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Partition function approach to non-Gaussian likelihoods: information theory and state variables for Bayesian inference

T0 review · 5 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper claims Bayesian inference is thermodynamics: a Bayes partition function governs sampler ensembles, so updates carry work, heat, entropy, and an effective complexity dimension.

desk verdict Genuinely useful n_eff diagnostic inside a thermodynamic analogy whose Jarzynski 'derivation' is definitional and whose Section 4 is internally inconsistent. read the letter →

arxiv 2411.13625 v2 pith:LI7NSIO2 submitted 2024-11-20 cond-mat.stat-mech astro-ph.COcs.ITmath.IT

Rebecca Maria Kuntz , Heinrich von Campe , Tobias Röspel , Maximilian Philipp Herzog , Björn Malte Schäfer This is my paper · ORCID
classification cond-mat.stat-mechastro-ph.COcs.ITmath.IT MSC 62F1582B3094A17
keywords BayespartitionfunctionthermodynamicanalogyforBayesianinferenceJarzynskiequalityevidenceeffectivedimensioninformationentropyRényisupernovacosmology
topics Dark Energy
open problems Dark Energy
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

This paper argues that the resemblance between the Bayesian evidence integral and a statistical-mechanics partition function is a working correspondence: a Bayes update can be modelled as a continuous transition between thermodynamic ensembles, so the language of thermodynamics — energy, work, heat, entropy, free energy — applies to sampling processes. The central object is the Bayes partition function $Z[T, J] = \int d\mu(\theta)\,[L(\theta)\pi(\theta)\exp(J_\nu\theta^\nu)]^{1/T}$, which recovers the Bayesian evidence at $T=1$, $J=0$ and generates posterior cumulants by differentiation. From a transferred Jarzynski equality the authors derive the work of a partial update as $W = -\lambda\ln L$; they build an information-theoretic Guggenheim scheme for Gaussian likelihoods that yields internal energy $U = nT/2$ and heat capacity $C = n/2$; and they show R\'enyi entropies arise as $q$-derivatives of the Bayes free energy. An effective dimension $n_{\mathrm{eff}} \leq n$ measures model complexity, with deviations from Gaussianity vanishing as temperature rises. A supernova cosmology example illustrates the formalism on a realistic non-Gaussian posterior.

What carries the argument

The load-bearing object is the Bayes partition function $Z[T, J] = \int d\mu(\theta)\,[L(\theta)\pi(\theta)\exp(J_\nu\theta^\nu)]^{1/T}$, a temperature- and source-weighted Bayesian evidence; for exponential-family likelihoods it becomes $\int d\mu(\theta)\,\exp(-(\Phi(\theta) - J_\nu\theta^\nu)/T)$, so it behaves like a canonical partition with potential $\Phi(\theta) = \chi^2(y|\theta)/2 + \phi(\theta)$. It carries the argument because every thermodynamic analogue in the paper — the free energy $G = -T\ln Z$, the information entropy $S = \partial_T(T\ln Z)$, the posterior cumulants as $\partial_J$ derivatives, and the heat capacity $C = \beta^2\partial_\beta^2\ln Z$ behind the effective dimension — comes from standard partition-function identities, while the insertion $Z_\lambda[T] = \int d\mu(\theta)\,[L^\lambda\pi]^{1/T}$ provides the continuous prior-to-posterior interpolation on which the Jarzynski transfer and the work formula $W = -\lambda\ln L$ rest. The Jeffreys covolume $\sqrt{\det F}$ as integration measure makes all of these objects invariant under reparametrization.

What would settle it

Run a finite-time annealing protocol that moves a sampler ensemble from the prior ($\lambda = 0$) to the posterior ($\lambda = 1$) of a non-Gaussian likelihood, accumulating the log-likelihood increments along the protocol as the work $W$; if the exponential average $\langle e^{-W/T}\rangle$ of this measured work does not equal the ratio of the interpolated Bayes partition functions, $Z_1/Z_0$, then the Jarzynski transfer fails for the actual sampling process and the paper's 'work' is a definitional label rather than a physical quantity.

Watch

Extended reading notes

Core claim

Its central claim is that the Bayes partition function governs an ensemble of samplers in exactly the way a statistical-mechanics partition function governs a physical ensemble, making the divide between posterior sampling and thermodynamic equilibrium artificial. For exponential-family likelihoods the partition becomes $Z[T, J] = \int d\mu(\theta)\,\exp(-(\Phi(\theta) - J_\nu\theta^\nu)/T)$ with potential $\Phi = \chi^2/2 + \phi$ from the likelihood and prior, so the thermodynamic analogues — free energy $G = -T\ln Z$, entropy $S = \partial_T (T\ln Z)$, cumulants from $\partial_J$ — follow by standard identities. The paper transfers Jarzynski's equality to this setting, obtaining $W = -\lambda\ln L = \lambda\chi^2/2$ as the work expended at stage $\lambda$ of an update, and verifies the analogy by deriving $U = nT/2$ and heat capacity $C = n/2$ for a Gaussian likelihood, the same equipartition form as a monoatomic ideal gas with $n$ degrees of freedom. It further defines an effective dimension $n_{\mathrm{eff}} = 2C$ that is strictly positive and at most $n$, decreasing as even-order cumulants signal non-Gaussianity, with the deficit scaling as $|\Delta n| \sim T^{-k/2-1}$; at $T = 1$ the temperature derivative of the partition reproduces the surprise statistic, the KL divergence between prior and posterior. The cosmological example, supernova distance moduli constraining the matter density $\Omega_m$ and the dark energy equation-of-state parameter $w_0$, recovers the Bayesian evidence at $T = 1$ and confirms that a Gaussian approximation carries higher entropy than the true posterior at all temperatures.

Load-bearing premise

The thermodynamic reading rests on treating a Bayes update as a genuine process with a well-defined work variable; the paper defines that work through the free-energy difference, $W = -\lambda\ln L$, so the Jarzynski equality holds by construction, and the analogy would collapse if no real stochastic process connecting prior to posterior actually obeys that equality.

Editorial extensions

If this is right

  • The influence of data on an inference acquires a thermodynamic cost: at stage $\lambda$ of an update the work is $W = -\lambda\ln L$, so the $\chi^2$ value of the data directly quantifies the expenditure of a partial Bayes step.
  • Information entropy decreases as inference progresses in the Gaussian case, and the rate of decrease is governed by the variance of the $\chi^2$ potential plus its covariance with the prior, giving a design criterion for informative experiments.
  • For Gaussian problems the heat capacity $C = n/2$ identifies the parameter dimension as the number of degrees of freedom; for non-Gaussian problems $n_{\mathrm{eff}} = 2C$ gives a strictly positive, temperature-dependent measure of model complexity that drops as higher-order cumulants appear.
  • R\'enyi entropies of any order become $q$-derivatives of the Bayes free energy, with the Shannon entropy recovered in the limit $\alpha \to 1$, so the partition function is a generating object for generalized entropies.
  • Applied to supernova cosmology, the formalism recovers the Bayesian evidence at $T = 1$ and reproduces the entropy-maximising ordering between a Gaussian approximation and the true posterior, showing the thermodynamic quantities are numerically accessible from standard MCMC output.

Reading between the lines

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

  • If the thermodynamic reading is more than formal, any concrete annealing or tempering schedule between prior and posterior should satisfy a fluctuation identity: the exponential average of the accumulated log-likelihood increments should equal the evidence ratio, a testable prediction for actual MCMC samplers that would separate a genuine fluctuation theorem from a definitional one.
  • The effective dimension could serve as a model-comparison statistic alongside the Bayesian evidence: the temperature decay $|n - n_{\mathrm{eff}}| \sim T^{-k/2-1}$ identifies the dominant cumulant order, so heating a sampler ensemble isolates the leading non-Gaussianities.
  • Because every integral uses the invariant Jeffreys measure, the construction's natural scope is any Riemannian statistical manifold, not only exponential families; for strongly non-Gaussian models the Gaussian results would be the local, flat-geometry approximation and $n_{\mathrm{eff}}$ the curvature-corrected dimension.
  • The assignment of extensive and intensive variables (inverse Fisher information as 'volume', parameter dimension as degrees of freedom) is one consistent dictionary rather than a unique one; other pairings would produce different but equally consistent thermodynamic potentials.
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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

5 major / 5 minor

Summary. The paper proposes a Bayes partition function Z[T,J] = ∫ dμ(θ) [L(y|θ)π(θ) exp(J_ν θ^ν)]^{1/T} as a Gibbs-like partition function for Bayesian inference, and argues that a Bayes update is a continuous transition between thermodynamic ensembles. On this basis it derives information-theoretic analogues of entropy, work, heat, free energy, and thermodynamic processes; relates Rényi and Shannon entropies to partition functions; defines an effective dimension from the heat-capacity analogue; and illustrates the formalism on a supernova cosmology inference problem. The central claim is that thermodynamic state variables and processes are meaningful descriptions of sampling processes.

Significance. If the analogy were rigorously established, the paper would provide a useful dictionary between statistical mechanics and Bayesian inference, with potential applications to MCMC diagnostics and model comparison. Some components are correct and elegant: the Gaussian partition function gives U = nT/2, S = (n/2)ln(2πT)+1, and C = n/2; the Rényi-entropy result in Section 6 can be obtained from partition-function identities; and the effective-dimension diagnostic in Section 7 is an interesting proposal. The cosmology example is a concrete, if simple, illustration. However, the two load-bearing pieces of the thermodynamic analogy — the Jarzynski work relation and the isentropic-process thermodynamics — contain serious errors as written, and the entropy-derivative formula in Eq. (15) is quantitatively wrong. These issues must be corrected before the paper's central claims can be accepted.

major comments (5)
  1. [Sec. 2.1, Eqs. (20)–(25)] The derivation of W = -λ ln L from Jarzynski's equality is not a derivation. Jarzynski's equality, Eq. (20), is an identity for averages over the full phase space for a specified switching protocol, and the work W is a trajectory-dependent functional of the protocol, not a state function of θ. The paper instead writes equality of integrals over every sub-domain K in Eq. (24) and then equates integrands to obtain Eq. (25). This step is not justified by Jarzynski's equality; it effectively defines W by the relation e^{-W} = L^λ. Moreover, the statement that Eq. (20) 'assumes that the transition occurs through a series of equilibrium states' is incorrect: Jarzynski's equality is valid for arbitrarily fast, non-equilibrium protocols. To make the claim meaningful, the authors should either specify a concrete switching protocol (e.g., a sudden quench) whose work is exactly λχ²/2, or explicitly present W = -λ ln L as a definition and discuss its limitations as a physical work variable.
  2. [Sec. 4, Eq. (57)] The isentropic process contains an internal contradiction. The text sets d(JJ) = 0 and dF^{-1} ≠ 0, but Eq. (57) reports dH = dW = -½ F^{-1} d(JJ), which is identically zero under that premise. The correct differential of H includes a second term: dH = (n/2)dT - ½ F^{-1} d(JJ) - ½ JJ dF^{-1}. With dT = 0 and d(JJ) = 0, the isentropic work is dW = -½ JJ dF^{-1}, not the expression in Eq. (57). This error is not cosmetic: it invalidates the 'adiabatic' work claim and the comparison with p dV work. The authors must correct the differentials and re-derive the process equations consistently.
  3. [Sec. 2, Eq. (15)] The λ-derivative of the Shannon entropy at T = 1 is miscomputed. For L ∝ exp(-χ²/2) and π ∝ exp(-φ), the intermediate distribution is p_λ ∝ exp(-λχ²/2 - φ). Direct differentiation gives dS/dλ = -(λ/4)Var(χ²) - (1/2)Cov(χ², φ), not -λ Var(χ²) - Cov(χ², φ) as in Eq. (15). The sign and the qualitative claim of decreasing uncertainty survive, but the quantitative rate of information gain is incorrect. This formula is cited in the summary and should be corrected with the derivation shown.
  4. [Sec. 3, Eqs. (35) and (47)] The thermodynamic variables are not consistently defined in the Gaussian example. If F^{-1} is an extensive state variable, then the differential dG in Eq. (35) should contain a term involving dF^{-1} in addition to d(JJ); as written, the conjugate pair is ambiguous. Correspondingly, the total differential dH in Eq. (47) omits the -½ JJ dF^{-1} term that is required if F^{-1} is an independent variable. This is the root of the isentropic contradiction in Section 4. The authors should either treat F^{-1} as a function of other state variables, in which case the independent differentials must be stated, or include the F^{-1} differential explicitly throughout.
  5. [Sec. 6, Eq. (86)] The substitution in Eq. (86) is not correct as written. For the true posterior p(θ|y) ∝ exp(-Φ_J(θ)), one has p(θ|y)^{T0} ∝ exp(-T0 Φ_J), not Z[T0]^{-1} exp(-Φ_J/T0). The quantity Z[T0] = ∫ dμ exp(-Φ_J/T0) is a different integral. The final Rényi-to-Shannon identity is nevertheless obtainable by directly computing ∫ p^α = Z[1/α] / Z[1]^α, but the paper's route conflates a temperature-scaled distribution with a power of the posterior. This should be clarified, otherwise the derivation appears to be engineered rather than derived.
minor comments (5)
  1. [Abstract and Sec. 1] The name is misspelled as 'Jazinsky' in the abstract and in Section 1; it should be 'Jarzynski'.
  2. [Sec. 3, Eq. (48)] Eq. (48) reads 'dU = n/2'; the differential dT is missing and the equation should read dU = (n/2) dT.
  3. [Sec. 8, Figure captions] The figure captions contain garbled axis labels such as '10□1 100 101'; these should be typeset as 10^{-1}, 10^0, and 10^1.
  4. [Sec. 8, Eq. (28) paragraph] The paragraph following Eq. (28) is incomplete: after 'the values of the j's must be pairwise distinct' the sentence breaks off. The argument about the determinant vanishing for underconstrained models should be written out in full.
  5. [Sec. 7, Eq. (102)] The notation in Eq. (102) is under-specified: the set P and the blocks s ∈ π are not defined. Please define the pairings used in the cumulant expansion.

Circularity Check

1 steps flagged · score 6.0 of 10

Jarzynski transfer is definitional: W=-λ ln L is imposed by requiring Eq. (24) for every sub-domain, not derived from a switching process.

  1. self definitional [Section 2.1, Eqs. (23)-(25)]
    "For a parameter domain K⊆ Rn, Equation 20 leads to ∫K dµ(θ)(π e^{−W})^{1/T}/∫K dµ(θ)π^{1/T} = ∫K dµ(θ)(Lλπ)^{1/T}/∫K dµ(θ)π^{1/T} , which ultimately results in e^{−W}=Lλ ↔ W=−λ ln L = λχ2/2 , since the equality of the integrals in Equation 24 holds for any domain of integration K."

    Jarzynski's equality is a global statement about an ensemble average for a specified non-equilibrium protocol; it does not imply equality of integrals over every sub-domain K. By writing Eq. (24) for arbitrary K and then invoking 'any domain of integration K', the authors force the integrands to be equal pointwise, i.e. (π e^{-W})^{1/T}=(L^λπ)^{1/T}, which is exactly the claimed result e^{-W}=L^λ. No switching process or trajectory-dependent work variable is ever defined; W is chosen as the quantity that makes the Jarzynski relation hold for every K. Thus W=-λ ln L is not derived from a physical process but inserted as the defining ansatz, making the central 'work = log-likelihood' claim circular by construction.

full rationale

The paper's central advertised result, the transfer of Jarzynski's equality to Bayesian inference, is circular by construction. The derivation starts from Jarzynski's global average relation (Eq. 20), but immediately rewrites it as an equality of integrals over an arbitrary domain K (Eq. 24). Jarzynski's theorem supplies no such local identity; the only way to obtain Eq. 24 for every K is to assume that the integrands match, which is exactly e^{-W}=L^λ. No protocol with trajectory-dependent work is specified, and the paper's remark that Eq. 20 requires 'a series of equilibrium states' is inconsistent with Jarzynski's equality, which is valid for arbitrarily fast non-equilibrium switching. The 'work' formula is therefore a definition dressed as a prediction. Section 6's Rényi-to-Shannon result is a mathematical identity following from Bayes' theorem for a flexible family of partitions, so it is not circular even though the temperature dependence in Eq. (79) is deliberately chosen. Section 4 contains an internal inconsistency (the isentropic subsection sets d(JJ)=0 yet Eq. (57) reports -1/2 F^{-1}d(JJ)), but that is a correctness issue rather than a circularity. Self-citations to Röver et al. (2022), Schosser et al. (2024b), and Herzog et al. (2023) are present but are not the source of the main circularity. Accordingly, the score reflects one strong definitional circularity in the central claim, with the remainder of the paper retaining independent mathematical content.

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

The central claim rests on a set of analogical identifications (partition, Jarzynski work, temperature) and on self-cited results for positivity of C and the definition of the partition function. There are no numerically fitted free parameters, but several functional choices are made by hand to obtain the desired information-theoretic results.

assumptions (6)
  • domain assumption The Bayes partition function Z[T,J] = integral dmu [L pi exp(J theta)]^(1/T) is the correct analogue of a thermodynamic partition function.
    Introduced in Eq. (6), inherited from Roever et al. (2022); the entire analogy hangs on this identification.
  • ad hoc to paper Jarzynski's equality applies to Bayes updates.
    Assumed in Section 2.1; no microscopic dynamics or work protocol is constructed, so the applicability is postulated rather than derived.
  • standard math The invariant measure dmu(theta) = d^n theta sqrt(det F) is the appropriate integration measure.
    Eq. (5), the Jeffreys prior volume form, is standard in information geometry, but its use throughout parameter-space integrals is an assumption about the geometry.
  • domain assumption Specific heat C is strictly positive for Bayes partitions.
    Invoked in Section 7 via Schosser et al. (2024b), a self-cited result not proved in this paper; needed to guarantee n_eff > 0.
  • domain assumption A Gaussian likelihood is sufficient for the thermodynamic state-variable identifications.
    Section 3 limits the Guggenheim scheme to Gaussian likelihoods (Eq. 26); non-Gaussian generalization is acknowledged as highly non-trivial.
  • ad hoc to paper The asymmetric temperature scaling L^(1/T) pi^((T+1/T)/2) is a valid starting point for relative entropy derivations.
    Chosen in Eq. (79) so that T = 1 yields the KL divergence; the authors admit the precise choice does not influence the arguments, indicating it is a design choice rather than a derived result.
invented entities (1)
  • Effective dimension n_eff
    purpose: Measure of model complexity and deviation from Gaussianity in a posterior distribution.
    Defined in Eq. (97) as 2C_non-Gaussian; no external falsifiable handle beyond internal consistency, and the cosmology example only checks that entropy ordering matches the Gaussian maximum-entropy property.

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Cite this review

Pith. "Pith review of Partition function approach to non-Gaussian likelihoods: information theory and state variables for Bayesian inference." pith.science (2026). https://pith.science/paper/LI7NSIO2

@misc{pith2026241113625,
  author       = {Pith},
  title        = {Pith review of: Partition function approach to non-Gaussian likelihoods: information theory and state variables for Bayesian inference},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LI7NSIO2}},
  note         = {Machine review of arXiv:2411.13625}
}
read the original abstract

The significance of statistical physics concepts such as entropy extends far beyond classical thermodynamics. We interpret the similarity between partitions in statistical mechanics and partitions in Bayesian inference as an articulation of a result by Jaynes (1957), who clarified that thermodynamics is in essence a theory of information. In this, every sampling process has a mechanical analogue. Consequently, the divide between ensembles of samplers in parameter space and sampling from a mechanical system in thermodynamic equilibrium would be artificial. Based on this realisation, we construct a continuous modelling of a Bayes update akin to a transition between thermodynamic ensembles. This leads to an information theoretic interpretation of Jazinsky's equality, relating the expenditure of work to the influence of data via the likelihood. We propose one way to transfer the vocabulary and the formalism of thermodynamics (energy, work, heat) and statistical mechanics (partition functions) to statistical inference, starting from Bayes' law. Different kinds of inference processes are discussed and relative entropies are shown to follow from suitably constructed partitions as an analytical formulation of sampling processes. Lastly, we propose an effective dimension as a measure of system complexity. A numerical example from cosmology is put forward to illustrate these results.

Figures

Figures reproduced from arXiv: 2411.13625 by the authors.

Figure 1
Figure 1. The Guggenheim scheme of standard thermodynamics (at fixed N), on the left, and one possibility to construct the analogous Guggenheim scheme of information theory for a Gaussian likelihood (at fixed n and N), on the right. Intensive variables in blue, extensive variables in red. −2 0 2 θ −0.2 0.0 0.2 0.4 Φ( θ) − Jθ, L(y|θ) 1 T = 1, J = 0 1 T = 2, J = 1 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Schematic changes of a univariate Gaussian (shaded region) and its corresponding potential (solid line), as given in equation (29), with different values of T and J. For simplicity, F = 1 is fixed. In a Gibbs ensemble, the internal energy U, the enthalpy H and the entropy S of the parameter space configuration follow from the partition Z[T, p] := Z[T, p,N = 1] as H(S, p) = U + pV = T 2 ∂T lnZ[T, p] (37) U(S, V) = T … view at source ↗
Figure 3
Figure 3. Decrease of |n − neff| with temperature T and even order of non-Gaussianity k. Here, the scaling is T −k/2−1 (k+2) (k−1)! . differences between Gaussian and non-Gaussian landscapes become very pronounced at low T. Additionally, [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: shows the partition as a function of temperature. For T = 1, the Bayesian evidence can be recovered from the partition function. Using Bayes’ theorem, one can write the partition function as Z[T] = p( y) 1/T ∫ dµ(θ) p(θ| y) 1/T . (110) The evidence is derived numerical…
Figure 6
Figure 6. Figure 6: The thermodynamic information entropy S[T] as a function of temperature T is always higher for the Gaussian approximation (orange) than for its non-Gaussian counterpart (blue) - in accordance with the Gaussian distribution being entropy maximising for fixed variance. T…

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Pith tools

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