However, I think it is important to make a distinction between the validity of Bayesianism and the application of Bayesianism. I recall reading on lesswrong (although I cannot find the post at this moment) that the relation between Bayesianism and frequentism should be seen like the relation between Quantum Mechanics and classical physics
Quantum Mechanics isn’t consistent with General Relativity, our best explanation of gravity. Despite decades of trying, neither can be formulated as an approximation of the other. Even if one day physicists finally figure out a “Theory of Everything”, it would still be a model. It would be epistemically incorrect to claim it was “exact”.
Curiously, there is one interpretation of QM known as Quantum Bayesianism, which holds that wavefunctions are subjective and they are the fundamental concepts for reasoning about the world, and subjective probability distributions arise as approximations of wavefunctions under decoherence. That is, Bayesianism itself might be an approximation of a “truer” epistemic theory!
My humble opinion is that there is no ultimately “true” epistemic theory. They are all just models of what humans do to gain knowledge of the world. Some models can work better than others, often within certain bounds, but none of them is perfect.
I am very interested in Quantum Bayesianism (in particular Leifer’s work) because one of the things we have to do to be “quantum Bayesians” is figure out a physically neutral description of quantum mechanics, that is, a description of quantum mechanics that doesn’t use physical jargon like ‘time.’ In particular, physicists I believe describe spacelike and timelike separated entanglement differently.
That is, a Bell inequality violation system (that is where B and C are space separated) has this graph
A → B <-> C ← D
(where famously, due to Bell inequality violation, there is no hidden variable corresponding to the bidirected arc connecting B and C).
But the same system can arise in a temporally sequential model which looks like this:
A → B → D → C, with B <-> C
where an appropriate manipulation of the density matrix corresponding to this system ought to give us the Bell system above. In classical probability we can do this. In other words, in classical probability the notion of “probabilistic dependence” is abstracted away from notions like time and space.
Also we have to figure out what “conditioning” even means. Can’t be Bayesian if we don’t condition, now can we!
where an appropriate manipulation of the density matrix corresponding to this system ought to give us the Bell system above. In classical probability we can do this. In other words, in classical probability the notion of “probabilistic dependence” is abstracted away from notions like time and space.
Yes, but the notion of Bayesian inference, where you start with a prior and build a sequence of posteriors, updating as evidence accumulates, has an intrinsic notion of time. I wonder if that’s enough for Quantum Bayesianism (I haven’t read the original works, so I don’t really know much about it).
The temporal order for sequential computation of posteriors is just our interpretation, it is not a part of the formalism. If we get pieces of evidence e1, e2, …, ek in temporal order, we could do Bayesian updating in the temporal order, or the reverse of the temporal order, and the formalism still works (that is our overall posterior will be the same, because all the updates commute). And that’s because Bayes theorem says nothing about time anywhere.
My humble opinion is that there is no ultimately “true” epistemic theory. They are all just models of what humans do to gain knowledge of the world. Some models can work better than others, often within certain bounds, but none of them is perfect.
Quantum Mechanics isn’t consistent with General Relativity, our best explanation of gravity. Despite decades of trying, neither can be formulated as an approximation of the other. Even if one day physicists finally figure out a “Theory of Everything”, it would still be a model. It would be epistemically incorrect to claim it was “exact”.
Curiously, there is one interpretation of QM known as Quantum Bayesianism, which holds that wavefunctions are subjective and they are the fundamental concepts for reasoning about the world, and subjective probability distributions arise as approximations of wavefunctions under decoherence. That is, Bayesianism itself might be an approximation of a “truer” epistemic theory!
My humble opinion is that there is no ultimately “true” epistemic theory. They are all just models of what humans do to gain knowledge of the world. Some models can work better than others, often within certain bounds, but none of them is perfect.
I am very interested in Quantum Bayesianism (in particular Leifer’s work) because one of the things we have to do to be “quantum Bayesians” is figure out a physically neutral description of quantum mechanics, that is, a description of quantum mechanics that doesn’t use physical jargon like ‘time.’ In particular, physicists I believe describe spacelike and timelike separated entanglement differently.
That is, a Bell inequality violation system (that is where B and C are space separated) has this graph
A → B <-> C ← D
(where famously, due to Bell inequality violation, there is no hidden variable corresponding to the bidirected arc connecting B and C).
But the same system can arise in a temporally sequential model which looks like this:
A → B → D → C, with B <-> C
where an appropriate manipulation of the density matrix corresponding to this system ought to give us the Bell system above. In classical probability we can do this. In other words, in classical probability the notion of “probabilistic dependence” is abstracted away from notions like time and space.
Also we have to figure out what “conditioning” even means. Can’t be Bayesian if we don’t condition, now can we!
Yes, but the notion of Bayesian inference, where you start with a prior and build a sequence of posteriors, updating as evidence accumulates, has an intrinsic notion of time. I wonder if that’s enough for Quantum Bayesianism (I haven’t read the original works, so I don’t really know much about it).
The temporal order for sequential computation of posteriors is just our interpretation, it is not a part of the formalism. If we get pieces of evidence e1, e2, …, ek in temporal order, we could do Bayesian updating in the temporal order, or the reverse of the temporal order, and the formalism still works (that is our overall posterior will be the same, because all the updates commute). And that’s because Bayes theorem says nothing about time anywhere.
Exactly!