I need help: Device of imaginary results by I J Good

(As promised, I post description of my attempts here)

Case 1: Julius Caesar. The existence of Caesar seems very likely for me. Therefore, I will think about evidence that would convince me that Caesar is a myth.

I decided that the following three pieces of evidence will be enough for me to start doubting existence of Julius Caesar:

1) I don't know anything about the process of burial of the Roman emperors. Hence it wouldn't be inconsistent to assume that there is an official "emperor's tomb", a luxurious necropolis where the crypts of every emperor are located. Having assumed that this true, I imagine a discovery saying that the crypt of Julius Caesar is missing, or his tomb is empty, or the body inside couldn't belong to Julius, or something along those lines.

2) Similarly, the existence of a reliable independent Arabian historian similar to Herodotus wouldn't contradict my worldview. So I can assume that there indeed was such a historian. If his book about Roman Empire had failed to mention Julius Caesar at all, that would have been an evidence pointing to Julius being a myth.

3) Why can't I imagine the second Arabian history book failing to mention Caesar?

Having imagined all that, I decided that it would be enough for me to start doubting Julius Caesar's existence.

Let's turn to the probabilities.

1) P(no crypt|no Julius Caesar) ~ 1. P(no crypt|Julius Caesar) = ?

It doesn't look like there's some way to easily estimate that quality. I notice that I'm confused, but let's try anyway. It's possible to convert this probability into the relative frequency: there were cases of some monarch's bodies having been removed from their graves. I know two such cases: False Dmitriy I of Russian Tsars and Akhenaten of the Eighteenth dynasty of Egypt. Counting the other people of their lineage gives the approximate figure around 2/350 which I consider kind of satisfying, though it appears to be somewhat higher than desired. However, I had to consult Wikipedia to get this estimate, and that kind of violates the "no gathering other relevant real-world data" rule.

2) and 3) P(no mention of Caesar in the Arabic book|no Julius Caesar) ~ 1 P(no mention of Caesar in the Arabic book|Julius Caesar) = ?

I have trouble estimating that at all. I have no slightest idea how to estimate this without a lot of imaginary betting, and imaginary betting kind of defeats the whole point. Why shouldn't I estimate P(Julius Caesar existed) via betting?

Well, betting makes sense only when my utility function is linear in money, and that holds only when the probabilities are sufficiently close to 0.5. Maybe I should break my prior into several parts via Good's device and then estimate the parts via betting.

It seems plausible. However, I think that if Jaynes had meant his exercise to be done in that way, he would have explicitly told so.

I notice that I'm confused. Let's try to find my mistake:

1) The alternative hypothesis "Julius Caesar existed" is too vague, and hence it is difficult to come up with the conditional probabilities. It seems likely, but I can't come up with something better.

2) The experiment is bad. I should think about something else: for example, I draw a random portrait of Roman emperor from the urn containing portraits of every Roman emperor, how many non-Caesar's portraits are enough? Coming up with something like that seems very difficult for me. I'm very bad at selling non-apples.

This is a reason I wanted to find a new perspective and don't wanted to spoil anyone by anchoring him/her to my vision.

3) The evidence is bad. I should come up with some new evidence such that I can calculate the probability P(E|Julius Caesar)

4) There is a good way to estimate conditional probabilities, I just missed it.

I haven't worked much with the other cases, but it seems that it would be difficult to calculate their relevant probabilities as well (how can I calculate anything about geology in (3), for example?). So I think that I've misunderstood something, and maybe someone there can describe the correct way of doing this exercise to me.

I have the same reservation regarding the probability regarding propositions 1) and 2) as army1987. In particular, I find that the probability that all writings regarding the aforementioned people are true is exceedingly low for both, but the probability that some person existed bearing that name, who performed at least one action or bore one trait that was subsequently recorded is rather high. Considering that this is meant to be an exercise on evaluating one's priors (or at least that is how it appears to me), I would consider choosing one interpretation or the other and work from that. If you feel the need, simply try both interpretations or find a middle ground that you feel comfortable with. If this is not your issue with the propositions, then I would require more information on your attempts to solve the exercise in order to provide meaningful feedback.

(I decided not to share details of my attempt to solve this exercise unless asked. I don't think that my perspective is so valuable and anchoring would be bad.)

I think it would be good. It narrows down the issues required for the response. And it demonstrates effort to solve the problem on your part.

To be brief - "explain this to me, but I won't show any evidence of making the effort myself or attempting to make it easier for you to respond to me" - doesn't feel so respectful of the effort your asking of others, and it's so broad that I don't expect great bang for the buck with the effort required.

If you've got a question, make the issue as clear as you can make it, instead of treating us like guinea pigs in your experiment where you're trying to avoid bias.

I don't mean this in a big negative way (I don't do the emotion free writing tone so popular here). I see that you've spent effort detailing your question, are asking an honest one, and have a reasonable reason besides laziness not to elaborate further (avoid bias).

But asking for input is asking for a favor, and I tend to expect people asking for favors to make my fulfillment of their needs as easy and productive for me as possible.

Another way to express it, is that I'd be writing expressly in the dark of what the real issue is. Maybe it's a control thing. "Do this so I can analyze the results" doesn't have a lot of appeal.

I guess I wasn't brief. It's an interesting question, but as posed, it left me with a shrug.

In (1) and (2), how you count a character if it was inspired to a real person of the same name but a sizeable fraction of the stories about him are false?

(4) sounds like a shibboleth to test whether you've read the title text of http://xkcd.com/867/.

(5) is logically impossible, unless “see” means something other than ‘detect light’ and/or “darkness” means something other than ‘lack of light’.

(7), (8) and (9) in principle might be correct in certain circumstances and wrong in others.

A problem with this is that humans often exhibit base rate neglect, and so may overestimate small probabilities in this sort of exercise .

What evidence would convince you that there is a very small chance that Julius Cesear was a real person? What evidence would convince you that Achillies was a real person? (with whatever certainty you call "certain")