NaClhv

Here are some more examples from which you can estimate the Bayes' factors for an earnest, personal human testimony.

Imagine that you've promised to meet me on a particular date, but I don't show up to the appointment. You're understandably peeved, but then you get a phone call from me saying, "I just got into a car accident. I'm okay. But I'm really sorry that I couldn't make it to our meeting today. Can we still meet?"

Now, would you believe my story? Did I really get into a car accident on the day of our appointment? What would you assign as the probability that I'm telling the truth?

The average driver gets into a car accident roughly once in 18 years. That's about once every 6500 days. So the prior probability for getting into a car accident on a particular day is 1/6500. If you choose to believe me - say, you think there's less more than a 90% chance that I really was in an accident - then you've changed the odds for my car accident from 1/6500 to 10/1, and you've therefore granted my phone call a Bayes' factor of 65000 - or nearly 1e5.

Remember our calculations from earlier: even with a Bayes's factor of just 1e4, there's already a 99.999% chance for the resurrection to be real. In other words, if you would believe that I got into a car accident, you ought also to believe in the resurrection. Otherwise you're being inconsistent. If you wish to disbelieve the resurrection, you must also be the kind of person who says, "I don't believe you. I think you're lying about the car accident. You need to give me additional evidence before I believe that something that unlikely happened".

Ah, but maybe the people who are skeptical of the car accident are right? Maybe we should be more skeptical in general? It might be the polite thing to do to believe someone in such situations, but how do we know that that's actually the mathematically right thing to do?

Well, this is where the fact that this actually happened to me comes into play. I once got into a car accident on my way to a wedding. I was not hurt, nor was my car seriously damaged - but the whole affair did cause me to miss the entire wedding ceremony. I only managed to show up for the reception. That day, I told numerous people that I had gotten into a car accident, and gave it as my excuse for missing the ceremony. Not a single one of these people doubted me in the slightest: they all believed me. And they were right to do so, because I had in fact gotten into a car accident.

In fact, I've never heard of anyone, anywhere falsely using the "I had a car accident" excuse for missing an appointment. There are simply no reports of it that I know of. This is in spite of the fact that I have heard of numerous car accidents, and have been in one myself, and have heard it used as a genuine excuse before. All this, combined with the great deal of trust that the others correctly put in me when I told them of my car accident, tells me that the earlier 90% chance for the accident is too conservative. If I were to hazard a guess, I would say that such car accident stories are trustworthy about 99.9% of the time. That means that the posterior odds for the car accident are about 1e3, and the Bayes' factor from an earnest, personal testimony about a car accident is about 1e7 - although this is admittedly somewhat speculative.

So, if someone tells you about their car accident on a particular date, the Bayes' factor for their testimony should at least be 1e5 as a lower bound, and probably (but more speculatively) around 1e7.

Now, what if someone claims to have gotten into two car accidents in one particular day? The prior odds for such an event, assuming independence, is about 1e-7.6. Now, I have not heard anyone make this claim exactly, but I have heard of somewhat comparable events, like two tire blowouts happening on the same day (this, too, actually happened to me once). The comparison is difficult to make, as there are strong dependence factors and statistics on blowouts are harder to come by. However, going on my intuition, and my experience with similar events like blowouts, I would be willing to believe someone who claimed to have had two car accidents on a particular day, or at least give them even odds that they're not lying. This gives their testimony a Bayes' factor of about 1e8. While this is not a solid measure of the Bayes' factor on its own, it does validate my earlier estimation of the Bayes' factor being around 1e7.

Now, what about the Bayes' factor in the following scenario?

You're talking to a friend that you haven't seen in a year, and you're exchanging news about mutual acquaintances. You ask, "how's Emma doing?" Your friend then replies and says:

"Oh, you haven't heard? Emma... is dead. She was killed in a car accident. And you know how she was really close to her mom? Well, when her mom heard the news of Emma's death, she committed suicide - they say that they had the funeral ceremony for both of them together."

You may have guessed that this, too, actually happened to me. A friend of mine told me this tragic story about a girl we both knew. Don't be too concerned - the name of the girl has been changed, and this happened long ago - long enough ago that all the parties involved must have gotten well past the shock and the grief.

But, let us turn back to the question at hand. Should I trust my friend, on this very unlikely story? The yearly car accident fatality rate is about 1 per 10,000. The suicide rate is about the same. My friend's story, therefore, has a prior odds of about 1e-8 of being true. There is some dependence factors which increase the odds (a mother is more likely to commit suicide after her daughter's death), but the specifics of the story (the specific cause and timing of the suicide) would again decrease the odds. Let's say that they basically cancel each other out.

I'll go ahead and tell you that I did believe my friend. I did not really doubt his story. If I had to put down a number for my degree of belief, I would say that I gave his story about a 1e3 odds of being true. So the odds for this sequence of events went from a prior of 1e-8 to a posterior of 1e3, and therefore the Bayes' factor for my friend's testimony is about 1e11.

But was I right to trust my friend? Maybe I should have said back to him, "I don't believe you. Your story is just too ludicrous"? Well, as it turns out, I did get independent verification for a good chunk of this story later on. I really was right to trust my friend. Given that this is only a single instance of verification, this only validates that I was right to trust my friend, but not necessarily that I was correct to give the story a posterior odds of 1e3. So, at a minimum, I was definitely justified in giving my friend at least 1e8 for the Bayes' factor as a lower bound, and I feel that the correct value should actually be closer to 1e11.

So, here is the summary of the Bayes' factor evaluations thus far. Using publicly available statistics (car accident and fatality rates, suicide rates), and empirical events in my own life which I have personally experienced, lived through, and verified, I obtained two separate Bayes' factors for an earnest, personal testimony. In a story about a car accident on a given day, the lower bound on the Bayes' factor for that story should be 1e5, and the actual is probably closer to 1e7. In a tragic story about the unlikely death of a mutual acquaintance, the lower bound on the Bayes' factor for that story should be 1e8, and the actual value is probably closer to 1e11.

We see that in each case, even the minimum possible Bayes' factor exceeds 1e4. Recall that a Bayes' factor of 1e4 for an earnest, personal testimony would already put the probability of the resurrection at 99.999%. The more likely values we calculated in these specific cases, of 1e7 and 1e11, agrees very well with the value of 1e8 that I've used for the general case.

There are more calculations to come in the next week.