3 ms·

An Overview of Bayesian Inference (2019)

121 points by luu 7y ago

ur-whale 7y ago
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?
- shoo 7y ago
  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 Probability theory isn’t true (quantum physics) http://www.stat.columbia.edu/~gelman/research/unpublished/bayes_holes_2.pdf http://www.stat.columbia.edu/~gelman/research/unpublished/ba...
  - deepnet 7y ago
    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 https://arxiv.org/abs/1605.08177
- gjm11 7y ago
  There is a thing called "Quantum Bayesianism": https://en.wikipedia.org/wiki/Quantum_Bayesianism https://en.wikipedia.org/wiki/Quantum_Bayesianism
- deepnet 7y ago
  "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-by-its-founder-20150604/ https://www.quantamagazine.org/quantum-bayesianism-explained...
  - analog31 7y ago
    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.
  - kgwgk 7y ago
    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 https://www.perimeterinstitute.ca/personal/cfuchs/Oviedo.pdf
  - kgwgk 7y ago
    > 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".
- Strilanc 7y ago
  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.
- kgwgk 7y ago
  The Entropic Dynamics approach to Quantum Mechanics - Ariel Caticha - https://arxiv.org/abs/1908.04693 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 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=ZGVmYXVsdGRvbWFpbnxjYXBpcmVsYWZpc2ljYXF1YW50aXN0aWNhfGd4OjVkNWNjYTcxZTk1ODk2MGQ https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsd...

121 points by luu 7y ago

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?

shoo 7y ago
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/bayes_holes_2.pdf http://www.stat.columbia.edu/~gelman/research/unpublished/ba...
- deepnet 7y ago
  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 https://arxiv.org/abs/1605.08177
gjm11 7y ago
There is a thing called "Quantum Bayesianism": https://en.wikipedia.org/wiki/Quantum_Bayesianism https://en.wikipedia.org/wiki/Quantum_Bayesianism
deepnet 7y ago
"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-by-its-founder-20150604/ https://www.quantamagazine.org/quantum-bayesianism-explained...
- analog31 7y ago
  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.
- kgwgk 7y ago
  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 https://www.perimeterinstitute.ca/personal/cfuchs/Oviedo.pdf
- kgwgk 7y ago
  > 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".
Strilanc 7y ago
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.
kgwgk 7y ago
The Entropic Dynamics approach to Quantum Mechanics - Ariel Caticha - https://arxiv.org/abs/1908.04693 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 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=ZGVmYXVsdGRvbWFpbnxjYXBpcmVsYWZpc2ljYXF1YW50aXN0aWNhfGd4OjVkNWNjYTcxZTk1ODk2MGQ https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsd...