An Overview of Bayesian Inference (2019) | Hacker News Reader

An Overview of Bayesian Inference (2019)

jaydaigle.net

121 pointsluu10 comments

Slightly OT, but I wonder if anyone in the know could shed a light on where Baye's theorem is applied (if anywhere) in quantum mechanics?

In particular, has anyone tries to apply the bayesian framework to the whole "the wave collapses upon measurement" shebang?

You may find this interesting:

Gelman, Yao - Holes in Bayesian Statistics

> Every philosophy has holes, and it is the responsibility of proponents of a philosophy to point out these problems. Here are a few holes in Bayesian data analysis: (1) the usual rules of conditional probability fail in the quantum realm, (2) flat or weak priors lead to terrible inferences about things we care about, (3) subjective priors are incoherent, (4) Bayes factors fail in the presence of flat or weak priors, (5) for Cantorian reasons we need to check our models, but this destroys the coherence of Bayesian inference.

In particular, you may appreciate the example and discussion in section 2:

> Probability theory isn’t true (quantum physics)

http://www.stat.columbia.edu/~gelman/research/unpublished/ba...

contrarily "...we obtain that quantum mechanics is the Bayesian theory in the complex numbers."

from Quantum mechanics: The Bayesian theory generalised to the space of Hermitian matrices by Alessio Benavoli, Alessandro Facchini, Marco Zaffalon

https://arxiv.org/abs/1605.08177

There is a thing called "Quantum Bayesianism": https://en.wikipedia.org/wiki/Quantum_Bayesianism

"Quantum Bayesianism, or QBism as Fuchs now calls it, solves many of quantum theory’s deepest mysteries. Take, for instance, the infamous “collapse of the wave function,” wherein the quantum system inexplicably transitions from multiple simultaneous states to a single actuality. According to QBism, the wave function’s “collapse” is simply the observer updating his or her beliefs after making a measurement."

https://www.quantamagazine.org/quantum-bayesianism-explained...

I'm not really sure this helps. There is still the puzzle about how the updating of beliefs at some indefinite time in the future causes the collapse of a wavefunction involving photons that have fled the scene at the speed of light, for instance. It usually takes some ridiculously hard experimentation to support a new theory about fundamental quantum phenomena.

It's not that simple. When a measurement is done on a pure state (where the knowledge of the state is already as good as it gets) the observers are not just updating their beliefs, they are changing the system. On the other extreme, in some cases one may have measurements that improve our (imperfect) knowledge of the physical state without disturbing it.

"Quantum measurement is nothing more, and nothing less, than a refinement and a readjustment of one’s initial state of belief. [...] Let us look at two limiting cases of efficient measurements. In the first, we imagine an observer whose initial belief structure ρ = |ψ⟩⟨ψ| is a maximally sharp state of belief. By this account, no measurement whatsoever can refine it. [...] The only state change that can come about from a measurement must be purely of the mental-readjustment sort: We learn nothing new; we just change what we can predict as a consequence of the side effects of our experimental intervention. That is to say, there is a sense in which the measurement is solely disturbance."

https://www.perimeterinstitute.ca/personal/cfuchs/Oviedo.pdf

> the infamous “collapse of the wave function,” wherein the quantum system inexplicably transitions from multiple simultaneous states to a single actuality.

By the way, the "collapse of the wave function" describes as well the transition from one state to another state. I don't understand what do you mean by "multiple simultaneous states".

Bayesian inference applies to quantum mechanics in the "obvious" way. You have some hypothesis space (of possible wavefunctions). You plan to perform an observation (a measurement). Different hypotheses assign different probabilities to the various possible outcomes. You perform the observation, get a result, and perform a standard Bayesian update based on the outcome.

There are two caveats somewhat unique to quantum mechanics.

First, measurement has a known kickback effect on the state. So you always have to be careful to distinguish inferences of what the value of the state was before measurement vs inferences of the state after measurement. In general, after the observation has been made and the back-action accounted for, you know more about the post-measurement state than the pre-measurement state.

Second, many apparently different probability distributions of wavefunctions are observationally indistinguishable. For example, a qubit with a 50% chance of being definitely ON and a 50% chance of being definitely OFF is observationally indistinguishable from a qubit known to be in a state of the form 1/sqrt(2)|ON> + e^(i theta)/sqrt(2) |OFF> where theta is a uniform random variable ranging from 0 to 2pi. Those two hypotheses give identical predictions for all possible measurements. So people usually work with equivalence classes of probability distributions, represented by mathematical objects called density matrices, instead of individual probability distributions. Density matrices massively simplify everything, to the point where you might not even recognize the Bayes bits happening underneath.

The Entropic Dynamics approach to Quantum Mechanics - Ariel Caticha - https://arxiv.org/abs/1908.04693

He wrote a monograph on the approach: "Entropic Inference and the Foundations of Physics" which should be available at https://www.albany.edu/physics/ACaticha-EIFP-book.pdf but that link doesn't work now. You can find it here: https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsd...

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