It would have been kind of impossible to work on AI in 1850, before even modern set theory was developed. Unless by work on AI, you mean work on mathematical logic in general.
Kurros
Of course acting on beliefs is a decision theory matter. You don’t have terribly much to lose by buying a losing lottery ticket, but you have a very large amount to gain if it wins, so yes 1⁄132 chance of winning sounds well worth $20 or so.
Hmm, do you know of any good material to learn more about this? I am actually extremely sympathetic to any attempt to rid model parameters of physical meaning; I mean in an abstract sense I am happy to have degrees of belief about them, but in a prior-elucidation sense I find it extremely difficult to argue about what it is sensible to believe a-priori about parameters, particularly given parameterisation dependence problems.
I am a particle physicist, and a particular problem I have is that parameters in particle physics are not constant; they vary with renormalisation scale (roughly, energy of the scattering process), so that if I want to argue about what it is a-priori reasonable to believe about (say) the mass of the Higgs boson, it matters a very great deal what energy scale I choose to define my prior for the parameters at. If I choose (naively) a flat prior over low-energy values for the Higgs mass, it implies I believe some really special and weird things about the high-scale Higgs mass parameter values (they have to be fine-tuned to the bejesus); while if I believe something more “flat” about the high scale parameters, it in turn implies something extremely informative about the low-scale values, namely that the Higgs mass should be really heavy (in the Standard Model—this is essentially the Hierarchy problem, translated into Bayesian words).
Anyway, if I can more directly reason about the physically observable things and detach from the abstract parameters, it might help clarify how one should think about this mess...
That sounds to me more like an argument for needing lower p-values, not higher ones. If there are many confounding factors, you need a higher threshold of evidence for claiming that you are seeing a real effect.
Physicists need low p-values for a different reason, namely that they do very large numbers of statistical tests. If you choose p=0.05 as your threshold then it means that you are going to be claiming a false detection at least one time in twenty (roughly speaking), so if physicists did this they would be claiming false detections every other day and their credibility would plummet like a rock.
To me, the simulation hypothesis definitely does not imply a supernatural creator. ‘Supernatural’ implies ‘unconstrained by natural laws’, at least to me, and I see no reason to expect that the simulation creators are free from such constraints. Sure, it means that supernatural-seeming events can in principle occur inside the simulation, and the creators need not be constrained by the laws of the simulation since they are outside of it, but I fully expect that some laws or other would govern their behaviour.
It defined “God” as supernatural didn’t it? In what sense is someone running a simulation supernatural? Unless you think for some reason that the real external world is not constrained by natural laws?
When you prove something in mathematics, at very least you implicitly assume you have made no mistakes anywhere, are not hallucinating, etc. Your “real” subjective degree of belief in some mathematical proposition, on the other hand, must take all these things into account.
For practical purposes the probability of hallucinations etc. may be very small and so you can usually ignore them. But the OP is right to demonstrate that in some cases this is a bad approximation to make.
Deductive logic is just the special limiting case of probability theory where you allow yourself the luxury of an idealised box of thought isolated from “real world” small probabilities.
edit: Perhaps I could say it a different way. It may be reasonable for certain conditional probabilities to be zero or one, so long as they are conditioned on enough assumptions, e.g. P(“51 is a prime” given “I did my math correctly, I am not hallucinating, the external world is real, etc...”)=1 might be achievable. But if you try to remove the conditional on all that other stuff you cannot keep this certainty.
Thanks, this seems interesting. It is pretty radical; he is very insistent on the idea that for all ‘quantities’ about which we want to reason there must some operational procedure we can follow in order to find out what it is. I don’t know what this means for the ontological status of physical principles, models, etc, but I can at least see the naive appeal… it makes it hard to understand why a model could ever have the power to predict new things we have never seen before though, like Higgs bosons...
Hmm, interesting. I will go and learn more deeply what de Finetti was getting at. It is a little confusing… in this simple case ok fine p can be defined in a straightforward way in terms of the predictive distribution, but in more complicated cases this quickly becomes extremely difficult or impossible. For one thing, a single model with a single set of parameters may describe outcomes of vastly different experiments. E.g. consider Newtonian gravity. Ok fine strictly the Newtonian gravity part of the model has to be coupled to various other models to describe specific details of the setup, but in all cases there is a parameter G for the universal gravitation constant. G impacts on the predictive distributions for all such experiments, so it is pretty hard to see how it could be defined in terms of them, at least in a concrete sense.
Are you referring to De Finetti’s theorem? I can’t say I understand your point. Does it relate to the edit I made shortly before your post? i.e. Given a stochastic model with some parameters, you then have degrees of belief about certain outcomes, some of which may seem almost the same thing as the parameters themselves? I still maintain that the two are quite different: parameters characterise probability distributions, and just in certain cases happen to coincide with conditional degrees of belief. In this ‘beliefs about beliefs’ context, though, it is the parameters we have degrees of belief about, we do not have degrees of belief about the conditional degrees of belief to which said parameters may happen to coincide.
Refering to this:
“Simply knowing the fact that the entropy is concave down tells us that to maximize entropy we should split it up as evenly as possible—each side has a 1⁄4 chance of showing.”
Ok, that’s fine for discrete events, but what about continuous ones? That is, how do I choose a prior for real-valued parameters that I want to know about? As far as I am aware, MAXENT doesn’t help me at all here, particularly as soon as I have several parameters, and no preferred parameterisation of the problem. I know Jaynes goes on about how continuous distributions make no sense unless you know the sequence whose limit you took to get there, in which case problem solved, but I have found this most unhelpful in solving real problems where I have no preference for any particular sequence, such as in models of fundamental physics.
The statistics also remains important at the frontier of high energy physics. Trying to do reasoning about what models are likely to replace the Standard Model is plagued by every issue in the philosophy of statistics that you can imagine. And the arguments about this affect where billions of dollars worth of research funding end up (build bigger colliders? more dark matter detectors? satellites?)
You don’t think people here have a term for their survey-completing comrades in their cost function? Since I probably won’t win either way this term dominated my own cost function, so I cooperated. An isolated defection can help only me, whereas an isolated cooperation helps everyone else and so gets a large numerical boost for that reason.
Lol, I cooperated because $60 was not a large enough sum of money for me to really care about trying to win it, and in the calibration I assumed most people would feel similarly. Reading your reasoning here, however, it is possible I should have accounted more strongly for people who like to win just for the sake of winning, a group that may be larger here than in the general population :p.
Edit: actually that’s not really what I mean. I mean people who want to make a rational choice to maximum the probability of winning for its own sake, even if they don’t actually care about the prize. I prefer someone gets $60 and is pleasantly surprised to have won, than I get $1. I predict that overall happiness is increased more this way, at negligible cost to myself. Even if the person who wins defected.
In this case, Feynman is worth listening to slowly. There is something about the way he explains this that the transcript does not do justice to.
Yeah I think integral( p*log(p) ) is it. The simplest problem is that if I have some parameter x to which I want to assign a prior (perhaps not over the whole real set, so it can be proper as you say—the boundaries can be part of the maxent condition set), then via the maxent method I will get a different prior depending on whether I happen to assign the distribution over x, or x^2, or log(x) etc. That is, the prior pdf obtained for one parameterisation is not related to the one obtained for a different parameterisation by the correct transformation rule for probability density functions; that is, they contain logically different information. This is upsetting if you have no reason to prefer one parameterisation or another.
In the simplest case where you have no constraints except the boundaries, and maybe expect to get a flat prior (I don’t remember if you do when there are boundaries… I think you do in 1D at least) then it is most obvious that a prior flat in x contains very different information to one flat in x^2 or log(x).
I can’t disagree with that :p. I will concede that the survey question needs some refinement.
Hmm, I couldn’t agree with that later definition. Physics is just the “map” after all, and we are always improving it. Mathematics (or some future “completed” mathematics) seems to me the space of things that are possible. I am not certain, but this might be along the lines of what Wittgenstein means when he says things like
“In logic nothing is accidental: if a thing can occur in an atomic fact the possibility of that atomic fact must already be prejudged in the thing.
If things can occur in atomic facts, this possibility must already lie in them.
(A logical entity cannot be merely possible. Logic treats of every possibility, and all possibilities are its facts.)” (from the Tractatus—possibly he undoes all this in his later work, which I have yet to read...)
This is a tricky nest of definitions to unravel of course. I prefer to not call anything supernatural unless it lies outside the “true” order of reality, not just if it isn’t on our map yet. I am a physicist though, and it is hard for me to see the logical possibility of anything outside the “true” order of the universe. Nevertheless, it seems to me like this is what people intend when they talk about God. But then they also try to prove that He must exist from logical arguments. These goals seem contradictory to me, but I guess that’s why I’m an athiest :p.
I don’t know where less “transcendant” supernatural entities fit into this scheme of course. Magic powers and vampires etc need not neccessarily defy logical description, they just don’t seem to exist.
I agree that in the end, banishing the word supernatural is probably the easiest way to go :p.
But don’t you think there is an important distinction between events that defy logical description of any kind, and those that merely require an outlandish multi-layered reality to explain? I admit I can’t think of anything that could occur in our world that cannot be explained by the simulation hypothesis, but assuming that some world DOES exist outside the layers of nested simulation I can (loosely speaking) imagine that some things really are logically impossible there. And that if the inhabitants of that world observe such impossible events, well, they will wrongly concluded that they are in a simulation, but actually there will be truly supernatural happenings afoot.
I mention this somewhat pointless story just because religious philosophers would generally not accept that God is merely supernatural in your sense, I think they would insist on something closer to my sense, nonsense though it may be.
“Jonah was looking at probability distributions over estimates of an unknown probability (such as the probability of a coin coming up heads)”
It sounds like you are just confusing epistemic probabilities with propensities, or frequencies. I.e, due to physics, the shape of the coin, and your style of flipping, a particular set of coin flips will have certain frequency properties that you can characterise by a bias parameter p, which you call “the probability of landing on heads”. This is just a parameter of a stochastic model, not a degree of belief.
However, you can have a degree of belief about what p is no problem. So you are talking about your degree of belief that a set of coin flips has certain frequentist properties, i.e. your degree of belief in a particular model for the coin flips.
edit: I could add that GIVEN a stochastic model you then have degrees of belief about whether a given coin flip will result in heads. But this is a conditional probability: see my other comment in reply to Vanvier. This is not, however, “beliefs about beliefs”. It is just standard Bayesian modelling.