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REVIEW 4 major objections 4 minor 73 references

How to implement the Bayes' formula in the age of ML?

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This chapter argues that the Bayes update in nonlinear filtering can be implemented by solving a max-min optimal transport problem whose solution is a map from prior to posterior samples, yielding a likelihood-free algorithm that captures…

desk verdict A clear expository synthesis of an OT-based likelihood-free Bayes update, worth refereeing, but its advertised error bound does not yet cover the neural network implementation. read the letter →

arxiv 2411.09653 v1 pith:25HFK7LK submitted 2024-11-14 eess.SY cs.SYmath.OC

Amirhossein Taghvaei , Prashant G. Mehta This is my paper · ORCID
classification eess.SYcs.SYmath.OC
keywords Bayesformulanonlinearfilteringoptimaltransportposteriorsamplingfeedbackparticlefiltercurseofdimensionalitysimulation-basedinferencemachinelearning
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper's central claim is that Bayes' formula — the step that turns a prior belief into a posterior after an observation — can be recast as a max-min optimal transport problem. The solution is a map $T(\cdot,y)$ that pushes samples from the prior $P_X$ onto the conditional posterior $P_{X|Y}(\cdot|y)$, for every observation $y$ at once. Because the objective is evaluated on samples from the joint law $(X,Y)$, the method is simulation-based: no likelihood formula is ever needed, which suits modern machine-learning practice. If the claim holds, nonlinear filters gain a principled way to represent multimodal posteriors, avoiding both the Gaussian bias of the ensemble Kalman filter and the weight degeneracy of particle filters. The chapter also shows that in the continuous-time limit the same optimization recovers the feedback particle filter, so the new update is an exact time discretization of an earlier algorithm.

What carries the argument

The load-bearing object is the block-triangular transport map $(x,y)\mapsto(T(x,y),y)$ selected by the quadratic-cost Monge problem under the coupling constraint $(T(X,Y),Y)\sim P_{X,Y}$. The max-min objective (11) is its Kantorovich dual: $f$ is a c-concave potential, meaning $x\mapsto\tfrac12|x|^2-f(x,y)$ is convex, and $T$ is the map that transports the independent coupling $P_X\otimes P_Y$ toward the joint law $P_{X,Y}$. This structure does three jobs: it pins down the unique Bayes-consistent map, it converts the consistency condition into an optimization amenable to neural-network parameterization, and it yields a quantitative error certificate through the optimality gap $\epsilon$ and the strong-convexity constant $\alpha$. In the recursive filter, the same step is iterated, and uniform geometric stability of the filter (Definition 2) converts the per-step bound into the global bound (27).

What would settle it

On the static example $Y=\tfrac12 X\odot X+\lambda_w W$ with $n=2$, train $f$ and $T$, compute the optimality gap $\epsilon$ from (12), estimate $\alpha$ as the minimum eigenvalue of the Hessian of $x\mapsto\tfrac12|x|^2-f(x,y)$ across $y$, and compare the empirical bounded-Lipschitz distance between $T(\cdot,y)_\#P_X$ and $P_{X|Y}(\cdot|y)$ with $\sqrt{4\epsilon/\alpha}$; a violation would refute Proposition 5's claim.

Watch

Extended reading notes

Core claim

The core discovery is the characterization of the Bayes update as a conditional Monge problem: among all maps $T$ with $(T(X,Y),Y)\sim P_{X,Y}$, choose the one minimizing $\mathbb{E}[\tfrac{1}{2}|T(X,Y)-X|^2]$. Kantorovich duality turns this into the max-min problem $\max_{f\text{ c-concave}_x}\min_T J(f,T;P_{X,Y})$ with $J(f,T;P_{X,Y})=\mathbb{E}_{(X,Y)\sim P_{X,Y}}[f(X,Y)]-\mathbb{E}_{(X,Y)\sim P_X\otimes P_Y}[f(T(X,Y),Y)+\tfrac{1}{2}|T(X,Y)-X|^2]$. For absolutely continuous priors with convex support, the pair $(f,T)$ exists and is essentially unique, and $T(\cdot,y)$ is the quadratic-cost optimal transport map from $P_X$ to $P_{X|Y}(\cdot|y)$. When the pair is only approximately optimal, the bounded-Lipschitz error between the transported prior and the true posterior is at most $\sqrt{4\epsilon/\alpha}$, where $\epsilon$ is the optimality gap and $\alpha$ is the strong-convexity constant of $x\mapsto\tfrac12|x|^2-f(x,y)$; in a recursively stable filter this becomes a time-uniform bound $C/\lambda\sqrt{4\epsilon/\alpha}$.

Load-bearing premise

The error guarantees rest on two unverified conditions: the filter must be uniformly geometrically stable, meaning past errors decay at a fixed exponential rate from any starting point, and the learned $f$ must make $x\mapsto\tfrac12|x|^2-f(x,y)$ strongly convex for every $y$; the paper's experiments do not check either.

Editorial extensions

If this is right

  • Bayes updates become likelihood-free: a simulator that draws $(X_i,Y_i)$ from the joint model is enough to train $f$ and $T$, so the method applies when the likelihood has no closed form.
  • Multimodal posteriors are representable: in the static and dynamic examples, the OT filter keeps both modes of a bimodal posterior, whereas SIR collapses into one mode and EnKF forces a Gaussian.
  • The error is controlled by training quality: whenever the learned pair has gap $\epsilon$ and $f$ is $\alpha$-strongly c-concave, the posterior error is no larger than $\sqrt{4\epsilon/\alpha}$, and $C/\lambda\sqrt{4\epsilon/\alpha}$ for uniformly stable filters.
  • The OT update is an exact time discretization of the feedback particle filter: the leading terms of a small-time expansion of the max-min objective reproduce the FPF gain and Poisson equation, with a divergence-free correction that does not change the density evolution.
  • With a resampling stage, the finite-particle version carries an additional $1/\sqrt{N}$ sampling error, keeping the usual Monte Carlo rate while avoiding weight-based degeneracy.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The optimality gap (12) could be tracked during training as a live diagnostic: with an estimate of $\alpha$, a trained filter would come with a data-dependent, checkable error bound rather than an unverified heuristic.
  • The learned map is amortized: after training on joint samples, conditioning on a new observation $y$ costs one forward pass through $T$, which points toward cheap conditional generation in inverse problems and simulation-based inference.
  • The bias-variance split the paper sketches suggests that the OT approach can beat the curse of dimensionality exactly when the transport map has exploitable structure; high-dimensional image-style observation models are the natural stress test of that expectation.
  • If the uniform geometric-stability assumption is relaxed to asymptotic stability, the bound likely loses its time-uniformity, predicting that on detectable-but-not-minorizing systems the error stays finite over each horizon but may drift with time.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. This chapter argues that the Bayes update can be implemented by solving a max-min optimal transport problem over a transport map T and a c-concave potential f, with T(.,y) the OT map from the prior P_X to the posterior P_{X|Y}(.|y). After a historical survey covering the Kalman filter, SIR particle filters, the ensemble Kalman filter, and the feedback particle filter, it presents the OT formulation (Eq. 11), an error bound under an assumed optimality gap and alpha-strong convexity (Prop. 5), a recursive OT filter with a stability-based error bound (Prop. 7), an asymptotic connection to the feedback particle filter (Prop. 6), and numerical demonstrations on a static bimodal example, a dynamic bimodal example, and the Lorenz-63 model.

Significance. If the stated guarantees applied to the algorithm actually implemented, the chapter would provide a useful simulation-based (likelihood-free) methodology for posterior sampling and nonlinear filtering that avoids the weight degeneracy of SIR and the Gaussian bias of EnKF. The chapter's strengths are its unified discrete/continuous treatment, its historical synthesis, the clear statement of the semi-dual max-min formulation, the explicit bounded-Lipschitz error bound in Eq. (13), and the numerical comparisons using MMD and MSE. However, the formal guarantees currently rest on assumptions that are neither imposed nor verified in the experiments, and there is an apparent sign inconsistency in the central objective; these issues need to be resolved before the central claim is fully supported.

major comments (4)
  1. [§4, Eq. (11), Eq. (14), Prop. 6] As printed, Eq. (11) defines J(f,T) = E_{P_{X,Y}}[f(X,Y)] − E_{P_X⊗P_Y}[f(T(X,Y),Y) + c(T(X,Y),X)], which gives the transport cost a negative sign. With this sign, the max-min problem is not the semi-dual of the constrained Monge problem in Eq. (10): in the elementary case P_X=δ_0, P_Y=δ_1, P_{X|Y}=δ_1, the printed objective yields value 0 (or is unbounded for f at the c-concavity boundary) instead of the OT cost 1/2. The displayed empirical objective in Eq. (14) has the cost term with a plus sign, and the expansion in Prop. 6 also uses the plus sign. Please correct the sign in Eq. (11) so that the cost term is added, e.g., J = E_{P_{X,Y}}[f(X,Y)] − E_{P_X⊗P_Y}[f(T(X,Y),Y)] + E_{P_X⊗P_Y}[c(T(X,Y),X)].
  2. [§4 Remark 3; §5.2 Prop. 5 and Prop. 7] The quantitative error bounds in Eq. (13) (Prop. 5) and Eq. (27) (Prop. 7) both assume that x ↦ 1/2|x|^2 − f_t(x,y) is alpha-strongly convex for all y and t. The implemented algorithm in Eq. (15) optimizes over an unrestricted neural-network class F, and Remark 3 explicitly concedes that c-concavity is not imposed, suggesting only an a posteriori check. No such check is reported for the experiments in Sec. 4.2 or Sec. 6. Consequently, Eqs. (13) and (27) do not cover the algorithm actually run, and the numerical success is not protected by the chapter's central quantitative claim. The chapter should either impose c-concavity/alpha-strong-convexity (e.g., via input-convex network architectures), verify and report the condition a posteriori, or state a version of Prop. 5 that holds without this assumption.
  3. [§5.2 Def. 2, Prop. 7, Remark 5; §6] The filter error bound in Prop. 7 also assumes uniform geometric stability of the filter (Def. 2). Remark 5 correctly acknowledges that this condition is strong and is only guaranteed under a minorization condition. However, the numerical sections that support the OT filter, including the dynamic bimodal example (28) and the Lorenz-63 example, do not check or discuss whether these models satisfy the stability assumption. If the chapter aims to present the OT filter as an algorithm with a proven guarantee, Assumption 1 of Prop. 7 should be checked for the reported examples or explicitly stated as an unverified hypothesis in the experimental sections.
  4. [§4.3, Prop. 6, Eq. (18)] Proposition 6 is stated as a result, but the asymptotic form f(x;y)=phi(x)y+psi(x)Δt and T(x,y)=x+K(x)y+u(x)Δt is assumed rather than derived. The appendix proof is a formal Taylor expansion for functions of that form; it does not establish that the solution of the max-min problem (11) has this asymptotic structure, nor does it give conditions for the remainder to be O(Δt^3) uniformly. Please label Eq. (18) explicitly as an ansatz/assumption and state the smoothness and limit-exchange conditions needed for the expansion, so that the claimed recovery of the FPF update is a clearly qualified statement.
minor comments (4)
  1. [§4.1, Eq. (14)] The notation for the shuffled sample is inconsistent: the text writes “i=2,...,n” where it should be “i=1,...,N,” and in the cost term it is unclear whether the second argument of c should be the shuffled sample X^i or the paired sample Xi. Please clarify so that Eq. (14) is an unambiguous empirical estimate of the objective in Eq. (11).
  2. [§6.0.2] The heading “Lorentz-63” should be “Lorenz-63” to match the standard name of the model.
  3. [§5.2, Remark 5] The minorization constant epsilon in the condition a(x|x') ≥ epsilon rho(x) uses the same symbol as the optimality gap epsilon in Prop. 7. Please use different symbols to avoid confusion.
  4. [References] The reference “Doucet A.and Johansen AM” has a formatting error (missing space and second initial format), and the Acknowledgments section is empty; the latter should either be filled or removed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the OT formulation is a standard semi-dual reformulation, and the error bounds are conditional on explicitly stated assumptions.

full rationale

The derivation is self-contained in the way that matters for circularity. The consistency condition (8) is the definition of the conditional law, and the replacement (9) is justified directly by the definition of conditional expectation rather than by assuming the conclusion. The max-min objective (11) is the ordinary semi-dual optimal-transport problem between the independent coupling P_X⊗P_Y and the joint P_X,Y; Prop. 4 grounds existence, uniqueness, and the OT-map identification in Carlier et al. (2016, Thm. 2.3) as reported by Al-Jarrah et al. (2024b), so the uniqueness claim is not imported as an unverified author fiat. Prop. 5 is a conditional statement: Eq. (13) holds only when x maps to 1/2|x|^2 - f(x,y) is alpha-strongly convex. Remark 3 explicitly admits that the neural-network class F does not impose this condition and says it should be checked a posteriori; the numerical sections report no such check. That is a real applicability gap between theorem and implementation, but it is not circularity: the bound is not fitted, and the failure mode is an unverified hypothesis, not an input that has been renamed as an output. Prop. 7 combines geometric filter stability (Def. 2) with Prop. 5 and cites the proof to an overlapping-author paper, but it is a standard epsilon-alpha estimate rather than a conclusion that presupposes itself. The numerical comparisons are against SIR, EnKF, and a high-accuracy SIR proxy for the posterior; no fitted parameter is relabeled as a prediction. The chapter therefore has no circular step that reduces its central claim to its inputs.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The theoretical formulation uses standard OT results, while the error bounds add strong regularity and stability assumptions that are not verified in the numerics. The only fitted quantities are the neural network weights used in the numerical demonstrations, which are not reported. No new physical entities or forces are introduced.

free parameters (1)
  • Neural network weights and training hyperparameters for (f,T) = not reported in chapter
    The numerical experiments in Sec. 4.2 and Sec. 6 train f and T on data; the final weights and hyperparameters are deferred to Al-Jarrah et al. 2024b, so the demonstrations are not reproducible from this text.
assumptions (6)
  • domain assumption The prior P_X is absolutely continuous with convex support, P_{X|Y} has a density, and E[|X|^2] < infinity (Prop. 4 assumptions)
    Needed for existence and uniqueness of the OT map in Prop. 4; invoked in Sec. 4, Step 3.
  • standard math Standard OT duality and existence of optimal transport maps under quadratic cost (Carlier et al. 2016, Thm 2.3)
    Used to derive the max-min objective (11) from the conditional Monge problem (10).
  • domain assumption The filter is uniformly geometrically stable, d(T_{t,s}mu, T_{t,s}nu) <= C(1-lambda)^{t-s} d(mu,nu) (Definition 2)
    Assumption 1 of Prop. 7; the paper admits this is strong and only guaranteed under a minorization condition.
  • ad hoc to paper The max-min optimality gap eps is uniformly bounded and x -> 1/2|x|^2 - f_t(x,y) is alpha-strongly convex for all y (Assumptions 2-3 of Prop. 7)
    The error bound (27) depends on these; the paper does not verify them for the neural-net implementations, and Remark 3 says convexity is not enforced.
  • domain assumption Simulation-based observation model: one can sample Y ~ h(.|x) but cannot necessarily evaluate the likelihood
    The OT formulation is designed to avoid explicit likelihood evaluations; stated in Sec. 3.
  • ad hoc to paper For Prop. 6, the OT solution has the asymptotic form f(x;y) = phi(x)y + psi(x) Delta t and T(x,y) = x + K(x)y + u(x) Delta t as Delta t -> 0
    The asymptotic expansion (19) assumes this ansatz; it is used to recover the FPF vector fields.

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Pith. "Pith review of How to implement the Bayes' formula in the age of ML?." pith.science (2026). https://pith.science/paper/25HFK7LK

@misc{pith2026241109653,
  author       = {Pith},
  title        = {Pith review of: How to implement the Bayes' formula in the age of ML?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/25HFK7LK}},
  note         = {Machine review of arXiv:2411.09653}
}
read the original abstract

This chapter contains a self-contained introduction to the significance of Bayes' formula in the context of nonlinear filtering problems. Both discrete-time and continuous-time settings of the problem are considered in a unified manner. In control theory, the focus on optimization-based solution approaches is stressed together with a discussion of historical developments in this area (from 1960s onwards). The heart of this chapter contains a presentation of a novel optimal transportation formulation for the Bayes formula (developed recently by the first author) and its relationship to some of the prior joint work (feedback particle filter) from the authors. The presentation highlights how optimal transportation theory is leveraged to overcome some of the numerical challenges of implementing Bayes' law by enabling the use of machine learning (ML) tools.

Figures

Figures reproduced from arXiv: 2411.09653 by the authors.

Figure 1
Figure 1. This exposition is concerned with particle filter algorithms that seek to approximate the posterior with the empirical distribution of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Neural net architectures for the function classes [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Numerical results for the static example in Sec. 4.2. (a) top-left: Samples [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Numerical results for the dynamic example 28. The left panel shows the trajectory of the particles [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Numerical results for the Lorenz 63 example. The left panel shows the trajectory of the unobserved component of the true state and [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.