NaClhv

And now, time for a short interlude in this series.

I've been in communication with Aron Wall of Undivided Looking. It's a great blog that people should check out, which covers much of the same subject matters as my blog. I've asked him for feedback on my series, and he graciously replied back with a very lengthy post. There are some thing in my series that he disagreed with, and there are some things in his reply that I disagreed with, and there are things that came up that merited further discussion - so I thought it best that I reply back in a few of my own posts.

Greetings

Aron,

Thanks for taking the time to read over my series and post your reply. It's been very helpful. Your reply was in fact a great deal more than what I had anticipated, so I thought it appropriate to respond in a few of my own posts.

Things I learned

Your explanation for exceedingly small probabilities being unreasonable, due to the need to take even crackpot theories into account at such extremes, makes sense. I gladly abandon my suggestion, in the comments of your blog, that scientific laws or the existence of the Roman empire can be asserted with 1e100+ odds. I was actually pretty uncomfortable with saying those things, but I couldn't put my finger on why I felt that way at the time - so this is in fact a relief.

In my own series, I have not very much strongly asserted that the probability I obtained (1e32) is definite. So I will take your suggestion that I will not take this value seriously, instead using it to demonstrate that Jesus plainly rose from the dead if we make a set of likely of assumptions. I will handle the other, very unlikely cases separately.

Fortunately, I'm still in the middle of my series, so I can add in things like this without too much trouble. For that reason, I'm glad we're having this discussion now, rather than after the end of the series.

Probabilities and Bayes' factors

As for a great deal of your other comments - I think there's some confusion between a probability and a Bayes' factor. Both of us have used words like these somewhat imprecisely, and clarifying our usage will resolve a number of things. So:

That 1e54 factor I cited is a Bayes' factor, not a probability. It is a ratio of in-model probabilities - and each model may just be a simple, computable hypothesis (e.g. "a random shuffle") or a complex aggregate thereof ("all the different possible ways that a deck of cards may be shuffled"). As such, I more or less stand by that number and am not bothered at all by its large magnitude - with the caveat that the value is for a specific model, and not for the resurrection hypothesis in the aggregate. Essentially, I'm saying that I'm ignoring crackpot theories for the time being (conspiracy theories, malicious spirits playing jokes on us, etc.). As a Bayes' factor, that 1e54 should not be thrown out simply because it is too large, although it is unreasonable to use it to calculate a final, aggregated probability. If I understood your point correctly, this should not be a problem.

The five sigma probability of 1e-6, although I called it a probability, should also really be interpreted as a Bayes' factor. The words get in the way here, because in null hypothesis significance testing, that five sigma value is a probability. But in the Bayesian framework it should really be a Bayes' factor between the null hypothesis and the "whatever result you got" hypothesis.

1e8 as the Bayes' factor for a human testimony

Furthermore, the 1e8 factor for the strength of a human testimony is also a Bayes' factor, not a probability. It is NOT saying that people lie only 1 out of 1e8 times, which would be clearly absurd as you pointed out. Rather, it's saying that your prior odds should be adjusted by that much based on a human testimony.

This difference resolves your counterexample of "how many times in my life have I been lied to?" Yes, there has been 1e5 situations where someone was tempted to lie, and you've maybe been lied to around 1e3 times. That works out to lying rate of 1e-2, or a truth to lie posterior odds of 1e2.

But each of these lies probably had a typical prior odds of 1e-3 ("I got into a car accident", "I got locked out of my house", "I went to Harvard", etc). You yourself have said that one in a million events happen all the time, so a prior odds of 1e-3 is not at all out of the ordinary. This is especially the case when someone is making a positive assertion that something happened ("I once bowled a perfect game"), rather than trying to deny something or get out of something ("I have to wash my hair").

So, that's a prior of 1e-3, and a posterior odds of 1e2, which gives us a Bayes' factor of 1e5. The other 3 orders of magnitude can be made up for by adding the "earnest, sincere" condition, and the fact that we're specifically taking about scenarios where people are tempted to lie to you. So, the value of 1e8 as the Bayes' factor for a human testimony is in good accord with your lying example.

That 1e8 number also still bears out through your license plate example. Would you doubt someone who's earnestly, sincerely claiming to have a license plate like 6DVL666? If the Bayes' factor for a human testimony is really about 1e8, you should have some doubt, but not a very strong doubt, that this person was telling the truth - this seems to be about the right level of skepticism required here. I don't think, for example, you'd find 10,000 or even 1000 liars to a single truth-teller in this scenario.

I'm quite sure of this because all the examples I gave in my series - winning the lottery, being struck by lightning, having a PhD in physics from Harvard, having a loved one killed in 9/11, etc. - all involve cases like the 6DVL666 license plate, and not like the 4ZIW623 license plate. They're all special, interesting, positive claims that someone might want to choose to lie about, where all the improbability is in the main claim and not in the details - and I still get values around 1e8. So in every example that I have examined or you have brought up, this value seems to hold up.

(To be continued in the next post)