Epistemology aside… I’d like to hear more about this Red Dwarf Luck Virus. Is it FDA approved? If not can I still get it on Amazon? If not, is it available through a reliable merchant you can recommend on the Dark Web?

“It has strictly zero prior probability, because it is not in the hypothesis space. But that is just a matter choice.” — Now I don’t know what you are talking about. Only the logically impossible can even possibly have a zero prior. Everything else has a nonzero prior. Most, a vanishingly small one. We can ignore most hypotheses not because their priors are zero, but because their priors, even in sum, are near enough to zero.

The rest of your comment is unintelligible. It does not sound like you actually understand what Einstein showed with regard to classical mechanics that defined quantum mechanics. Or that you are even aware of how a fortiori reasoning works.

Unless you know any of those has a non negligible prior, it has a negligible prior.

It has strictly zero prior probability, because it is not in the hypothesis space. But that is just a matter choice.

we know this because we aren’t constantly surprised that all our hypotheses fail

The reason that our hypotheses are not regularly failing is that our experiences are produced by a mechanism that exhibits symmetry (e.g. as yesterday, so today). What we can only make intelligent guesses at, however, is the nature and origin of that symmetry. A perfect example is the birth of quantum theory. For your statements about logical undeniability of the of the hypothesis space to be valid, there would need to be a clearly defined threshold where the accumulated evidence suddenly makes the inclusion of the quantum-mechanical model in the hypothesis space appropriate. No such threshold can be deduced – any attempt would be analogous to the fabled 0.05 alpha level in null-hypothesis significance testing: entirely arbitrary.

Perhaps it would help you to try to think like a computer scientist. How would you actually build an algorithm to do probability analyses? Where would the content of your background knowledge, b come from? How would you convert raw experiences into numbers?

What I’m talking about is the infinity of propositions that are not in the hypothesis space.

Unless you know any of those has a non negligible prior, it has a negligible prior. Knowledge cannot be born from ignorance.

And we know this because we aren’t constantly surprised that all our hypotheses fail because of failing to assign a non negligible prior to something we didn’t think of.

That only happened when we used lousy methods. Since the Scientific Revolution, we discovered that with good methods we are so good at detecting hypotheses with non negligible priors, we rarely miss them. The only time we do miss them is when we know we are: i.e. when we don’t know what the correct explanation is for something.

But if in b is no knowledge that set Y of hypotheses even cumulatively has a prior high enough to not be washed out by your margins of error, you can logically disregard them.

If you can legitimately say that the integral over sub-space Y is negligible, then it means you have already evaluated these hypotheses. I.e. they are already part of your hypothesis space. What I’m talking about is the infinity of propositions that are not in the hypothesis space.

Everything else is already subsumed under the probability you are wrong, which will the probability you are wrong given b.

No hypothesis space can contain the proposition ‘the truth is something outside this hypothesis space.’ It is mathematically impossible. What we might do, as a crude method, is to define a hypothesis, H_x, ‘something other that all these other hypotheses,’ but then, strictly speaking, there is no way to assign a prior distribution, or to calculate all the likelihoods. The final cover-your-back proposition is too vague to permit mathematical analysis. We can stick in some approximate probability for H_x, using e.g. empirical frequencies with which the hypothesis space is found to be too narrow, but again, this would be an estimate, contingent upon some higher-level probability model.

It also comes from logic, set theory, present information, processed information, etc., most of which deductively provable.

As long as any part comes from non-deductive inference, then you cannot claim deductive status for the conclusion. Under my limited hypothesis space, H_1 might be genuinely very highly probable. But another might arbitrarily consider a different hypothesis space that happens to contain the true hypothesis (while mine does not). In this case, he may very well find my H_1 improbable. That H_1 is probable under my model would be a deductively true fact. But it is a fact about a conjunction of some proposition about the world AND my assumed probability model. I.e., a construction of my mind. The existence of other models giving contrary conclusions concerning H_1 is enough to demonstrate that the very highly probable status of H_1 is not a deductively provable fact about external reality.

I hope I’ve succeeded in making my point more clearly.

Yes, it’s a fair note that photons and space-time are not the simplest things imaginable; just far simpler than a machine generating a coherent intelligent consciousness.

As a physicist, though, I have to say that I cannot think of a sense in which either photons or spacetime are “ontologically simple.” Actually, the serious quantum field theory folks will tell you that when spacetime is curved, particle number becomes a tricky concept, and we retreat from the idea that photons are objects. Instead, we have to talk about photon events in our particle detectors, and expectation values for “operators” associated to different regions of spacetime. We slide, it seems, from the mental image of a photon as a tiny BB with some numbers attached, to the notion of having experiences with a photon-ish character.

Of course, nothing is clear-cut in the philosophy of physics, and a great many physicists get by without caring about any of this stuff, but when you find physicists themselves saying things like, “the notion of ‘particle’ is highly nontrivial and problematic in this setting and is to be understood in a metaphorical sense” — well, something odd is going on! For my own part, I’m willing to say that we’re wrapping around to the idea of perception events as undeniable raw data or experiences, but many physicists would doubtless disagree. David Mermin writes,

There is now no generally agreed-upon key to dissolving the puzzlement that quantum mechanics engenders today in many of us. […] I would hope that within the next 150 years, such a key might be found that almost everybody would agree clarifies the character of the theory, in contrast to today’s state of affairs, where no school of thought commands more than ten percent of the population, except for those who maintain—but can they really mean it?—that there is nothing to be puzzled about.

“Regarding your first paragraph in the above reply, you cannot say that the scope of the hypothesis space is epistemically irrelevant. To evaluate a hypothesis, you must compare its performance to that of some set of rivals. This is what Bayes’ theorem does, and the output of Bayes’ theorem will depend on the exact choice of that set.” — Of course. But if in b is no knowledge that set Y of hypotheses even cumulatively has a prior high enough to not be washed out by your margins of error, you can logically disregard them. That’s my point. Only when information enters b (by the iterative method it will do so by beginning as a novel e) that one of those hypotheses should be assigned a non-negligible prior, is there are logical warrant to consider it. Everything else is already subsumed under the probability you are wrong, which will the probability you are wrong given b.

“Concerning the part that determines the hypothesis space, it comes from your own imagination, and what seems to you reasonable. There is nothing deductively provable about it’s contents.” — It also comes from logic, set theory, present information, processed information, etc., most of which deductively provable. For example, that a certain experience-set is recollected as rare with regard to its converse is an undeniable fact of present experience. Which logically entails conclusions in probability (rare = improbable). Likewise the binary fact that either your recollection in that respect is accurate to within a chosen margin of error, or it is not; giving you a foundational h and ~h to compare with each other, and accumulate evidence to test, with respect to trust in reliability of memory to provide data.

“This process must involve some form of random search, tentatively trying out novel propositions, to see how they perform.” — Hardly any truth of the world has been discovered by a random search. There is a reason scientists and historians zoom in weirdly fast on valid hypotheses. It’s not because they are psychic or spirits are guiding them or they swallowed a Red Dwarf Luck Virus. It’s because they use information available to ascertain the most probable hypotheses right from the start and then begin testing them, and then revising with feedback. That this works is itself data. (Incidentally, data that makes “external world” hypotheses increasingly more probable than Cartesian Demons.) Of course, most of this was done automatically by the brain interacting with the world for its first five or so years (hence children begin with antirealism, then learn realism is more likely, e.g. by discovering object permanence).

Regarding your first paragraph in the above reply, you cannot say that the scope of the hypothesis space is epistemically irrelevant. To evaluate a hypothesis, you must compare its performance to that of some set of rivals. This is what Bayes’ theorem does, and the output of Bayes’ theorem will depend on the exact choice of that set.

You take b, the ‘background knowledge,’ to be something that appears fully formed from somewhere, and claim that if a hypothesis isn’t in b, then we don’t need to worry about it. But where does b come from? Concerning the part that determines the hypothesis space, it comes from your own imagination, and what seems to you reasonable. There is nothing deductively provable about it’s contents. The problem is not one of simply examining b to see what hypotheses are inside it (what would I use for that, a microscope?), but rather one of estimating what hypotheses we should consider.

You say that new hypotheses can enter b, once they are discovered. But how are such discoveries made? By applying methods that, again, at least approximate Bayes’ theorem. This process must involve some form of random search, tentatively trying out novel propositions, to see how they perform. The difference between some hypothesis being outside b at some point and then later inside b is simply that up to that point in time, you had never tried thought experiment. To say that the contents of b are an undeniable fact is to misunderstand the process.

Respectfully,
Tom

“We can not say it is an undeniable fact that H_1 belongs in the search space, while H_2 does not.” — This confuses real space with epistemic space. The hypotheses we don’t know whether they belong (or full on don’t know) are not in b (background knowledge, a condition in every term of a Bayesian equation), therefore they are irrelevant epistemically. They become relevant only when discovered, and then they are in b, causing an update. Remember, Bayes’ Theorem tells us what P(h) is only given what we have presented to us (in b). It is therefore a conditional probability, one that is necessarily true (within an a fortiori error margin) given the undeniables accessible to us at that time. The question then is: how often are we wrong about that. For which we have data…

Among those undeniables will be data relating to the condition we are in. For example, on previous occasions in our experience (hence present as recollections now; and, by iteration transformation, likewise data reported from others who have come before us, putatively centuries worth) statistically after a poor search of the possibility space we end up buying into too many hypotheses that fail, but after a diligent search of the possibility space, we far more often end up with successful hypotheses rather than overlooked ones (which we know from their success in practice, which is massively improbable as a coincidence: e.g. the convergence and predictive power principles in Sense and Goodness, Chapter II.3.1, pp. 51-53). So we know that if we perform a diligent search of the possibility space, we will not commonly overlook the correct hypothesis and buy into a false one. (This is true even for the Newton-Einstein case, both of whom discovered correct hypotheses, the one simply extending the other’s into new conditions; likewise Classical Mechanics is an emergent property of Quantum Mechanics.)

If it helps, you can do this in two stages: instead of working from a real-world hypothesis right away, just assume you are a brain in a vat, and what you are trying to figure out are the rules used by the external computer to feed you information (including recollected memories). You will end up with a set of rules that will be undeniably true to a high probability. Then, once you’ve sussed all of that you can, you can test then the hypothesis “a computer is doing this” against “this is actually all being caused by a physical world that acts that way” and find the latter is informationally much less complex and thus has a much higher prior. Indeed, if the computer is eternally infallible, then it wouldn’t even matter: the physical world then does exist…as the output of a computer (SaG, Chapter II.2.1.2, pp. 31-32). And if it’s not, there will be information eventually that gives it away (and here, the closer you imagine the computer to being eternally infallible, so as to increase the rarity of give-aways, the more complex and specified that computer has to be, and thus the more complex the hypothesis, and thus the lower its prior probability).

“The point at which we make that decision, and the exact route to that position are governed to some extent by the data, but also by intuition and personal taste.” — That’s not epistemic, that’s pragmatic. We apportion research and vetting time to risk, and risk varies to values (some people care more about some risks than others). So you are here confusing the true probabilities (the ones we could in principle deductively arrive at if we could be absolutely thorough with all undeniable data) with the ones we actually generate, which are a fortiori (probabilities far lower than actual, but just high enough to risk acting on, or too low to).

Bayesian epistemology is all about a fortiori reasoning. See Proving History, index, “a fortiori.” So, for example, although I could determine nearly the actual probability of my apartment building being destroyed by a meteorite tomorrow (the data is available), all I need to know is that it is unlikely enough for me to worry about, so I can run a quick and crude a fortiori calculation: one divided by the number of days I’m sure this spot has not been hit by meteorites (let’s say, 3,650,000 days, or 10,000 years; I could do better with a Laplacean formula, but the difference doesn’t matter, this is a fortiori). I then know for a fact that the probability of my apartment being destroyed by a meteorite tomorrow is less than (and I can show also simply that it is very, very much less than) 1 in 3,650,000, which is too unlikely to care about or take any precautions against. I could, if I wanted, walk that calculation all the way down to the undeniables of my direct experience, as the foundation for all the data I used to run that top calculation. Thus, I can be deductively certain that the probability of my apartment being destroyed by a meteorite tomorrow is less than 1 in 3,650,000. Without ever knowing (or even caring) what the actual probability is (death by meteorite in general is closer to 1 in 75 million). And this is how we live every part of our lives. All knowledge works this way.

This is also how we eliminate fringe hypotheses like H_1897634. We don’t need to know their prior. We only need to know that it is lower than some value that removes it from our math because it washes under the margins of error we are already working with. Meanwhile, our being wrong about that is already included in the converse probability to the one we assign the hypothesis we conditionally decide to believe is warranted: e.g. if we choose H1 and say it is 95% likely, then there is a 5% chance some other hypothesis is true, and that could well be H_1897634; since we are already saying so, no contradiction is generated, and since our P(H1) is conditional on b, our P(H1) is deductively certain (potentially actually; but typically a fortiori), until the contents of b change, e.g. we discover something we overlooked about H_1897634. All knowledge statements (beyond the undeniables) actually reduce to “the P of h is x given all we know right now.” Not “the P of h is x” or even strictly speaking “x is true.”