## Wednesday, August 24, 2011

### The Problem of the Single Case

I was looking at some exchanges I had with skeptics on Debunking Christianity, and one aspect of my views that was difficult to get across to them was the idea that, without attributing blatant irrationality to anyone, we can allow that different people are going to be able to assess the antecedent probability of something like the Resurrection of Jesus in different ways. These people are accustomed to working in scientific contexts where Bayes' theorem is used as a forecasting tool, and I take it what happens in those scientific contexts is that there are frequencies that are thought to determine what the antecedent probability of something is. So, we can look at how frequently something has happened in the past, and we can determine how likely it is to occur in the future. There is therefore a single, determinable answer as to how likely something is to occur.

However, to do this,  you have to subsume events within a reference class, and ask how likely that type of event is to occur. In the case of historical events, however, all of them are at least in one sense completely unique. How frequent are Kennedy assassinations? The guy could only be assassinated once. So, we receive a report that Kennedy was assassinated. We could argue that since the event was unprecedented, the probability of that event was zero, while the probability of false newspaper reports is considerably higher than zero. Therefore, we ought to disbelieve the report and assume that the newspaper report was erroneous.

On the other hand, political leaders are assassinated from time to time, so if we subsume the Kennedy assassination into the reference class of assassinations of political leaders, it becomes considerably less improbable. If we subsume it under the category of assassinated Presidents, we know that of the 34 Presidents that preceded JFK, three of them were killed by an assassin's bullet.

Extraordinary claims require extraordinary evidence. OK, can we measure the extraordinariness of the Kennedy assassination? How?

This is from the linked Stanford Encyclopedia essay on interpretations of probability:

Finite frequentism gives an operational definition of probability, and its problems begin there. For example, just as we want to allow that our thermometers could be ill-calibrated, and could thus give misleading measurements of temperature, so we want to allow that our ‘measurements’ of probabilities via frequencies could be misleading, as when a fair coin lands heads 9 out of 10 times. More than that, it seems to be built into the very notion of probability that such misleading results can arise. Indeed, in many cases, misleading results are guaranteed. Starting with a degenerate case: according to the finite frequentist, a coin that is never tossed, and that thus yields no actual outcomes whatsoever, lacks a probability for heads altogether; yet a coin that is never measured does not thereby lack a diameter. Perhaps even more troubling, a coin that is tossed exactly once yields a relative frequency of heads of either 0 or 1, whatever its bias. Famous enough to merit a name of its own, this is the so-called ‘problem of the single case’. In fact, many events are most naturally regarded as not merely unrepeated, but in a strong sense unrepeatable — the 2000 presidential election, the final game of the 2001 NBA play-offs, the Civil War, Kennedy's assassination, certain events in the very early history of the universe. Nonetheless, it seems natural to think of non-extreme probabilities attaching to some, and perhaps all, of them. Worse still, some cosmologists regard it as a genuinely chancy matter whether our universe is open or closed (apparently certain quantum fluctuations could, in principle, tip it one way or the other), yet whatever it is, it is ‘single-case’ in the strongest possible sense.

So, if we can't measure the extraordinariness of the Kennedy assassination, how can we measure the extraordinariness of the Resurrection?

Hiero5ant said...

Wait, what?

Why are you citing a problem for Frequentist interpretations of probability as though it affected Bayesians?

Victor Reppert said...

Well, of course there are plenty of ways of being a Bayesian. But what I keep getting excoriated for is my view that antecedent probabilities are subjective (or, if not completely subjective, at least they are bound to differ from person to person without any side being irrational). I have even been told I shouldn't even be talking about Bayesianism without affirming clear and definite prior probabilities based on frequencies. I have just been arguing that these sorts of constraints on Bayesianism work in some scientific contexts, but certainly don't work in all contexts.

unkleE said...

1. You have isolated the clear difference regarding the resurrection of Jesus. Unbelievers say it is impossible because we know, with certainty, dead men don't come back to life. They are including Jesus' resurrection in the class of all people's deaths, and we can agree with them that in all cases, and under natural conditions, the antecedent probability is zero.

But believers include the resurrection in the class of God raising the son of God from death, and in that class, we have 1 event, 1 success = 100%. Different initial classes based on different assumptions lead to different assessments of probability.

2. But the single case can be assessed by theoretical considerations. Take the beginning of the universe. Our universe only happened once, and we have no evidence (yet) of any others. A clear single case. Yet Roger Penrose assessed the sample space of all possible universes (he called it the "phase-space volume) and came to the conclusion that there were 10^10^123 possible universes there, and hence the probability of our universe occurring by chance was 1 in 10^10^123.

Morrison said...

Rational discussion are impossible at Debunking Christianity, for two reasons...

1. If you make a truly serious posts that casts doubt on an argument John made, it will deleted.

2. In the alternative, and John has been caught doing this because he left up the tag, "altered by moderatate" your post will be modified.

And then you will be banned.

John knows we know this.

Jesse Parrish said...

Like this?

But yes, I agree with the criticisms of frequentism, and I do not think that Bayesians can or should restrict themselves to probabilities subject to frequency interpretations.

Credence is bigger than that.

unkleE:

No, I do not have to ascribe a 0 probability to the Resurrection.

unkleE said...

Jesse Parrish said: "No, I do not have to ascribe a 0 probability to the Resurrection."

What antecedent probability would you give to it? (I'm not after a number, just a couple of words!)

And since you have said it isn't zero, is that because you think there is some small possibility it could happen by natural causes, or because you give some probability to God doing it? Or both?

These aren't trap questions or attempts to argue, just wanting to understand your view. Thanks.

Steven Carr said...

How frequent are Kennedy assassinations?

There were two, weren't there?

UNKLE
But believers include the resurrection in the class of God raising the son of God from death, and in that class, we have 1 event, 1 success = 100%.

CARR
So your expectation that your god will raise ordinary people is zero?

How do we know Jesus was the Son of God?

Easy. He was resurrected.

How do we know Jesus was resurrected?

Easy. He was the Son of God?

How do we know Muhammad was a Prophet?

Easy. He was given scriptures.

How do we know Muhammad was given revelation?

Easy. We can expect Allah to give revelation to his Prophet.

Religion is so simple. Almost too simple.

One Brow said...

I agree that ultimately, using a probability argumebnt for events like the resurrection is pointless.

Anonymous said...

Hey all, I have a question:

I never hear about Bayes being used for doing history outside of talks about the resurrection. Why is that? Or am I mistaken? It's quite possible that I just haven't read enough.

Thanks!

B. Prokop said...

Steven! How are you? We've missed you here. I hope all has been well with you!

As to your posting: No one, and I do mean NO ONE, has ever used such infantile reasoning as you have caricatured in your ridiculous fake dialogs. The ultimate in straw man arguments! Please, pull-eeeeez, don't tell me that that is how you genuinely believe people such as Lewis, Chesterton, or Aquinas (or even anyone contributing to this website) would argue their case.

Yer gonna have to do better than that if you want to compete with the Big Dogs. Otherwise, you'd better stay on the porch!

Boz said...

"How frequent are Kennedy assassinations"

To determine a bayesian prior probability, you are asking the wrong question.

As an analogy, see: "how frequent are (boz-breaking-the-4-minute-mile-for-the-first-time) events?". this is also a single-case.

The correct question is: "How frequent are assassinations", and "how frequent are sub-4 minute mile athletes".

Then we can use our Uninformative prior, with a uniform distribution.

:

Using a uniform prior probability:

(1) under the category:humans

(3) under the category:US Presidents

All three approaches will yield the same final probability. So it doesn't matter which you choose. the difference is that some evidence is moved to/from "background knowledge". The total evidence is still the same.

:

"Extraordinary claims require extraordinary evidence. OK, can we measure the extraordinariness of the Kennedy assassination? How?"

The value of the prior probability. "A specific person was assassinated" is more extraordinary (lower prior) than "A specific US president was assassinated". This is pretty straightforward.

"A specific person was assassinated" requires more evidence than "a specific political leader was assassinated", which requires more evidence than "a specific US president was assassinated"

:

For a bodily ressurection of a person via supernatural means, if we assign a prior probability of, say, 1 in a million, we are also saying that roughly 7,000 humans alive today have been ressurected via supernatural means.

Boz said...

Victor said: "I have even been told I shouldn't even be talking about Bayesianism without affirming clear and definite prior probabilities based on frequencies. I have just been arguing that these sorts of constraints on Bayesianism work in some scientific contexts, but certainly don't work in all contexts"

I agree that single-case events are very difficult to determine a prior probability for. The prior distributions have long tails. If you have a prior probability of 1.3%, and I have a prior of 1.4%, there is likely no way to resolve that small disagreement. when the two priors are many orders of magnitude apart, then productive discussion can be had on that disagreement.

So Victor, roughly what prior probability would you assign to a person being bodily ressurected by supernatural means?

For mathamatical reasons, it can't be 0. I would charitably say roughly 1 in a trillion.

Victor Reppert said...

Boz: You are assuming that there is a correct reference class to put events in. Why assume that? How do you go about doing it in a nonarbitrary way?

Boz said...

Boz: You are assuming that there is a correct reference class to put events in. Why assume that? How do you go about doing it in a nonarbitrary way?

Victor Reppert: I agree with you that There is no correct reference class. The decision is arbitrary.

However, through the magic of Bayes' Theorem, this is not a problem! Because all reference classes lead to the the same final(posterior) probability. This can be shown in the Kennedy example.

Every piece of evidence can be considered before, or after we calculate the prior probability. (But not both before AND after).

under the category:humans , the prior probability that Kennedy would be assassinated = (all assassinations / all humans who ever lived) = ~1m / 50b = 0.00002. Now consider ONLY the piece of evidence that he was a current US president. This makes the posterior probability 4/44 = 1/11. (we have yet to consider any other evidence)

under the category:serving US presidents , the prior probability = 4/44 = 1/11. (next, we will consider the other evidence)

so, it all evens out, like magic!

---

So, going back to the original question: "how can we measure the extraordinariness". Extraordinariness is measured by the prior probability.

Kennedy, a person, was assassinated, is very extraordinary. 0.00002

Kennedy, the current US president, was assassinated, is not extraordinary. 0.090909

Gregory said...

I wonder what the "prior probability" is--all things being 'sort of' equal--that Bayesian theory would have this kind of "Delphic" ontological status in a closed, physical universe?

Tony Hoffman said...

S Wilde: "I never hear about Bayes being used for doing history outside of talks about the resurrection. Why is that? Or am I mistaken? It's quite possible that I just haven't read enough."

Modern History is (at best) a soft science, but it is committed to methodological naturalism. (New Testament studies is basically the study of history, without the methodological naturalism restriction.) A Bayes analysis of historical questions, per se, is not one that historians commonly employ (not that I've seen, anyway), but Bayes seems to have become the accepted way to respond to the question (asked by Christians) of why it is that historians do not allow that supernatural occurrences are the best explanations for historical events.

A person who claims that a supernatural event is the best explanation for anything described in the Bible is not a historian, because there are obviously better, naturalistic explanations for everything described -- starting with the fact that it's way more common for people to make stuff up, be fooled, etc., than it is for donkeys to talk, people to walk on water, etc.

Anonymous said...

Victor, why the hell don't you tell your people that what Morrison says about me deleting and/or altering posts is not true from your experience? You KNOW that if I delete or edit out personal attacks it's because of what Morrison must have said, because I have NEVER done that with anything YOU have said.

Why won't you say this? If someone on my Blog accused you of doing the same thing I would say that from my experience you have never done that with anything I said on your blog.