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Schrödinger Bridges via the Hacking of Bayesian Priors in Classical and Quantum Regimes

Clive Cenxin Aw, Peter Sidajaya

Abstract

Bayes' rule is widely regarded as the canonical prescription for belief updating. We show, however, that one can arbitrarily preserve pre-specified beliefs while appearing to perform Bayesian updates via "prior hacking": engineering a reference prior distribution such that, for a fixed channel and evidence, the update matches a chosen target distribution. We prove that this is generically possible in both classical and quantum settings whenever Bayesian inversions are well-defined (with the Petz recovery map as the quantum analogue to Bayes' rule), and provide constructive algorithms for doing so. We further establish a duality between prior hacking and Schrödinger bridge problems (a key object in statistical physics with applications in generative modelling), yielding in the quantum setting a unique, inference-consistent selection among candidate bridges. This formally establishes the Bayes-like updating that Schrödinger bridges are performing with respect to the process as opposed to the reference prior, both in classical and quantum settings.

Schrödinger Bridges via the Hacking of Bayesian Priors in Classical and Quantum Regimes

Abstract

Bayes' rule is widely regarded as the canonical prescription for belief updating. We show, however, that one can arbitrarily preserve pre-specified beliefs while appearing to perform Bayesian updates via "prior hacking": engineering a reference prior distribution such that, for a fixed channel and evidence, the update matches a chosen target distribution. We prove that this is generically possible in both classical and quantum settings whenever Bayesian inversions are well-defined (with the Petz recovery map as the quantum analogue to Bayes' rule), and provide constructive algorithms for doing so. We further establish a duality between prior hacking and Schrödinger bridge problems (a key object in statistical physics with applications in generative modelling), yielding in the quantum setting a unique, inference-consistent selection among candidate bridges. This formally establishes the Bayes-like updating that Schrödinger bridges are performing with respect to the process as opposed to the reference prior, both in classical and quantum settings.
Paper Structure (26 sections, 9 theorems, 89 equations, 8 figures, 3 algorithms)

This paper contains 26 sections, 9 theorems, 89 equations, 8 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Classical Setting
  3. The Classical Formalism
  4. Classical Prior Hacking
  5. Prior Hacking for Notable Classes of Classical Maps
  6. Classical Schrödinger Bridges & Prior Hacking
  7. Quantum Setting
  8. Quantum Prior Hacking
  9. Notable Qubit Channel Cases
  10. Quantum Schrödinger Bridges & Prior Hacking
  11. Concluding Discussion
  12. Hadamard product
  13. The Iterative Proportional Fitting (IPF)/Sinkhorn problem
  14. Proofs
  15. Proof for Theorem \ref{['thm:classical-prior-hacking-channel-forall']}
  16. ...and 11 more sections

Key Result

Theorem 1

(the Sinkhorn problem idel2016reviewmenon1968matrixbrualdi1968convexmenon1969spectrum) Given a tuple $(\mathcal{E},p,q)$, where $\mathcal{E}$ is a $d \times d'$ transition matrix and probability distributions $p \in \Delta^{d-1}$ and $q \in \Delta^{d'-1}$, the following are equivalent:

Figures (8)

  • Figure 1: A cartoon illustration of prior hacking, which is detailed in Section \ref{['sec:classical-prior-hacking-introduce']}.
  • Figure 2: Image plots for a symmetric bistochastic channel acting on a trit space. For details on plot features, see Section \ref{['sec:c-example-text']}. Here, prior-hacking depends on $q$, in contrast to Figure \ref{['fig:cerase']}.
  • Figure 3: Image plots for a generic erasure channel acting on a trit space. For details, see Section \ref{['sec:c-example-text']}. Here, prior-hacking is shown to be independent of $q$, in contrast to Figure \ref{['fig:cdepolar']}.
  • Figure 4: Image plots for a $(0,1)$-absorber channel acting on a trit space. For details, see Section \ref{['sec:c-example-text']}. Notice how one can only perform a Bayesian update toward states with weights on the $2$-state that are higher than that of $q$. Thus, prior-hacking is not always possible for any given $p,q$, as per Theorem \ref{['thm:classical-prior-hacking-channel-forall']}.
  • Figure 5: The relationship between prior hacking and Schrödinger bridge (Theorem \ref{['thm:CSB-is-CPH']}). The map obtained through prior hacking, $\hat{\mathcal{E}}_{\gamma}$, is the same map as the reverse map of the Schrödinger bridge with reference prior $p$, $\hat{\mathcal{F}}_{p}$. With $p \to \rho$ and Bayes' rule upgraded to the Petz recovery, this commutativity diagram applies also to the quantum regime (Theorem \ref{['thm:QSB-is-QPH']}), but for the inference-consistent QSB only.
  • ...and 3 more figures

Theorems & Definitions (19)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Conjecture 1
  • Theorem 5
  • ...and 9 more