuing to talk uselessly about that real likelihood it makes more sense to talk about our evaluation of likelihood - a Bayesian approach.

Outside of professional scientific and statistical work, I would say that the most useful and most important reason for using a Bayesian approach to probability, rather than one of the 'objectivist' approaches, is that it helps avoid the black and white thinking of only considering things in terms of believe or not believe.

Consider a list of questions such as the following for example:
- How probable is it that I exist?
- How probable is it that New York exists?
- How probable is it that Pnhom Penn is the capital of Cambodia?
- How probable is it that the Titanic disaster was an accident?
- How probable is it that William Shakespeare wrote Hamlet?
- How probable is it that Socrates existed?
- How probable is it that 9/11 was an inside job?
- How probable is it that Zeus existed?
- How probable is it that I don't exist?

In 'objective' terms, each of these obviously has either a 1 or 0 probability of being true, but (having never been to New York or Cambodia) only two of those answers are actually certain to me. There is a fantastically slim but non-zero probability that everyone I've talked to and everything I've seen in print, TV, movies, songs and online has been wrong about New York. There's a slightly higher probability that I'm wrong about the capital of Cambodia (even disregarding the spelling ), because I've seen far fewer reports about that.

But more importantly, saying something like "There's a 90 or 95% probability that Socrates existed" is far more useful than simply saying "I believe that Socrates existed" or (even more uselessly) "The probability that Socrates existed is either 1 or 0 and I'm not certain which." Evaluating the probability of a statement being true is a coherent and useful way to use probability, in the many circumstances where objectivist probabilities are all but useless.

In fact it's more than just a useful way of thinking, it's a way of thinking which helps avoid the pitfalls of presuppositionalism and dogmatism because instead of working from the position that some prior thing is 'true,' we're constantly reminding ourselves of that level of uncertainty implicit in all our knowledge: In the case of this particular subject for example, the fact that the reliable, consistent observational patterns we see in our world still only represent a tiny fraction of what goes on.


- How probable is it that the patterns reliably observed in the visible spectrum on 1 or 2% of the populated land surfaces of our planet are representative of real and universal constraints on the nature of reality?

[Mithrae]

And in the early 20th century, presumably you would have argued...

• The fact that there has never been a verified black hole ever recorded certainly suggests that the probability that black holes are possible is more likely to be zero than 100%. So there is a very high probability that black holes never occur.

Why would you think that?

I am not arguing that a probability can even be assigned to supernatural events.

Not objectivist probability, I agree. But what do you think about my response to Liam above? In the many cases where objectivist probability does us little good, there is still a useful way to use probability, by evaluating the probabilistic strength of a proposition rather than futilely trying to talk about the real likelihood of an event.

I would argue the use of Bayesian probability is not only more useful in those cases, but hugely important and beneficial in countering the habits of black and white thinking.

[liamconnor]

[Replying to post 14 by Mithrae]

I am not sure I agree with everything you say. But I will say this, the following raises a question which has been bothering me. It has to do with probability in general. I have thought of bringing this up for some time now, but was not sure on what subforum: apologetics, philosophy, science?

Your quote:

Yes, in most cases we don't know the real likelihood of an event occurring with any great precision, either in fore- or hindsight; people's actions, deities' actions, political and economic outcomes, weather events... Many of the things which concern us most, in other words.

Indeed. A news report shows a probability of 30% rain occurring in my area tomorrow. But tomorrow it rains. That means there was a 100% chance of raining, right?

So what good is probability? Obviously it means only, "Based on our incomplete knowledge, there is a x% probability of z happening".

But "our knowledge" is a rather loose factor here.


In history, it is none the less.

Based on our knowledge of Julius Caesar prior to his crossing the Rubicon, what is the antecedent probability of him crossing the Rubicon?


Well, how the hell should we know?! I guess it was 100% because that is in fact what he did. But that only means that certain probabilities are assigned after the fact, and therefore are given a 0 or 100%. How helpful is that?

[Mithrae]

Indeed. A news report shows a probability of 30% rain occurring in my area tomorrow. But tomorrow it rains. That means there was a 100% chance of raining, right?

So what good is probability? Obviously it means only, "Based on our incomplete knowledge, there is a x% probability of z happening".

Exactly. Well, assuming weather is deterministic.

But that's why I started a new topic on this, because to greater or lesser extents probability is a part of virtually all our expectations and beliefs, yet so easy to trip ourselves up over. I know I often have! I'm glad I took the time to Wiki it the other day, because highlighting and understanding the distinction between 'objectivist' probabilities - both frequentist and propensity - and the Bayesian probability which even those paragons of truth and epistemic rigour on our nightly weather reports use is very important.

Especially since the apparent mis-use of frequentist probability is such a common argument seen on these forums.

[marco]

Indeed. A news report shows a probability of 30% rain occurring in my area tomorrow. But tomorrow it rains. That means there was a 100% chance of raining, right?

No. There was no certainty that it would rain. The 30% chance occurrence did occur, which is hardly surprising. We can use "conditional probability" to amend our estimates; the probability of picking an ace is 1 in thirteen, but if we have seen the king of hearts is the top card then our chance of picking an ace is changed. It becomes slightly better: 4 in 51.

It would be rather meaningless to try to estimate the a priori probability of Caesar crossing the Rubicon. When an event has occurred one no longer talks of the probability of its occurring.

The probability of throwing a head and a tail with two coins is a half. But in quantum statistics that probability is one third. Much depends on what information we have and how we use it.

What's the probability the Resurrection actually happened? It makes no sense mathematically to place some numerical value to this.