Bayesian Epistemology (2022)
plato.stanford.edu
plato.stanford.edu
She tells a terrific story with a fascinating large cast of characters including Laplace,Bayes,Fisher,Pearson,Jeffries,Savage,Turing and many others. Engagingly told, highly recommended. Could the takeaway "Do you want to solve a practical problem or do you want scientific rigor?"
Another example is "The Code Book" by Simon Singh, which compiles various historical stories related to codes, cyphers, and cryptography.
I think the Bayesian linear regression is cool, but I'm not a big fan of the Bayes Factors tests, since they are very sensitive to the choice of priors.
If we just switch from p-values (with 0.05 cutoff) to Bayes Factors (with hard cutoffs), then we'll have many of the same problems stats currently has...
https://jonathanweisberg.org/pdf/VarietiesvF.pdf
Another point to notice is that "Bayesian epistemology" generally means "Bayesianism as discussed by philosophers". There is also "Bayesian statistics", i.e. as discussed by statisticians. Introductions to either of them show surprisingly little overlap between the two "Bayesianisms".
Here's a fun one https://sci-hub.3800808.com/10.1038/302687a0
Here's the truth table for implies (if) in logic.
| A | B | If A then B |
|---+---+-------------|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
This is where you need to add temporal logic so that the scientific understanding of A casually implies B can be represented in logic.
In short the paper does nothing of the sort of what it says it does because it fundamentally uses the wrong tool for the job.
B = (A or B) and (if A then B)
| A | B | x A B | y A B | z A B | If A then B |
|---+---+-------+-------+-------+-------------|
| T | T | T | T | T | T |
| T | F | F | F | F | F |
| F | T | F | F | T | T |
| F | F | F | T | F | T |
Or to put it another way: No one outside a logic course seriously thinks that A implies B means the same thing as not A or B.
Consider the rule: If a user has write access, then they must be an admin.
If you write a function to enforce this, you’ll naturally follow logical implication—checking if the user has write access and raising an error if they aren’t an admin, while doing nothing if they don’t have write access.
If for whatever reason B is stuck to T in all cases no one will think that A implies B, they will think that B is independent of A.
Yet that would still fit with a world in which A logically implies B.
You would need B to be dependent on a third variable which you're not controlling in the experiment that allows the value of B to change T or F outside your control.
Any good experimentalist will tell you to fix your broken experiment if that kept happening.
It's only when you have no control over the variables and you collect statistics that the view in the paper makes sense.
That's the difference between experimental and observational science.
And the philosophy of science is decades to centuries away from understanding that in any meaningful way.
First, I think Popper did not fundamentally disagree with the Bayesian approach: ultimately his critique of Bayesianism is a small adjustment to Bayesianism. When presented with a set of competing hypotheses { h_0, h_1, h_2... }, Bayesianism says that we assign a probability to each, whereas Popper points out that experiments and observation never really actually add credence to a hypothesis, they rather only decrease the probability of a hypothesis.
In theory, decreasing the probability of hypothesis h_0 does not increase the probability of hypothesis h_1, because the set of hypotheses is infinite--that is to say for any n there is always a potential h_{n+1} which has not been thought of by scientists. We know that the sum of the probabilities of hypotheses must be 1, but since the set of hypotheses is an infinite set, decreasing the probability of h_0 does not necessarily increase the probability of h_1, because the probability of any h_n in the set could be increasing.
I think Popper is right in theory, but in practice, I think this is less important than philosophers think it is. Pragmatically, we can treat the set of hypotheses as finite, operating only on the hypotheses humans have... hypothesized. We have no way to operate on hypotheses nobody has thought of, so we just operate on the set of hypotheses we have thought of. Since the sum of this set of probabilities must be 1, decreasing the probability of one hypothesis does increase the probability of all the other hypotheses in the set. Where Popper's critique becomes important is if we keep decreasing the probabilities of all the hypotheses in the set--this indicates that the hypothesis which is true is not in the set (i.e. nobody has come up with the correct hypothesis to test). This indicates a need for new hypotheses. But in a lot of cases, experimentation keeps decreasing the probability of all the hypotheses except one in the finite set of hypotheses humans have thought of, while numerous attempts to decrease the probability of that one hypothesis fail. While Popper would say this does not increase the probability of that one hypothesis, operating as if it does increase the probability of that hypothesis seems to work. We've been able to go to the moon, eradicate smallpox, and build intelligent-seeming machines, all based on hypotheses that we "increased the probability of" in this way, even though increasing the probability of a hypothesis is theoretically impossible.
This all goes back to philosophers' favorite navelgazing claim: that nothing is knowable. Ultimately, I think this is a dishonest argument which even philosophers don't believe. I've offered to punch many a philosopher in the face: after all, it's not knowable that it's going to hurt. But strangely, philosophers who claim to believe that nothing is knowable DO seem to know that will hurt, and none have taken me up on my offer.*
* Unlike the philosophers, I do believe in (probablistic) knowability, and I'm highly confident that punching them in the face would hurt them, so if anyone actually takes me up on this offer, I (probably) won't punch them. So far, no one has called my bluff.
> We have no way to operate on hypotheses nobody has thought of, so we just operate on the set of hypotheses we have thought of. Since the sum of this set of probabilities must be 1, decreasing the probability of one hypothesis does increase the probability of all the other hypotheses in the set.
We've been able to go to the moon because we understood general relativity, which absolutely could not have been created from a purely Bayesian approach for the reasons you made clear - it required new ideas.
> Where Popper's critique becomes important is if we keep decreasing the probabilities of all the hypotheses in the set--this indicates that the hypothesis which is true is not in the set (i.e. nobody has come up with the correct hypothesis to test). This indicates a need for new hypotheses.
I'll add that once relativity was hypothesized, it was added to the finite set of hypotheses that humans had hypothesized. Your "counterexample" is well within the scientific process I described.
The thing here is that whether or not the model of Bayesian epistemology is "useful", it's a fairly accurate description of how scientists approach the acquisition of knowledge, so when you attack that you're attacking science. And the thing is, outside of the brains of philosophers, science works--Bayesian epistemology does describe a process of knowledge acquisition that is effective. Planes still fly, smallpox is still eradicated, microchips still direct electricity to perform computation. Popper can say, "well ackchyually, we haven't actually increased confidence in the hypothesis of acquired immunity, we've only decreased confidence in the other hypotheses," but smallpox isn't going to say, "You know what, Popper, you're right," and reappear to decimate Europe. So de facto it sure seems like we do often know things with some degree of confidence when we eliminate other hypotheses, Popper notwithstanding.
And as I've said over and over: we need science more than ever. Polio just re-emerged in the US, and vaccine denialism is prevalent. While philosophers are sitting in their ivory towers saying "well ackchyually nothing is knowable" people are dying of diseases for which the prevention is not only knowable but known. So you'll excuse me not liking philosophers much here.
> By ‘Bayesian’ philosophy of science I mean the position that (1) the objective of science is, or should be, to increase our ‘credence’ for true theories, and that (2) the credences held by a rational thinker obey the probability calculus. However, if T is an explanatory theory (e.g. ‘the sun is powered by nuclear fusion’), then its negation ~T (‘the sun is not powered by nuclear fusion’) is not an explanation at all. Therefore, suppose (implausibly, for the sake of argument) that one could quantify ‘the property that science strives to maximise’. If T had an amount q of that, then ~T would have none at all, not 1-q as the probability calculus would require if q were a probability.
> Also, the conjunction (T₁ & T₂) of two mutually inconsistent explanatory theories T₁ and T₂ (such as quantum theory and relativity) is provably false, and therefore has zero probability. Yet it embodies some understanding of the world and is definitely better than nothing.
> Furthermore if we expect, with Popper, that all our best theories of fundamental physics are going to be superseded eventually, and we therefore believe their negations, it is still those false theories, not their true negations, that constitute all our deepest knowledge of physics.
> What science really seeks to ‘maximise’ (or rather, create) is explanatory power.
https://www.daviddeutsch.org.uk/2014/08/simple-refutation-of...
Deutsch is confused by this situation because he doesn't have the scientific background to understand the usefulness of negations of hypotheses.
Historically, for example, a lot of people believed the sun revolved around the earth. If we treat this as T, then ~T is "the sun does not revolve around the earth".
~T certainly lacks details, but to say it's a "bad explanation" is rather silly. Obviously it's an incomplete explanation, which is why Galileo presented a full explanation ("the earth revolves around the sun") rather than just saying, "the sun does not revolve around the earth". But in fact, "the sun does not revolve around the earth" was the part that was controversial because it was the bad explanation being presented by the church (who happened to be closer to philosophers than scientists).
Basically, Deutsch is just making a straw man argument. In Deutsch's mind, the fact that "the sun does not revolve around the earth" is an incomplete theory of heliocentrism is somehow a refutation of all science, when in fact that's simply not the sort of hypothesis scientists even explore typically.
Bayesian epistemology is an attempt to model why science works--it relies on science, not the other way around.
The reverse is not true--scientists have written a lot of philosophy--and since they tend to base their philosophy in reality rather than logic based on speculation, it tends to be better philosophy than philosophers.
Ctrl+F "astro" finds nothing on this page: https://en.wikipedia.org/wiki/David_Deutsch
I can assure you Deutsch is no confused by anything in that matter and it's obvious you don't know who he is.
He is literally the guy who created quantum computation and IS a scientist.
And no that's not his argument against the the sun is revolve around the earth.
"The sun is powered by nuclear fusion" is a clear explanation of the phenomena we observe. "The sun is not powered by nuclear fusion" would be an explanation if we observed phenomena that were inconsistent with the sun being powered by nuclear fusion.
> I can assure you Deutsch is no confused by anything in that matter and it's obvious you don't know who he is.
Well, I do now, and I assure you he is still confused.
> He is literally the guy who created quantum computation and IS a scientist.
So, he's a programmer.
What hypotheses is he known for testing? What makes him a scientist in your mind?
> And no that's not his argument against the the sun is revolve around the earth.
I'm not sure who you think said it was; I certainly didn't.
The reason I'm being very blunt about this, is because bullshit like this is actively harmful. Science is fucking important. Science is what resulted in the technology you're using to read this. Science is, with non-negligible probability, the basis of medicine that prevented you from dying before the age of 5 to be able to read this. When philosophers posit that they can inspect the their own navels and find deep truths about the world, they are undermining one of the fundamental pillars of society that holds up so much of the positive changes humans have been able to make.
We need to call this what it is--nonsense and misinformation--and stop amplifying its signal.
> By ‘Bayesian’ philosophy of science I mean the position that (1) the objective of science is, or should be, to increase our ‘credence’ for true theories, and that (2) the credences held by a rational thinker obey the probability calculus. However, if T is an explanatory theory (e.g. ‘the sun is powered by nuclear fusion’), then its negation ~T (‘the sun is not powered by nuclear fusion’) is not an explanation at all. Therefore, suppose (implausibly, for the sake of argument) that one could quantify ‘the property that science strives to maximise’. If T had an amount q of that, then ~T would have none at all, not 1-q as the probability calculus would require if q were a probability.
Of course "the sun is not powered by nuclear fusion" IS an explanation, it's just not an explanation of a phenomenon we observe, which is why most scientists don't believe "the sun is not powered by nuclear fusion". If we observed something about the sun that was not consistent with the hypothesis that it is powered by nuclear fusion, "the sun is not powered by nuclear fusion" would indeed be an explanation of what we were observing.
This is all sidestepping the absurdity that Deutsch doesn't seem to understand that "none at all" has a mathematical representation, 0, meaning that if p = 1 - q = 0, then q = 1. This is not difficult math here, folks.
> Also, the conjunction (T₁ & T₂) of two mutually inconsistent explanatory theories T₁ and T₂ (such as quantum theory and relativity) is provably false, and therefore has zero probability. Yet it embodies some understanding of the world and is definitely better than nothing.
Uh sure, which is why nobody with a brain takes the conjunction of those two things. This isn't a criticism of Bayesian philosophy of science, it's a straw man argument.
> Furthermore if we expect, with Popper, that all our best theories of fundamental physics are going to be superseded eventually, and we therefore believe their negations, it is still those false theories, not their true negations, that constitute all our deepest knowledge of physics.
Deutsch apparently doesn't know what an approximation is, and instead thinks of correct/incorrect as a binary. Relevant https://hermiene.net/essays-trans/relativity_of_wrong.html
A lot can go wrong with a person's brain between the publication of Deutsch-Josza (1985, 1992) and the writing of the linked post (2014).
I'd also note that the algorithm described is more a work of math than an example of experimental science. It's not creating a hypothesis and testing it, it's writing an algorithm for a (then-theorized) computer system. So I wouldn't say that this lends credence to his ideas on the validity of hypotheses.
As I say, I also disagree with his philosophical (and political) viewpoints. But your dismissal above is based on a pretty shallow reading alongside unfounded personal attacks.
Here is a decent paper by a statistician about this issue: https://arxiv.org/abs/1006.3868
What this is at its essence is that science has allowed us to evolve, learn to kill lions and bears, create agriculture, build ships, cure diseases, travel to the moon, build AI, etc. And all this time while science has been empowering humans and saving lives, science has been under attack by philosophy. You have a scientist saying, "I observe that solar and lunar patterns are more consistent with the earth revolving around the sun" and a philosopher saying "ackchyually the bible says the sun revolves around the earth". When evidence (collected through scientific methods) for a hypothesis becomes overwhelming, the last refuge of ignorance is the philosopher saying, "ackchyually, you don't know that because nothing is truly knowable".
Epistemology is an attempt to understand how we know things, and Bayesian epistemology is probably the best description of how we know things based on science. It's a description, based on observation of how scientists practice science, of how science works.
So when philosophers come in and say Bayesian epistemology doesn't work, they're saying science doesn't work. It's yet another attack on science by philosophers.
And as I said in my other post, Popper's criticism of Bayesian epistemology is actually smart: he does understand what he's talking about, it just doesn't, ultimately, matter much, because the practice of science de facto works, in practice, even if the philosophical model says it doesn't. If all the nuance of Bayesian epistemology and Popper's ideas isn't captured, it's easy for it just to become a straw man argument for philosophers to say that science doesn't work. When it comes down to it, the way people talk about Popper and Bayesian epistemology is just a more sophisticated version of "ackchyually, you don't know that because nothing is truly knowable".
I'm not defending Bayesian epistemology, per se. I'm defending science, as it's practiced, because as I said, science is fucking important. Now, more than ever, in the era of anti-vaxxers and climate change denial, we desperately need people to believe in science.
Is this supposed to be a meaningful sequence of words?
> it has nothing to do with animals learning how to survive in their environment.
As an animal who would have died in childbirth and taken my mom with me were it not for science, I disagree.
This is wrong, and it's a bit ironic you are so adamant on a point that is bad philosophy and leads to bad science as a way of insisting that philosophy has no relevance for science.
To underscore the bad science you are led to in terms of assumed truth, let alone hypothesis: there is very little evidence or justification or explanation that any of the processes used by the invertebrate here execute calculation that obeys the very specific axioms of probability and updates to a state in accordance with Bayes' theorem. Stimulus response is not Bayes' theorem. Updating a state from new inputs is not Bayes' theorem.
A lot has changed since invertebrates started doing that. Not only have we evolved more senses than the first invertebrates, we've also developed methods such as Bayesian inference to combine the results of multiple observations, as well as numerous methods for removing confounding variables such as control groups and regression analysis. Unsurprisingly this has led us to discover a lot more with science, with a lot more accuracy, than invertebrates.
And yes, updating a state from new inputs is not literally Bayes theorem, which is why nobody said it was. However, the process of updating a belief confidence from new inputs as it is done today can be modeled today using Bayesian inference. No, invertebrates don't do that--which is again, why I never said they did.
It's a bit tiresome to be corrected by people who clearly don't seem to understand that Bayes theorem, Bayesian inference, and Bayesian epistemology are all named after the same guy because they're all built on each other in that order. Yes, they aren't all the same thing, but if you're jumping in with that as if it's a correction, you certainly don't understand the concepts.
Which instantly refutes your position, because the alleged "bullshit" he quoted was written by a scientist, David Deutsch, not a by philosopher. Meanwhile, the defenders of "Bayesian philosophy of science" and (largely synonymous) "Bayesian epistemology" are mainly philosophers.
Am I correct in assuming you will now account for the above mistake and change your opinion to "Bullshit like this is exactly why I think philosophers are better philosophers than scientists are"?
If you insist on calling Deutsch a scientist, fine, go ahead. That doesn't change my argument in any way, but you'll have to actually read the post you're responding to to know that.
> Am I correct in assuming you will now account for the above mistake and change your opinion to "Bullshit like this is exactly why I think philosophers are better philosophers than scientists are"?
Probably the better thing to do would be to simply remove that paragraph since people like you don't read past the first thing you can find to disagree with. The rest of the post stands on its own fine, but I can't edit it now.
Science must work despite it, or you don't have science.
And any other singularity you get from assuming the odds of a hypothesis is infinitely smaller than the odds of it being false is unrealistic. You shouldn't assume that.
This inductive case against scientific knowledge should only serve to decrease our second-order credence in the proposition that we have assigned the highest credence to the scientific hypotheses that most closely correspond with reality. It does nothing to change the fact that, conditional on evidence we currently have, we may very well have correctly proportioned credence.