Bayesianism versus Critical Rationalism

If your first probability estimate is wrong, without any error bar -- but simply wrong in an unknown way -- then you're screwed, right?

Think of it this way - if it's wrong in an utterly unknown way, then the wrongness has perfect symmetry; there's nothing to distinguish being wrong one way from being wrong in another. By the axiom that you shouldn't make up information, when the information is symmetric, that part of the distribution ("part" as in you convolve the different parts together to get the total distribution) should be symmetric too. And since the final probability estimate is just the average over your distribution, the symmetry makes the problem easy - or if the problem is poorly defined or poorly understood, it at the very least gives you error bars - it makes the answer somewhere between your current estimate and the maximum entropy estimate.

No, because the universe has a state/law, not a probability distribution over states. A theory/universe/statement is either true or false, a probability distribution over theories is not, though it can be scored for accuracy in various ways. A probability distribution over theories is not a statement about the actual state of the universe.

Similarly, the universal prior is in no way "true"; it's a distribution, not a statement at all. You shouldn't even expect it to be "true" since it's meant to be updated. What is important about it is that it has various nice properties such as eventually learning any computable distribution.

More often a matter of birth. Agents usually start somewhere.

A few details about the process.

Hang on, Popper's philosophy doesn't depend on any foundations? I'm going to call shenanigans on this. Earlier you gave and example of Popperian inference:

Consider a theory, T, that all swans are white. T is a universal theory.

No confirming evidence can prove T is true. You can see 5 white swans or 500 or 50 million. Still might be false.

But if you see one black swan it is false.

This is an asymmetry between confirmation and falsification when applied to universal theories. It does not hold for all theories.

Consider the negation ~T. At least one swan is not white. This theory cannot be refuted by any amount of observations. But it can be confirmed with only one observation. ~T is a non-universal theory and not the kind science is after.

Unquestioned assumptions include, but are not limited to the following:

• The objects under discussion actually exist (Solomonoff Induction does not make this assumption)

• "There is no evidence which could prove T" is stated without any proof, what if you got all the swans in one place, what if you found a reason why the existence of a black swan was impossibile?

• Any observation of a black swan must be correct (Bayes Theorem is explicitly designed to avoid this assumption)

• You can generalise from this one example to a point about all theories

• "Science is only interested in universal theories". Really? Are palaeontology and astronomy not sciences? They are both often concerned with specifics.

You must always begin with assumptions, if nothing else you must assume maths (which is pretty much the only thing that Bayes Theorem and Solomonoff Induction do assume).

I think that things like unquestionable foundations, or an infinite regress, are flaws. I think we should reject flawed things when we have better options. And I think Bayesian Epistemology has these flaws. Am I going wrong somewhere?

To be perfectly honest I care more about getting results in the real world than having some mythical perfect philosophy which can be justified to a rock.

What I'm not convinced of is the claim that this basic bit of math solves any major epistemological problem.

Stating that you believe Bayes' theorem but doubt that it can actually solve epistemic problems is like saying you believe Pythagoras' theorem but doubt it can actually tell you the side lengths of right angled triangles, it demonstrates a failure to internalise.

Bayes' theorem tells you how to adjust beliefs based on evidence, every time you adjust your beliefs you must use it, otherwise your map will not reflect the territory.

Regarding Solomonoff induction, I think you are now attempting to justify it by argument. But you haven't stated what are the rules for what counts as a good argument and why. Could you specify that? There's not enough rigor here. And in my understanding Bayesian epistemology aims for rigor and that is one of the reasons they like math and try to use math in their epistemology. It seems to me you are departing from that worldview and its methods.

Does Popper not argue for his own philosophy, or does he just state it and hope people will believe him?

You cannot set up rules for arguments which are not themselves backed up by argument. Any argument will be convincing to some possible minds and not to others, and I'm okay with that, because I only have one mind.

Epistemologies should be able to deal with all categories of knowledge, not just predictive knowledge. For example they should be able to deal with creating non-emprical, non-predictive moral knowledge. Can Solomonoff induction do that? How?

Allow me to direct you to my all time favourite philosopher

Popperian epistemology doesn't assign probabilities like that, and has a way to act.

My first wikipedia quote (Every admissible statistical procedure is either a Bayesian procedure or a limit of Bayesian procedures.) was somewhat technical, but it basically meant that any consistent set of actions is either describable in terms of probabilities or nonconsequentialist. How would you choose the best action in a Popperian framework? Would you be forced to consider aspects of a choice other than its consequences? Otherwise, your choices must be describable using terms of a prior probability and Bayesian updating (and, while we already agree that the latter is obvious, here we are using it to update a set of probabilities and, on the pain of inconsistency, our new probabilities must have that relationship to our old ones).

Explanations should be given for why each possibility is a good idea; the hypothetical you give doesn't have enough details to actually reach an answer.

Definitely use all the evidence when making decisions. I didn't mean for my example to be complete. I was wondering how a question like that could be addressed in general. What pieces of information would be important and how would they be taken into account? You can assume that the less relevant variables, like which disease is more painful, are equal in both cases.

Prediction is a goal of science, but it is not the primary one. The primary goal is explanation/understanding.

I may have been unclear here. I meant prediction in a very broad sense, including, eg., predicting which experiments will be best at refining our knowledge and predicting which technologies will best improve the world. Was it clear that I meant that? If you seek understanding beyond this, you are allowed but, at least for the present era, I only care about an epistemology if it can help me make world a better place.

About Solomonoff, if I understand you correctly now you are starting with theories which don't say much (that isn't what I expected simpler or shorter to mean). So at any point Solomonoff induction will basically be saying the minimal theory to account for the data and specify nothing else at all. Is that right? If that is the case, then it doesn't deal with choosing between the various possibilities which are all compatible with the data (except in so far as it tells you to choose the least ambitious) and can make no predictions: it simply leaves everything we don't already know unspecified.

No, not at all. The more likely theories are those that include small amounts of theory, not small amounts of prediction. Eliezer discusses this in the sequences here, here, andhere. Those don't really cover Solomonoff induction directly, but they will probably give you a better idea of what I'm trying to say than I did. I think Solomonoff induction is better covered in another post, but I can't find it right now.

I thought the theories were supposed to specify everything (not, as you say, "the rest can be anything") so that predictions could be made.

Sorry, I was abusing one word 'theories' to mean both 'individual descriptions of the universe' and 'sets of descriptions that make identical predictions in some realm (possibly in all realms)'. It is a very natural place to slip definitions, because, for example, when discussing biology, we often don't care about the distinction between 'Classical physics is true and birds are descended from dinosaurs.' and 'Quantum physics is true and birds are descended from dinosaurs.' Once enough information is specified to make predictions, a theory (in the second sense) is on equal ground with another theory that contains the same amount of information and that makes different predictions only in realms where it has not been tested, as well as with a set of theories for which the set can be specified with the same amount of information but for which specifying one theory out of the set would take more information.

That fragment is misleading because Popper means "preferred" in a methodological sense, not considered to have a higher probability of being true, or considered more justified. It's not a preference about which theory is actually, in fact, better.

I'm not sure how one would act based on this. Should one conduct new experiments differently given this knowledge of which theories are preferred? Should one write papers about how awesome the theory is?

No, he means they are more important and more interesting. His point is basically that a theory which says nothing has a 100% prior probability. Quantum Mechanics has a very low prior probability. The theories worth investigating, and which turn out important in science, all had low prior probabilities (prior probability meaning something like: of all logically possible worlds, for what proportion is it true?) They have what Popper called high "content" because they exclude many possibilities. That is a good trait. But it's certainly not a guarantee that arbitrary theories excluding stuff will be correct.

All of this is present is Bayesian epistemology.

Consider Bayes theorem, with theories A and B and evidence E:

P(A|E) = P(E|A) P(A) / P(E)

Let's look at how the probability of a theory increases upon learning E, using a ratio.

P(A|E) / P(A) = P(E|A) / P(E)

P(B|E) / P(B) = P(E|B) / P(E)

Which one increases by a larger ratio?

[P(A|E) / P(A)] / [P(B|E) / P(B)] = [P(E|A) / P(E)] / [P(E|B) / P(E)] = P(E|A) / P(E|B)

The greater P(E|A) is compared to P(E|B), the more A benefits compared to B. This means that the more theory A narrowly predicts E, the actual observation, to the exclusion of other possible observations, the more probability we assign to it. This is a quantitative form of Popper's preference for more specific and more easily falsifiable theories, as proven by Bayes theorem.

The theories worth investigating, and which turn out important in science, all had low prior probabilities (prior probability meaning something like: of all logically possible worlds, for what proportion is it true?)

That's basically what Solomonoff means by prior probability.