Sunday, November 28, 2010

The Case Against Frequentism

From my Infidels paper on miracles. 

 

IV. Probability and its Empirical Foundations

According to Hume, probabilistic beliefs concerning the intentions of a supernatural being are inadmissible in reasonings concerning matters of fact because these beliefs fail to be grounded in experience. This insistence has been enunciated by Bayesian theorists, and it is the frequency theory. But the frequency theory has fallen on hard times, and most Bayesian theorists do not accept it, largely because of difficulties related to the problem of the single case.
The problem is this. Frequencies give us information as to how often event-types have occurred in the past. But we often want to know the probability of particular events: this coin-toss, this horse-race, this piece of testimony to the miraculous, etc. If we are to accept Hume's conclusion that testimony to the miraculous ought never to be accepted, we need to show more than just that rejecting testimony to miracles in general is a good idea because false miracle claims outnumber true ones. Many Christians are skeptical of miracle claims put forward by televangelists, but nonetheless believe that the evidence in support of the resurrection of Jesus, and perhaps in support of some modern miracles, is sufficient to overthrow our ordinary presumption against accepting miracle reports.
Frequentists have attempted to assess the prior probability of individual purported events by assimiliating them some class of events. Thus, we assess the probability of a particular coin-toss as 1/2 in virtue of its membership in the class of coin-tosses. But the question is which class the relevant reference class is. The claimed resurrection of Jesus falls into many classes: into the class of miracles, into the class of events reported in Scripture, the class of events reported by Peter, the class of events believed by millions to have occurred, into the class of events basic to the belief-system of a religion, etc. Of course it is what is at issue between orthodox Christians and their opponents whether the class of miracles in the life of Jesus is empty or relatively large.
Wesley Salmon attempts to solve this problem by defining the conception of an epistemically homogeneous reference class. A class is homogenous just in case so far as we know it cannot be subdivided in a statistically relevant way. Thus, according to Salmon, if Jackson hits .322 overall but hits .294 on Wednesdays, the Wednesday statistic is not to be treated as relevant unless we know something about Wednesday that makes a difference as to how well Jackson will bat. Thus, according to Salmon, the relevant reference class is the largest homogeneous reference class; we should try to get a sample as large as we can without overlooking a statistically relevant factor.[13]
There are two difficulties with this method as an attempt to satisfy Hume's strong empiricist requirements for properly grounded probability judgments. First, questions of statisical relevance cannot be fully adjucated by appeal to frequencies. Second, the very heuristic of selecting the largest homogeneous reference class cannot be read off experience.
On the first point, consider the situation of a baseball manager who must choose between allowing Wallace to bat or letting Avery pinch-hit for him. Wallace has an overall batting average of .272, while Avery's is .262. But the pitcher is left-handed, and while Wallace bats .242 against left-handed pitching, Avery bats .302. Nevertheless, the pitcher is Williams, and while Avery is 2-for-10 against Williams, Wallace is 4-for-11. Have these batters faced Williams too few times for this last statistic to count? And can this be straightforwardly determined from experience? What is needed is a judgment call about the relevance of this statistical information, and this judgment cannot simply be read straightforwardly from frequencies. The frequentist's epistemology for probabilistic beliefs, insofar as it is an attempt to conform to empiricist/foundationalist constraints, seems impossible to complete.
On the second point, is the heuristic of selecting the smallest homogeneous reference class justified simply by an appeal to experience? Admittedly it makes a certain amount of common sense. But this attempt to go from a statistical "is" to an epistemological "ought" seems to suffer from with the same (or worse) difficulties that getting "ought" from "is" suffers from in ethics, and here again Hume's empiricist/foundationalist assumptions impose an impossible burden on probability theory.
The frequency theory seems clearly to be the theory of priors that Hume would have adopted had he been involved in the contemporary Bayesian debate on prior probabilities. But even this theory fails to adjudicate the issue concerning miracles in Hume's favor or in favor of the defenders of miracles, because it lacks the resources within itself to select the appropriate reference class. This inability to provide determinate answers to questions of probability is what makes this theory inadequate for resolving the question of miracles. Therefore Hume cannot justify his claim that it is never rational to believe testimony to any miracle on the grounds that miracles are less frequent in experience than false miracle reports.[14]

13] Salmon, The Foundations of Scientific Inference (Pittsburgh: University of Pittsburgh Press, 1967), pp. 90-93.
[14] These objections were suggested to me by in conversation by Patrick Maher.

43 comments:

John W. Loftus said...

Let's put miraculous claims into three categories, 1)Those I myself have personally experienced; 2) Those that a contemporary person claims to have experienced who can testify to them, and 3) Those that are claimed by ancient people where we do not have any access to the people involved or can read a first hand account of it.

The numerical value of (1) is zero, even as a former believer and even as a pentecostal in my early years.

The numerical value of (2) is also zero. I have heard of such stories but I was never there to investigate these claims for myself, and since I have checked on a few of these stories and found them false it seems reasonable to think the ones I wasn't able to investigate for myself are false too, even if they might have happened.

Since the numerical value of (1) and (2) is zero then I find nothing problematic about claiming the numerical value of (3) is also zero.

This best represents the historian. He or she must assume the present is the key to understanding the past. And that's why histories must continually be re-written as facts are produced which shed more light on things previously neglected.

The historian looks into the past from his present perspective, He cannot do otherwise. Again, he cannot do otherwise.

Therefore given that we do not experience the kinds of miracles reported in the Bible historians must account for these stories from the perspective of what their own experience has taught them.

They cannot do otherwise. So it just isn't rational to believe they occurred even if they did, even if they did.

Victor Reppert said...

Were you around for the discussion of contemporary miracles a few years back?

John W. Loftus said...

No, probably not. All I need to claim is that I have never personally experienced one even as a pentecostal in my early years. And if these other contemporary miracles have taken place then why was it that when I investigated the ones I did they were all false.

I'll link to something to show you how I did such an analysis before long. I've gotta find it though.

John W. Loftus said...

I found it. All I had was this guys testimony which I deconstructed from the things he himself told me apart from any additional testimony from the other people involved.

You must watch the video to see whether or not what I said was reasonable. Enjoy as you have the time.

Victor Reppert said...

Prior to Obama's election, an African-American had never been elected President. In my experience, I had never seen this happen. So the likelihood of an African-American being elected president is zero, and I should assume that all reports saying that he was elected to the Presidency are made-up stories.

John W. Loftus said...

Ahhhh yes, you profess not to know what an extraordinary claim is.

Such a delusional man you are and another reason I blast you and you're cohorts since nothing, literally nothing penetrates that brain of yours.

With that I'm unsubscribing.

Sheesh, and you teach philosophy. I'd get you fired for being so ignorant if I could

Victor Reppert said...

I'm claiming that events are ordinary or extraordinary relative to an existing belief-system. Any attempt to establish that a claim is extraordinary simpliciter founders on the problem of the single case. Of course, I should expect it to take more evidence to convince a naturalist of a miracle than it does to convince someone who believes in the existence of a God who might have a reason to bring that event about.

Victor Reppert said...

The point being missed here, of course, is that there are many things that I accept on the testimony of others that are not part of my own limited experience. The fact that I haven't seen something doesn't give me much grounds for saying that it didn't happen.

The reason you hold the position you do with respect to miracles is that you believe in something akin to the causal closure of the physical. Of course, the AFR is an argument that the physical isn't closed.

What I learned in classes on epistemology and probability theory (from atheist professors, mostly) is that there is no privileged "neutral ground" from which to build up our epistemic positions. All we can do is start from where we are, investigate the facts, and hope our positions converge someday.

But don't hold your breath on that.

John W. Loftus said...

I felt bad about my response so I thought I'd come back to set you straight. In the years 1900 or 1930 or 1960 it would be unreasonable to think an African-American would be the president. That means exactly what I said, it would be unreasonable to think this even if it took place.

Then if we travel to 1980 or 1990 or 2000 it becomes more reasonable to think this could happen.

As we travel through the year 2008 we see it becomes more and more reasonable to believe it, and it did. That's because evidence was mounting that it will happen.

That's the way it is. Take me back to 1900 and I will maintain it can't happen. And I would be perfectly reasonable to say that at that time.

With regard to miracles the same reasoning applies. In today's world with my experience I am perfectly reasonable to say the miracles in the Bible did not happen even if they did. The reason? Because they do not happen today.

You maintain they did, but that claim is unreasonable at this point in time and that too is my point.

John W. Loftus said...

...and stop putting words in my mouth.

John W. Loftus said...

Here's what you are arguing for Vic. You're in the position of someone in the year 1900 who says that it's probable a negro (remember back then?) will be the president in a century. You have no evidence for this claim of yours except faith. Your contemporaries would have every right to laugh at you especially if you centered your whole life around such a faith claim (what that would be in this case escapes me).

So likewise people can laugh at your claim that the biblical miracles took place too.

And this is true regardless of the fact that in 1900 you were right or that today you are right about the Biblical miracles.

Do you understand this. Please say you do.

John W. Loftus said...

While I wait for your answer let me add that your god demands that people in today's world believe or be damned Well, at least most believers think this). But he has place us in a situation just like you would be in 1900 who said a negro would be president in a century. Does not not sound ridiculous to you as it does to me, that I should believe despite the present day evidence to the contrary?

That's you. That's your god. You deserve your fantasy god.

Victor Reppert said...

Talk about putting words in someone's mouth! I haven't believed in Believe or be Damned since 1974! I am an inclusivist (much like Sennett), with some sympathies with Talbott's universalism.

There is evidence concerning the resurrection. It isn't just the nature of the miracle claim, it facts of history that make any attempt to reconstruct in naturalistically very difficult. And there are lots of bad theories out there.

The evidence may not be extraordinary enough. But I do think the history of the founding of Christianity renders belief in the resurrection possible for some reasonable persons. But that's a very long debate.

Mr Veale said...

John

Don't go...I'm enjoying this (and people swear at me on infidel sites.)

My thoughts (which come across as a bit grumpy, but it's late, and I'm not rewriting them now)

1) By its very nature, a miracle of the Resurrections import will be a unique historical event. So the vast majority of the human race will be separated from it over time. (If it happened last Tuesday, 1000 years from now lots of people will be separated from it by time.)
So I'm not sure that your divisions are relevant.
2) If you want to do a bit of reading on events that don't fit in with Scientific predictions, you could check out Ben Goldacre's "Bad Science" on placebos. I also think that NDE's count. And the inexplicable healings at Lourdes etc. They don't prove a darn thing about Theism, or miracles. But they suggest that there is a class of events containing events that are not explicable, and are not likely to be explicated any time soon, by modern science. That would seem to be the relevant reference class for assessing extraordinary claims.
5) I've cited two major studies of history and historians. So instead of stipulating what history is, and what it can and cannot say, is there any chance that you might engage with the relevant literature? NT Wright, Craig Keener and CS Evans have all addressed this issue from a Christian perspective. Can we engage with their arguments? And your condescending attitude to pre-modern societies does suggest that you may not be best placed to give advice on ancient history.
6) Again - the McGrew's have addressed the issue of vanishingly low prior probabilities. Can we discuss their arguments?
7)I find it interesting that you begin to argue that the Christian God is not worthy of worship. Now that's a valid argument. But I can't see how a person can argue that way without running the risk of letting non-rational forces affect their reasoning. Wouldn't it be inhuman not to let non-rational forces affect us when considering this type of argument?
8) Let's put Obama's election to one side. Suppose scouts of a stone age tribe, who have always been sober and reliable in their testimony, reported seeing man made flying devices. Suppose their tribe had no experience of aeroplanes. And that this tribe wouldn't have experience of them for decades to come.
Should the tribe discount the testimony of the scouts?

Most importantly...

9) Why don't you have your own wikipedia entry? You merit one, and you're referenced a lot on wikipedia. It seems a bit unfair if you ask me. I'd offer to write one, but I don't know enough about you yet. Which was why I thought I'd check wikipedia....

Graham

Mr Veale said...

One more thing---as Columbo said---the argument from the Resurrection, as I've set it out in any case, does not depend on "x seen a Resurrection".

It starts with claims that are *not* extraordinary like
-"x said she found an empty tomb" and
"y believed he seen a person z physically alive and well, when we know that z had died several days previously"

There is nothing extraordinary about an empty tomb, and as you say, people report strange things all the time.

Now priors, like one's stance on Theism, can and should affect how one assesses that evidence. I think that's where Vic is coming from - it's certainly Swinburne's position.

Or we can abandon that approach, like Tim, and argue that cumulatively the evidence is so extraordinary that it warrants an inference to a miracle.

But it is the co-occurrence of several strands of evidence that motivates the inference to a miracle.
Nobody, to my knowledge, is arguing that we should accept that a miracle happened because an otherwise reliable source says so. (And in Lydia McGrew's interview with Luke 'Prog' Muelhauser, this is made explicit. Such an argument would not work, and it is not how the case is being advanced)

Graham

Mr Veale said...

"Abandon" is the wrong word for what Tim is doing - apologies

GearHedEd said...

If I may...

For David Parker:

The assumption is that in Bayesian logic "1" represents success, or affirmation, or whatever positive result is appropriate to the question, where "0" is the negation.

Thus,

If A=>B, then if P(B)=1, P(A)=1

Is a true statement, and not a typo, since A can never be less than B (equal to or greater than), and if B is "1" (full success), then A MUST be "1" as well.

For Victor:

You said, "According to Hume, probabilistic beliefs concerning the intentions of a supernatural being are inadmissible in reasonings concerning matters of fact because these beliefs fail to be grounded in experience. This insistence has been enunciated by Bayesian theorists, and it is the frequency theory. But the frequency theory has fallen on hard times, and most Bayesian theorists do not accept it, largely because of difficulties related to the problem of the single case.
The problem is this. Frequencies give us information as to how often event-types have occurred in the past. But we often want to know the probability of particular events: this coin-toss, this horse-race, this piece of testimony to the miraculous, etc. If we are to accept Hume's conclusion that testimony to the miraculous ought never to be accepted, we need to show more than just that rejecting testimony to miracles in general is a good idea because false miracle claims outnumber true ones.

I think Hume is on firmer ground than that, ESPECIALLY as regards the single case.

Without appealing to the infinite as a real number (any finite number 'x' divided by infinity is exactly zero) of false miraculous claims, the math says that the more 'false' claims there are for each 'true" claim, the closer the Bayesian probability that any claim is true approaches zero.

In my opinion, then, Hume was stating that there is no reasonable way to accept miracle claims on the basis of claims alone, and that's why failing to ground the claims in experience (his?) is justified as a way to exempt them from (his?) reality.

Am I wrong to defend Hume thus?

GearHedEd said...

Probably. Looks like comparing apples and oranges, but I'm not a philosopher.

My game is math, and I was thinking along those lines.

That's what it looks like from here, but if I understand anything about Bayesian logic, it's an attempt to assign probabilities to philosophical (and other) questions in a mathematical logic format. The tricky part is setting up the initial conditions for a proper evaluation.

Like the computer maxim goes:

Garbage in, garbage out.

GearHedEd said...

Then again, if P(B) = 1 (complete success), and P(A)>=P(B), then P(A) must = 1 (first place); however, this may not be a good example since Johnny can only finish the race in one position (so the ">" symbol would be unnecessary in your example).

GearHedEd said...

This works, too:

A = John MAY HAVE finished the race in first place
B = John finished the race

If A=>B, if P(B)=1 (full success, i.e., finished first), P(A)=1 (DID finish the race in first place).

But again, John can only have finished the race in one spot, making ">" extraneous.

GearHedEd said...

But this is off topic.

Anonymous said...

To all your "philosophers": STOP USING PROBABILITIES WHEN YOU KNOW NOT EVEN THE FIRST THING ABOUT THE SUBJECT.

If you want to know about Bayes' Theorem, TAKE A MATH CLASS, not a philosophy class.

Philosophy is dead.

Tristan D. Vick said...

Victor-

Your Obama example misses it's mark. There is nothing in nature which says a black man getting elected President is impossible let alone improbable.

It's by no means miraculous that in a free Democracy a black man who runs for President has the possibility of being elected into the Presidency.

Whereas miracles, the sort of magic and mystical, as talked about by believers are not verifiable let alone possible (according to the laws of nature). Thus most miracles must be considered highly unbelievable.

As a philosopher you may want to brush up on your David Hume, but also perhaps Kant. Because I think you are confusing probability with plausibility, that is to say given the circumstances it was highly probable Obama would have a chance at getting elected, however, it was highly implausible that Mickey Mouse would take corporeal form and would be elected President.

Regardless, it comes down to a matter of proof and validation--otherwise all you have is conjecture minus a basis for support (in reality). If you say Jesus resurrected that's the same as me saying Mickey Mouse was elected President. You need some mighty hefty evidence to prove such a claim, because typically men don't resurrect and fictional mice don't become president. Whose to say it didn't happen? All we can say it is doesn't usually happen, we've never witnessed it ourselves, and so it's probably a better bet to assume it didn't happen.

Obama being elected President of the United State, however, has a strong basis in reality--1) it was a totally probable event, 2) it was totally possible, and 3) it happened. Moreover, all this can be verified and collaborated.

Now if miracles are to be believed, wouldn't you think that they must meet the same prerequisites? If not, then why the double standard?

Victor Reppert said...

Tristan: The Obama example was a counterexample to John, who goes from the lack of corroboration for miracles in his own experience to a zero probability. It's pretty much on open secret in the study of Hume's essay that a straightforward application of the mathematical probability theory he lays out would lead to, for example, skepticism about newspaper reports that person X won the lottery.

The title links to an full-length essay that I wrote on Bayesianism and miracles, which was published on Infidels. I also published a paper for the peer-reviewed International Journal for Philosophy of Religion in February 1989. So I think I have done some brushing up on Hume.

In fact, I explicitly argue that it's quite to be expected that the evidence for miracles like the Resurrection will be insufficient to persuade atheists.

The question is whether it is really possible to find a "neutral" or "unbiased" position from which to evaluate just how strong the antecedent improbability of a miracle really is. I don't see how you can evaluate the likelihood of miracles independently of a) the antecedent probability of theism and b) the overall likelihood that God, if He exists, would do something along these lines as part of his way of communicating with human beings.

The model of reasoning that I am inclined to accept is that everyone starts where they start, and then conditionalizes their belief systems on the evidence. And then, maybe, however many generations it takes, we can reach a consensus. With issues like God or Christianity I don't see the consensus coming anytime soon. Experimentally imagining how the evidence might look to someone with a different belief system, which is what's involved in the "outsider test" is something that is worth doing, but I don't think it represents the ultimate test of the adequacy of one's religious (or non-religious) beliefs.

Mr Veale said...

Re: extraordinary events and eyewitness testimony - according to BBC's "Horizon" scientists dismissed the possibility of Freak Waves, despite excellent eyewitness evidence of their occurrence. "Freak waves are the stuff of legend. They aren't just rare, according to traditional views of the sea, they shouldn't exist at all." Then on New Year's Day, 1995 a wave of 26m was measured hitting the Draupner oil rig in the North Sea off Norway.

Subsequent research vindicated the eyewitnesses. Freak waves are alarmingly common. As yet, there is no consensus on what causes these waves.

http://www.bbc.co.uk/science/horizon/2002/freakwave.shtml

Mr Veale said...

Re: Extraordinary events in history.

In "A Problem from Hell" Samantha Power tries to explain why the allies did not react to the German attempt to exterminate the Jewish people in World War II. Naturally, she points out that the allies suppressed evidence and intelligence of the Holocaust as they did not want to be dragged into any campaign that would interfere with their central objective - the military defeat of Germany. (So no bombers could be spared for Auschwitz etc.)

But Power notes, in an aside, that this does not quite explain all the facts. Enough evidence existed in the public realm to convince the average citizen in the West that the Holocaust was underway. And anti-semitism does not explain why the average Westerner was so disinterested in the plight of the Jews in Europe. Isaiah Berlin could only see a massive anti-Semitic pogrom. This was the analysis of many Zionists also.

So what explains their disbelief? ""A plot for outright annihilation had never been seen before and therefore could not be imagined. The tales of German cremation factories and gas chambers sounded far-fetched. The deportations could be explained: Hitler needed Jewish slave labour for the war effort...In the 1920's and 1930s, the press had debunked many of the allies wartime reports of German savagery, yielding "a hangover of skepticism" "

Power also tells the story of Raphael Lemkin, who fled Nazi Germany, and Soviet Poland, as he had read the evidence correctly. Hitler would invade the East, and annihilate the Jewish people. Jewish communities, whether intellectual or traditional, refused to believe him. Pogroms could be expected, and great suffering. But it simply made no sense for a modern nation to turn the engines of total war to an unreasonable goal - the elimination of an ethnic group that posed no threat. The Nazis had something to lose in making such an effort, and they would gain nothing in return.

Samantha Power "A Problem from Hell: America and the Age of Genocide" pp17-61, esp 34-36

Tragically, the Holocaust was a unique historical event. Genocides were common - Lemkin knew this. He studied the Turkish genocide of the Armenians, the Mongol incursions into the West etc. But never before had the machinery of an industrialised state been turned to such an irrational goal. Lemkin knew what was coming. Those who kept their thoughts within the realm of common experience did not.
For this reason (and so many, many others) I doubt that simple rules about 'analogy' capture the complexity of history. In fact "analogy" does not even feature in the index of "The Pursuit of History". It's easy to see why.
Graham

GearHedEd said...

Victor said,

"It's pretty much an open secret in the study of Hume's essay that a straightforward application of the mathematical probability theory he lays out would lead to, for example, skepticism about newspaper reports that person X won the lottery."

Except that we can see the grinning redneck holding the giant fake winnings check in all the publicity photos, so there is independent corroboration of the premise that someone won the lottery. Not so in the case of miracles.

GearHedEd said...

Victor also said,

"...The question is whether it is really possible to find a "neutral" or "unbiased" position from which to evaluate just how strong the antecedent improbability of a miracle really is. I don't see how you can evaluate the likelihood of miracles independently of a) the antecedent probability of theism and b) the overall likelihood that God, if He exists, would do something along these lines as part of his way of communicating with human beings."

This problem you have pointed out is at the root of why Bayesian logic can't prove miraculous claims: You have no basis for assigning any initial probability vlaues.

GearHedEd said...

Victor also said,

"...Experimentally imagining how the evidence might look to someone with a different belief system, which is what's involved in the "outsider test" is something that is worth doing, but I don't think it represents the ultimate test of the adequacy of one's religious (or non-religious) beliefs."

I'll go you one further: I don't think it's even possible to imagine how the evidence might look to someone with a different belief system, and is therefore impossible to assign any meaningful Bayesian values to the initial conditions, thus making probability analysis a futile exercise.

Mr Veale said...

"so there is independent corroboration of the premise that someone won the lottery. Not so in the case of miracles."
1)
Victor says - I have evidence that corroborates that a miracle took place
The sceptic says - no you don't the prior probability of a miracle is too low
Victor says - But in many cases, events with an astonishingly low prior probability do occur
The sceptic says - Yes, but we have evidence that corroborates that these events took place!
2)
The reference to the photograph is not helpful, given that written testimony is the primary source of our knowledge of the past. Suppose we only had written testimony that an individual won a lottery, and some evidence that their wealth increased.
Would we ceteris paribus accept the recipient's testimony that they were, indeed, incredibly fortunate?

(Or suppose it was an event with an even lower probability - a eccentric aristocrat decided to donate huge funds to a peasant chosen at random - would we accept testimony that such an event had occurred?)

3)
I'm not sure that objecting to subjective probabilities simply because they are subjective is a great strategy for the Frequentist.

4) In any case we do,commonly, have degrees of belief in various propositions. In the absence of an argument that this is irrational, Bayesians who reason from degrees of belief seem to be on safe ground.

5)
Another popular objection is "This isn't how I use Bayes Theorem in my day job." I'm not sure if that's what's behind Ed's arguments. But ignorance of Bayesian epistemology isn't an argument against any version of it. Neither is an air of condescension.

Graham

Mr Veale said...

"I don't think it's even possible to imagine how the evidence might look to someone with a different belief system"

That's a very odd claim.

History and the social sciences crucially depend on our ability to put ourselves in another person's shoes. To imagine how events would look to individuals with alien belief systems is essential to history.
And psychologists commonly infer how events appear to individuals with cognitive structures different to our own - young infants and even primates. Are you arguing that cognitive science, for example, is impossible?

And "you only think that because..." is levelled at believers and non-believers on this blog ad nauseam.

In any case, what on earth does this have to do with fixing priors? Who is recommending that we put ourselves in our opponents shoes before we fix a prior probability?

Graham

David Parker said...

"Who is recommending that we put ourselves in our opponents shoes before we fix a prior probability?"

Isn't that implied by John's OTF?

Mark Frank said...

You say that frequency theory has falled on hard times - but there is no theory of probability that is doing better. The philosophy of probability is unsettled.

However, whatever your philosophy, the probability of an event depends on what you know about it. It is not a fixed value of the event.

Take a dice throw. I throw a dice and it lands six uppermost. What is the probability of that event?

Mostly all we know about a dice is most dice are manufactured so that they are roughly symmetrical in shape and weight and that such dice when tossed repeatedly come down with each side in approximately equal frequency. However, we might know more about this dice. We might know that is assymmetrical. We might know that the person tossing it likes to have the one uppermost before tossing and that this leads to sixes being less frequent in the long run. In the extreme we might already know the result of the toss!

I believe this is the answer to what class of events a miracle belongs. The class is defined by the amount of relevant knowledge we have about the event (what is relevant is based on our experience of what is likely to influence the outcome).
So applying a Bayesian approach to a miracle e.g. water into wine.

We have an observation:

A small number of people report observing Jesus turning water into wine.

We have two competing hypotheses:

H1. The water did turn into wine.
H2. The reports were erroneous - either the observers were mistaken or they misled us.

The prior probability for H1 is based on what we know. For most people this is very little except there was some water and some people wanted it to turn to wine. The observed frequency of water turning into wine under those circumstances other than the case under consideration is zero.
The prior probability for H2 is a lot higher. Our knowledge includes that people often make false reports intentionally and unintentionally.

Of course you may believe that Jesus was capable of turning water into wine and wanted to do so - so you believe that you have relevant "knowledge" which raises the prior probability of H1 dramatically.

GREV said...

Someone on this thread said -- Philosophy is dead.

Oh that is good for a chuckle.

Mr Veale said...

I should also point out that historians have to capture unique and strange experiences all the time.

Tosh makes it clear that the historians first duty is to respect the otherness of the past. Many cultures have experienced the world in ways that we struggle to imagine (merely consider the way that we experience time in seconds, minutes and hours. Now compare this to societies that did not have mechanical clocks.)

The historian, according to Tosh, must use her imagination and experience to understand and re-create these experiences for the modern reader.
"Their purpose is not only to uncover the strangeness of the past but to explain it."

Graham

Mr Veale said...

The point being, of course, that banging on about "analogy" is pointless. Yes, there must be some 'point of contact' with history - 'a negotiation between familiarity and strangeness' - but historians seek to re-create unrepeatable experiences as a matter of course.

Graham

Mr Veale said...

Let's not get too caught up with the issues of priors. Here's a little thought experiment.

Suppose a group consisting of several investigative journalists, police officers, Martin Amis, Reza Aslan, EJ Hobsbawm, Simon Conway Morris, Richard Fortey, Ben Goldacre, Elton John, James Randi, Tim Haggard, Michael Shermer and Stephen Weinberg are all at one event, and all report that they observed a rock roll a short distance up a hill at that event. (Maybe they're all at an outdoor arts festival or something..use your imagination.)

Suppose the sceptics are reluctant to report this, as they were unable to explain the event (they could detect no trickery) and we have to do a little digging to discover their testimony. Say it pops up in personal journals and diaries, and a few of the record comments to journalists. But every sceptic on the list unequivocally testifies to the event in some way.
The few religious witnesses present could conceive of no religious significance. The stone just unexpectedly rolled up a hill for a few metres, then came to a stop. So they mention it in a few interviews and such, but they don't make a fuss in public. They, too, lack an explanation - and they're a little worried that this is a hoax at their expense.
Only the journalists and the police seem to get excited over the event. Say they had some photographic evidence, and could point to tracks left in the ground. Yet every effort to recreate the conditions of the event fails to reproduce the event. That is, geologists and physicists can't explain why this rock would behave this way, and they can't make any other rock behave this way.

Now, if he had that sort of testimony, with some corroboration, wouldn't John agree that "a rock had rolled up a hill at a public event"? Even though the prior probability is extraordinarily low on his assumptions - and on Vics.
The testimony is so improbable if the event did not happen, that it overwhelms the vanishingly low prior probability of the event.

Now the evidence for the Resurrection is not that good. But that's not my point. My point is that testimony can at least in theory provide evidence of a 'miracle/paranormal event'. Even when the prior probability is vanishingly low on everyone's reasoning


Or is John maintaining that he would deny that the event occurred in the face of the testimony?

Graham

Mark Frank said...

"My point is that testimony can at least in theory provide evidence of a 'miracle/paranormal event'. Even when the prior probability is vanishingly low on everyone's reasoning"

I agree. However, and I think this is Hume's point, the prior probability has to be greater than the prior probability of the alternative that the witnesses were either lying, somehow deluded or misinterpreted.

Also you need to be careful about the definition of the priors. We might conclude that the stone did indeed roll up hill. The interesting question is why.
One prior might be that he was right, another might be that the centre of the stone is iron and there was a large magnet at the top of the hill the witnesses were not aware of. A third is to recognise we don't yet know why.

Let us add one more thing to your example. Suppose immediately before the event an evangelist held an intense session among the assembled audience with much prayer and music and shouting and announced he would cause the stone to roll up hill. Now the priors really kick in - the religious types in the audience would find it far more plausible that supernatural forces caused the stone to roll. The Dawkins, Goldacre crowd would now find it far more likely there was some kind of mass delusion/trick to which they had been subject a la Darren Brown. Your own prior might change if you were told that the evangelist was not Christian but some never heard of sect that worshipped the old Norse Gods.

Mr Veale said...

Mark

I designed the thought experiment so that
(a) the form in which the testimony is given by each witness reduces the probability of lying
(b) the odds of all these witnesses lying about the same event would be vanishingly low
(c) there was no available naturalistic explanation for the purported event (i. journalists, ii. James Randi and Michael Shermer are present to rule out trickery & geologists and other scientists are witnesses iii. geologists cannot replicate the results even though they can replicate the conditions.) If you like, change the purported event to make naturalistic explanations more difficult.
(d) the physical traces rule out hallucination etc
(e) the point of the thought experiment is that we cannot give an explanation for the event, theological or otherwise. Yet even in the absence of any explanation, or any set of priors that would significantly raise the probability of the event, we would still infer that the event probably happened.

Graham

Mr Veale said...

Re: "Odin's evangelist"
- yes, I imagine that would provide good evidence that a supernatural event had occurred.
However here's something for Christians to ponder, especially those who are inclined to evidentialism. A Biblically informed epistemology would insist that we reject the truth of the Odinists claims. Even though we would have better evidence for Odinism than we have for the Resurrection.

Deut13
1 If a prophet, or one who foretells by dreams, appears among you and announces to you a sign or wonder, 2 and if the sign or wonder spoken of takes place, and the prophet says, “Let us follow other gods” (gods you have not known) “and let us worship them,” 3 you must not listen to the words of that prophet or dreamer. The LORD your God is testing you to find out whether you love him with all your heart and with all your soul. 4 It is the LORD your God you must follow, and him you must revere. Keep his commands and obey him; serve him and hold fast to him.

Matthew 7

15 “Watch out for false prophets. They come to you in sheep’s clothing, but inwardly they are ferocious wolves. 16 By their fruit you will recognize them. .... 21 “Not everyone who says to me, ‘Lord, Lord,’ will enter the kingdom of heaven, but only the one who does the will of my Father who is in heaven. 22 Many will say to me on that day, ‘Lord, Lord, did we not prophesy in your name and in your name drive out demons and in your name perform many miracles?’ 23 Then I will tell them plainly, ‘I never knew you. Away from me, you evildoers!

The point being that attaching a miracle to a message does not establish the truth of that message.

Graham

Mr Veale said...

BTW: It's not that there aren't answers available to the Christian evidentialist

Eg: Swinburne "Was Jesus God?"pp100-102 & 125-127

Mark Frank said...

"Yet even in the absence of any explanation, or any set of priors that would significantly raise the probability of the event, we would still infer that the event probably happened."

Graham - I don't think Hume or I would disagree with you. The issue is not the absolute value of prior beliefs but the relative value.

kilo papa said...

I wonder why Christians think of the resurrection as such an astonishing event?

If there is a God who had the astounding power to create everything in existence-billions of planets,galaxies-and the astonishing complexity of life, then wouldn't bringing a three day old corpse back to life be the equivalent of a two-bit magician pulling a rabbit out of a hat?

I once saw illusionist David Blaine bring a dead fly back to life. The resurrection of Jesus by the Christian god would be less impressive than that.