I would like to try double crux.

If you have an other being involved in any way the power isn't clairvoyance, it's telepathy. The problem is how to distinguish the telepath from the clairvoyant who interprets their unusual senses by hallucinating non-existent "voices" or some such.

Yes, I was responding to what I thought he meant instead of his literal wording there. That's why I included "in the sense that you probably mean". I don't think "low probability" by itself is sufficient to make an event a miracle, and was assuming there was more to it than that, like an apparent purpose for being that way, perhaps.

Please correct me if I am wrong, but I feel that there is a ... well, not mistake... assumption that is not necessarily true. What I mean is the following. Let us consider the space of all possible inputs and the space of all possible outputs for the Turing machine (yeah, both are infinitely dimensional, who cares). The data (our Universe) is in the space of outputs, theory to test in the space of inputs. Now, before any assumptions about data and theory, what is the probability for the arbitrarily chosen input of length n lead to output with length N (since the output is all the observed data from our Universe, N is pretty large) - this is what is prior probability, correct?

No? Perhaps you were trying to do something else, but the above is not a description of Solomonoff induction.

Where exactly is the faulty assumption here?

In Solomonoff induction, the observations of the universe (the evidence) are the inputs. We also enumerate all possible algorithms (the hypotheses modeling the universe) and for each algorithm run it to see if it produces the same evidence observed. As we gain new bits of evidence, we discard any hypothesis that contradicts the evidence observed so far, because it is incorrect.

The fact that we are looking always for something simpler is an assumption of simplicity. Our Universe apparently happened to be governed by the set of simple laws so it works. However, this is the assumption, or axiom. It is not corollary from some math - from math prior should be awfully complex hypothesis.

What probability should you assign to the proposition that the next observed bit will be a 1? How should we choose between the infinite remaining models that have not yet contradicted observations? That's the question of priors. We have to weight them with some probability distribution, and (when normalized), they must sum to a probability of 100%, by definition of "probability". We obviously can't give them all equal weight or our sum will be "infinity". Giving them increasing weights would also blow up. Therefore, in the limit probabilities must decrease as we enumerate the hypotheses.

To be more precise, for every ϵ > 0, there is some length l such that the probability of all programs longer than l is at most ϵ.

Otherwise, I can say that probability decays logarithmically with complexity, and you will say that it decays exponentially, and we will get totally different prior probabilities and totally different results.

Can you? It's not enough that it decays; it must decay fast enough to not diverge to infinity. Faster than the harmonic series (which is logarithmically divergent), for example.

Solomonoff's prior is optimal in some sense, but it is not uniquely valid. Other decaying distributions could converge on the correct model, but more slowly. The exact choice of probability distribution is not relevant to our discussion here, as long as we use a valid one.

the majority of the outputs with length N can not be produced by the input with length n (basically, only 2^n out of 2^N has any chance to be produced in such way). For the vast majority of these outputs the shortest input producing them will be just the algorithm that copies large part of itself to output - i.e., a prior[i] hypothesis is incredibly long.

The fact that we are looking always for something simpler is an assumption of simplicity. Our Universe apparently happened to be governed by the set of simple laws so it works. However, this is the assumption, or axiom. It is not corollary from some math - from math prior should be awfully complex hypothesis.

If the observation is in no way compressible, then there is no model simpler than the observation itself, and your prediction for the next bit can be no better than chance. Maybe you haven't observed enough yet, and future bits will compress.

But there can be no agents in a totally random universe, because there is no way to predict the consequences of potential actions. We can rule that case out for our universe by the anthropic principle.

If you put this assumption as initial axiom, it is quite logical to set incredibly low priors for God.

That's right. So what is your alternative? Give up on induction altogether? That's completely untenable.

That this is not the first time an icon has produced myrrh is actually evidence in favor of a hoax or natural event. In either case, it's happened before so it must not be all that difficult for the conditions to be right or to set it up.

As for the plane, at least several explanations are possible.

1. The event could actually be natural. Microscopic quantities of myrrh are quite detectible, and could be seeping from the paints for quite some time.
   • I can easily imagine the effect being encouraged by spraying a fine mist of oil, water, alcohol, or some other mild solvent over the icon. This could even be passed off as maintenance of some sort. "Just keeping it fresh. Travel is so hard on these paintings, you know."
   • The paint could be formulated with an unusually high myrrh content for some reason. This would go a long way to explain why not all myrrh-containing paints do this.
   • The guy who travels with it might not even know how it really works, but actually believe that it's a miracle.
2. Perhaps the myrrh could be applied directly to the icon in some way that doesn't really show up on an x-ray scanner. "Be careful please! Of course it smells like that. Don't you know what this is?"
3. The vessels containing the myrrh may be small enough that airport security doesn't care. A little goes a long way, and people travel with essential oils all the time. I can think of several ways to produce a myrrh distribution system by carving small channels in the frame. These would likely not even show up at airport security, and could contain enough myrrh to last for quite some time.
4. The supply of myrrh could be shipped by another means, or in checked luggage.
5. Charter planes don't have nearly the security considerations that public planes seem to require.
6. The myrrh supply doesn't necessarily have to travel at all. Myrrh is, in fact, reasonably easy to get in many places. I understand that christian book shops, for example, routinely sell the stuff. Here it is on Amazon. You could even repackage it into tiny vials for sale "to recoup travel costs". That would provide cover for your large orders.
7. The quantity of myrrh that the icons give off may have been vastly exaggerated over time to enhance the story, either by the originator of the story or by those retelling it. This is a widely known effect of word-of-mouth information transmission, and something that people are often perfectly happy to ignore when they encounter the truth.

Even combinations of several these explanations produce prior probabilities that are many orders larger than the probability for "God did it". The amount of complexity hiding in the word "God" is pretty extreme.

because of the amount and the longitude. Everything that could been inside should have gone away.

I don't think that word means what you think it means. Did you mean "latitude"? Hawaii has a more tropical climate because it is near the equator, not because it's near the prime meridian. Or did you mean "longevity"? Either way, you can't expect all oils to evaporate quickly like water, even in a warm or arid climate. It's true that there are some oils used in paint that dry quickly, but there are other oils that do not dry out, even after a long time.

The previous person was killed and tortured, the icon disappeared. The murderers were not found. It would be quite crazy idea, knowing this story, to make this mystification. To put your life under risk for what? For stupid hoax? You must be crazy to do it.

Or greedy. How about for money? Fame? There are known examples of hoaxes with motives like these.

And this person (current keeper) serves in police

A policeman might be even less afraid of criminals trying to kill him, because he already has to deal with them.

If I would need to do such a hoax, I would put some source of myrrh inside, and refill it periodically. It can be done of course. I could even believe that it can be done such that observer, taking the icon, would not notice any difference from the usual icon. But is it possible to avoid X-rays somehow? They travel by plane, I bet they do not put icon into luggage (it is too precious). So they must go with it as hand luggage. There the custom, using X-rays, observe small vessels inside the icons and asks what is it. And it is done, the hoax is over.

If I needed to perform such a hoax, I wouldn't have to modify the icon at all. The Roman Catholics tend to use statues instead of icons, and they have examples of those weeping too. I heard of one case where someone was caught applying the "tears" with a squirt gun. You don't need to carve channels or secret compartments for a hoax. It's enough to have a spray bottle and vegetable oil, which are available for purchase pretty much anywhere. And if you use a non-drying oil, you don't even have to re-apply it! It will stay "wet" long after water would have dried out.

-different myrrh-streaming icons, as long as it passed the check by church officials not only on the local level

I don't trust church officials any further than I can throw them. They have a fundamental conflict of interest. Their loyalty is to the Church, not the truth. The fact that someone is a priest makes me even less likely to trust them. The Roman Catholics have been rocked by child sexual abuse scandals which have been all over the news in this country, and worse, they attempted to cover them up to maintain the Church's reputation. A cursory web search reveals the Eastern Orthodox seem to have similar problems for similar reasons. The Churches cannot be trusted to be honest with us.

These officials are clearly not rationalists (if they were, I would not expect them to be religious!) I don't expect most of them to even be scientists. But even if they were, scientists can still be deceived.

"I feel that for me" feels very wishy-washy for me.

A feeling is not a statistic. It is very easy to misjudge frequencies and probabilities due to various well-known cognitive heuristics and biases. Credence calibration is a skill, and most people are bad at it.

Are you writing these questions and answers down? Or are you relying on memory--the availability heuristic?

Are you tracking the cases when the answers don't come to you? When the answer was wrong? If not, that sounds like confirmation bias. You need both sides to compute a likelihood ratio to evaluate evidence.

If you keep seeking until you find the right answer, then obviously questions can only be "answered", or "not answered yet". That's just persistence, and I don't understand what you think God has to do with it. Do you track the time it takes to get answers?

Have you seriously considered alternative hypotheses? I would expect that simply being mentally open to answers makes you more likely to notice them, even in unexpected places.

Did you use experimental controls, like asking for God's guidance in some cases and not in others to see if it made any difference? How about asking pagan gods or deceased ancestors? How about asking your own subconscious mind to see if it's just openness?

Did you look at base rates for other people?

Did you ask for any kind of peer review of your methods? Are you keeping score with the kind of scientific rigor that could be peer-reviewed?

You claim in your profile to have a scientific background. The scientific methods that I am describing should not be news to you. You know this stuff already, don't you? Science has developed these processes to avoid certain well-established epistemological failure modes. But religion tends to get mentally compartmentalized as a "separate magisterium". I'm suggesting that you also try to avoid these failure modes when thinking about God.

FYI, while I still do not believe in God, I was... actually just pretty satisfied by Scott Alexander's Answer to Job?

[edit: epistemic status: haven't actually thought seriously about it tho. Also, someone just gave a plausibly convincing counter argument]

God is such a complex claim that I would need this part solved before even allowing that God exists within the space of all logically consistent hypotheses to be considered. I suppose that's less of a crux and more part of a prerequisite definition, come to think of it.

What if we use a specific example of evil, in lieu of hashing out a complex definition that we can all accept.

Let's use one provided by the church itself and still preached from thousands of pulpits today, which cannot reasonably be divorced from the question of God.

Let's use Hell.

Without getting too deeply into the weeds, Hell is at least one of:

• What happens to evil people when they die
• An evil (and, I might add, eternal) thing that happens to some people when they die
• A... let's go with "place" here for simplicity's sake, where evil resides

Either omnipotent God can stop this and yet allows it to continue, or omnibenevolent God is powerless to stop it.

I don't think we need a definition of evil at all. If we take omnipotence to be "able to do literally anything" and rephrase omnibenevolence to be "unable to abide evil" the meaning doesn't change but the paradox becomes evident. The two qualities are mutually exclusive regardless of your definition of "evil".