Ex., if I believe in elves (perhaps I am living in the World of Greyhawk) and you tell me that there is an elf outside that wishes to speak to me, this is not an extraordinary claim. It may not be a true claim, but it is one which falls within the context of my worldview.

Ex., if I do not believe in elves (perhaps I am living in the real world) and you tell me that there is an elf outside that wishes to speak to me, this is an extraordinary claim because it is in contradiction to my worldview. Even were it true, I would be liable to dismiss it. I would certainly not accept pointy ears and a smarmy attitude as “proof” that the person on my doorstep was an elf.

Likewise, in the real world, some of the claims made re: Relativity or QM have been, so far as we know, accurate, but have required “extraordinary proof” to be accepted as so for the simple reason that they violated the until-then most popularly held scientific viewpoint.

IOW, “Extraordinary claims require extraordinary proof” is missing the phrase “to be accepted by those to whom the claim seems extraordinary”, which is required for the statement to be sensible.

Hence, “Extraordinary claims” means nothing more than “claims which go against other things that I have accepted as true, and therefore which would require me to alter my belief system should I accept them”, which has little or nothing to do with whether said claims are true. It justifies a “no amount of evidence will change my mind” attitude all too often, though perhaps not in this case.

Exactly! They look at hndreds of thousands of people who smoke or drink or whatever and figure out who had a serious illness and what it was. Then they assign a probability to you or to me to figure out our premiums. But it's all based on a very very large sample.

Again, from wikipedia (the page on Probability Theory – embedding the url caused a problem):

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion. Although an individual coin toss or the roll of a die is a random event, if repeated many times the sequence of random events will exhibit certain statistical patterns, which can be studied and predicted.

And if you follow the link to event, you'll see that an event is with respect to a larger sample space. Think about a UFO crashing on Earth and the alien is dead. Doctors are able to determine (this is hypothetical, after all) that the alien did not die in the crash but rather it died of congestive heart failure (and that is, presumably, why he crashed). Then another alien crashes and he, too is dead but not from the crash. Before doing an autopsy, what is the probability that this second alien died of congestive heart failure? There is, obviously, absolutely no way to determine this because there is no sample space from which to draw the conclusions. Heart disease in this species may be very common or very uncommon. Perhaps it's caused by space travel so it's more likely that the second alien did die from it. But no matter what, there is no way to assign a probability that the second alien died from heart disease.

I see the resurrection as the same situation. It is such a unique event that there is no sample space from which to draw any knowledge.

Exactly! They look at hundreds of thousands of people who smoke or drink or whatever and figure out who had a serious illness and what it was. Then they assign a probability to you or to me to figure out our premiums. But it's all based on a very very large sample.

Again, from wikipedia (the page on Probability Theory – embedding the url caused a problem):

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion. Although an individual coin toss or the roll of a die is a random event, if repeated many times the sequence of random events will exhibit certain statistical patterns, which can be studied and predicted.

And if you follow the link to event, you'll see that an event is with respect to a larger sample space. Think about a UFO crashing on Earth and the alien is dead. Doctors are able to determine (this is hypothetical, after all) that the alien did not die in the crash but rather it died of congestive heart failure (and that is, presumably, why he crashed). Then another alien crashes and he, too is dead but not from the crash. Before doing an autopsy, what is the probability that this second alien died of congestive heart failure? There is, obviously, absolutely no way to determine this because there is no sample space from which to draw the conclusions. Heart disease in this species may be very common or very uncommon. Perhaps it's caused by space travel so it's more likely that the second alien did die from it. But no matter what, there is no way to assign a probability that the second alien died from heart disease.

I see the resurrection as the same situation. It is such a unique event that there is no sample space from which to draw any knowledge.

I could be misunderstanding, but I don't think the resurrection is one of the variables–that's what they're trying to measure the probability of, in light of the variables. The variables would be things like the empty tomb, the beginning of the church, etc. What is the likelihood that all of these present variables would occur at the same time if there were no resurrection? Or does the resurrection actually have the most explanatory power (as opposed to other theories)?

I don't think probability theory is only used for repeatable, mathematical things. For example, actuaries determine things like the probability that a person will get a serious illness for health insurance companies. They look at the variables present (they smoke, they drink, etc.) and determine risk of illness. They have ways of assigning values to the variables, but I have no idea what those ways are, not being a statistician.

Every random variable has a probability distribution which is a function telling how often each random value comes up. So, in the case of the two dice, the probability distribution is peaked at 7 and decreases towards 2 and 12. This is because it is much more likely to get a 7 than a 2 or 12 because there is more than one way to get a 7.

So, the probability distribution has nothing to do with reducing the randomness of a variable. It simply tells you how likely each possible random value is.

The key — from what I understand — is that a random variable has the necessity of repeatability. A random variable is one that can be measured again and again and again so that the probability distribution is generated. The toss of the dice, the flip of a coin are both random variables because you can do it again and again. Yes, you can talk about the likelihood of THIS roll producing a 5 and this seems like an isolated event but the analysis is based on this same event being repeatable.

So, perhaps I'm just not “getting it” but I don't see the resurrection of Jesus as a random variable at all. It happened once to one man. It's not repeatable. There's no probability distribution function because there's only one data point. I guess we could phrase it something like “the probability of a person being raised from the dead” and then the case of Jesus is one instance of this. But I'm not sure that this even works.

Anyway, I'm still not satisfied that the analysis method is valid for this case but I don't know enough (and don't have the time to learn probability theory right now) to figure it out. I'm going to ask someone who is much more versed in probability theory and then I'll get back to you.

Some more clarifying definitions from Wikipedia:

Conditional probability is the probability of some event A, given the occurrence of some other event B. Conditional probability is written P(A|B), and is read “the probability of A, given B”.

Joint Probability is the probability of two events in conjunction. That is, it is the probability of both events together.

Marginal probability is the probability of one event, regardless of the other event. Marginal probability is obtained by summing (or integrating, more generally) the joint probability over the unrequired event. This is called marginalization. The marginal probability of A is written P(A), and the marginal probability of B is written P(B).

Bayes' theorem…is a result in probability theory, which relates the conditional and marginal probability distributions of random variables. In some interpretations of probability, Bayes’ theorem tells how to update or revise beliefs in light of new evidence a posteriori.

In other words, it measures the likelihood of the resurrection given the available evidence. It does this in part by measuring the probability of the present variables occurring together randomly. If such a thing is unlikely, then the resurrection looks more likely.

I found a reference which states that Swinburne uses Bayes' Theorem for his result. I then read a little bit about Bayes' Theorem and it is applicable to random variables. It “relates the conditional and marginal probabilities of stochasitc events A and B” (from wikipedia).

So, my questions is: Is it even valid to apply Bayes' Theorem to this problem? Are the resurrection and our background knowledge of the world random variables?

In the same way, he translates the mathematical formula, Pr (R/B&E), into philosophical terms. The formula is intended to show the probability of the resurrection to explain the available evidence. In philosophical terms, this formula shows the explanatory power of the theory.

The philsophical phrases are not changing the way the formulas work, he's merely communicating the work the formulas are doing to an audience who speaks “philosophy” rather than “mathematics.”