Conditional-Marginal Entropy-Rate Objective

• The conditional-marginal entropy-rate objective is a framework that contrasts conditional and marginal entropy to isolate side-information effects in generative and thermodynamic systems.
• It finds applications in flow samplers, sequence modeling, and partially observed processes by decomposing uncertainty into component-specific and aggregate contributions.
• Empirical and theoretical results demonstrate its value in optimizing discretization schedules, diagnosing learnability, and bridging conditional and marginal entropy formulations.

Conditional-Marginal Entropy-Rate Objective denotes a family of entropic constructions in which a conditional quantity is compared against a marginal one, either as a difference, a decomposition, or an optimization criterion. In the most direct usage, it is a bridge-aware inference-time rate signal for flow and Schrödinger samplers,

$$r_{cm}(t)=\left| \mathbb{E}_{Z,X_t\mid Z}\!\left[\operatorname{div}_x v_t(X_t\mid Z)\right] - \mathbb{E}_{X_t}\!\left[\operatorname{div}_x \bar v_t(X_t)\right] \right|,$$

derived from the time derivative of $H(Z\mid X_t)$. In adjacent literatures, closely related constructions appear as marginal-versus-conditional entropy production in strongly coupled thermodynamics, forward-versus-backward conditional entropy comparisons for sequences, and entropy-rate maximization over feasible stationary block laws with fixed context marginals. The unifying motif is that conditioning isolates structure-specific or side-information-specific variation, while the marginal term removes population-level, endpoint-mixed, or context-aggregated contributions (Trentini et al., 15 May 2026).

1. Conceptual scope and formal variants

The phrase is not attached to a single universal formalism. The most explicit “objective” appears in bridge-aware generative sampling, where the relevant scalar is the magnitude of a conditional-minus-marginal entropy-rate contrast. A second explicit objective appears in partially observed stochastic processes, where the entropy-rate functional

$$J(u):=-\sum_{c\in A^r}\sum_{a\in A}u(c,a)\log\frac{u(c,a)}{\eta_u(c)}$$

is maximized over a feasible class of stationary $(r+1)$-block laws. A third operational form appears in sequential modeling, where the normalized forward-minus-backward cross-entropy gap

$\Delta H = \frac{1}{N}(H_M(S)-H_{\hat M}(\hat S))$

is used as a learnability and distribution-shift diagnostic. In thermodynamics, by contrast, the relevant quantities are local entropy productions rather than optimization objectives in the machine-learning sense (Wang, 2024).

Across these variants, the conditional term typically measures uncertainty or dissipation relative to retained side information, while the marginal term measures the corresponding quantity after that information has been averaged out or hidden. This suggests that “conditional-marginal” is best understood as a structural pattern rather than a single standardized objective name. A plausible implication is that the term is most precise when the underlying framework makes both the conditional process and the marginal process explicit, and less precise when it refers only to a decomposition or an analogy (Kiriukhin, 12 Apr 2026).

2. Thermodynamic decomposition into marginal and conditional entropy production

In strongly coupled bipartite thermodynamics, the relevant construction begins with the path-wise total entropy production for the joint system $(\mathcal X,\mathcal Y)$,

$$\Sigma_{\mathcal X,\mathcal Y} = \ln \frac{p(\tilde{x},\tilde{y}\mid \tilde{c}_x,\tilde{c}_y)} {p(\bar{x},\bar{y}\mid \bar{c}_x,\bar{c}_y)},$$

together with the detailed fluctuation-theorem form

$\diss_{\X,\Y} = \Delta s(x,y) - \beta\, \heat_{\X,\Y}(\ft{x},\ft{y}), \qquad \Delta s(x,y) = -\ln p(\rti{x},\rti{y}) + \ln p(\fti{x},\fti{y}).$

The paper then defines the marginal entropy production of subsystem $\mathcal X$,

$\diss_{\X} = \ln \frac{p(\ft{x})}{p(\rt{x})},$

and the conditional entropy production of $H(Z\mid X_t)$0 given $H(Z\mid X_t)$1,

$H(Z\mid X_t)$2

The resulting balance equations are

$H(Z\mid X_t)$3

$H(Z\mid X_t)$4

$H(Z\mid X_t)$5

These are the local thermodynamic identities from which the paper derives its local second law (Crooks et al., 2016).

The crucial conceptual step is causal intervention. The joint trajectory probability is factorized into interventional trajectory probabilities,

$H(Z\mid X_t)$6

where $H(Z\mid X_t)$7 denotes that one subsystem’s trajectory is treated as a fixed external influence on the other. This is not the same as the ordinary conditional probability $H(Z\mid X_t)$8. The distinction isolates feedback-related dissipation through the transferred dissipation terms $H(Z\mid X_t)$9 and $$J(u):=-\sum_{c\in A^r}\sum_{a\in A}u(c,a)\log\frac{u(c,a)}{\eta_u(c)}$$0, with

$$J(u):=-\sum_{c\in A^r}\sum_{a\in A}u(c,a)\log\frac{u(c,a)}{\eta_u(c)}$$1

where

$$J(u):=-\sum_{c\in A^r}\sum_{a\in A}u(c,a)\log\frac{u(c,a)}{\eta_u(c)}$$2

In this setting, the conditional-marginal split is not an optimization objective but a thermodynamic partition of dissipation into observable, conditioned, and feedback-mediated pieces (Crooks et al., 2016).

The averaged inequalities are

$$J(u):=-\sum_{c\in A^r}\sum_{a\in A}u(c,a)\log\frac{u(c,a)}{\eta_u(c)}$$3

summarized as

$$J(u):=-\sum_{c\in A^r}\sum_{a\in A}u(c,a)\log\frac{u(c,a)}{\eta_u(c)}$$4

The significance is that marginal and conditional forms remain valid local versions of the Second Law even under strong coupling, without the usual weak-coupling idealization (Crooks et al., 2016).

3. Sequence models, entropy-rate scaling, and time-reversal diagnostics

In sequence analysis, one line of work studies the conditional entropy rate directly as

$$J(u):=-\sum_{c\in A^r}\sum_{a\in A}u(c,a)\log\frac{u(c,a)}{\eta_u(c)}$$5

The constant entropy rate hypothesis, or constant conditional entropy, is

$$J(u):=-\sum_{c\in A^r}\sum_{a\in A}u(c,a)\log\frac{u(c,a)}{\eta_u(c)}$$6

The same paper formalizes uniform information density as

$$J(u):=-\sum_{c\in A^r}\sum_{a\in A}u(c,a)\log\frac{u(c,a)}{\eta_u(c)}$$7

distinguishes strong UID from full UID, and proves the logical hierarchy

$$J(u):=-\sum_{c\in A^r}\sum_{a\in A}u(c,a)\log\frac{u(c,a)}{\eta_u(c)}$$8

while also establishing that $$J(u):=-\sum_{c\in A^r}\sum_{a\in A}u(c,a)\log\frac{u(c,a)}{\eta_u(c)}$$9 and $(r+1)$0. The same source argues that CER is inconsistent with Hilberg’s law,

$(r+1)$1

and therefore incompatible with the observed decrease of conditional entropy with prefix length (Ferrer-i-Cancho et al., 2013).

A distinct but related construction compares forward and backward conditional entropies of a sequence. For a process over a finite alphabet with fixed context length $(r+1)$2, the main theorem is

$(r+1)$3

where $(r+1)$4 and $(r+1)$5 are the first and last $(r+1)$6-tuples, and $(r+1)$7 depends only upon $(r+1)$8. The proof identifies exact cancellation of the bulk terms and leaves only a boundary correction, so “the difference in average conditional entropy is $(r+1)$9.” The empirical objective is then

$\Delta H = \frac{1}{N}(H_M(S)-H_{\hat M}(\hat S))$0

Its interpretation is operational: $\Delta H = \frac{1}{N}(H_M(S)-H_{\hat M}(\hat S))$1 means the reverse direction is easier to learn, $\Delta H = \frac{1}{N}(H_M(S)-H_{\hat M}(\hat S))$2 means the forward direction is easier to learn, and large $\Delta H = \frac{1}{N}(H_M(S)-H_{\hat M}(\hat S))$3 indicates large directional asymmetry in learnability (Wang, 2024).

Taken together, these results delimit what a conditional-marginal entropy-rate objective can mean in sequence settings. Exact constancy of conditional entropy is rejected as an incomplete hypothesis, whereas forward-versus-backward normalized conditional-entropy gaps are proposed as practical diagnostics. This suggests that, for realistic sequential data with long-range dependencies, the relevant object is not a constant rate but a length-dependent or direction-dependent conditional-marginal comparison (Ferrer-i-Cancho et al., 2013).

4. Entropy-rate maximization for partially observed processes

A more literal optimization framework appears in entropy-rate selection under partial observability. The setup begins with finite hidden and visible alphabets $\Delta H = \frac{1}{N}(H_M(S)-H_{\hat M}(\hat S))$4 and $\Delta H = \frac{1}{N}(H_M(S)-H_{\hat M}(\hat S))$5, an observation map

$\Delta H = \frac{1}{N}(H_M(S)-H_{\hat M}(\hat S))$6

and the observational fiber

$\Delta H = \frac{1}{N}(H_M(S)-H_{\hat M}(\hat S))$7

To obtain a finite-dimensional problem, the framework uses stationary $\Delta H = \frac{1}{N}(H_M(S)-H_{\hat M}(\hat S))$8-block laws $\Delta H = \frac{1}{N}(H_M(S)-H_{\hat M}(\hat S))$9 on $(\mathcal X,\mathcal Y)$0, with context marginal

$(\mathcal X,\mathcal Y)$1

stationary-consistency constraints, and a feasible visible class

$(\mathcal X,\mathcal Y)$2

The entropy-rate functional is

$(\mathcal X,\mathcal Y)$3

which is $(\mathcal X,\mathcal Y)$4 for a stationary finite-state Markov process. The selector is

$(\mathcal X,\mathcal Y)$5

Here the marginal term is the context marginal $(\mathcal X,\mathcal Y)$6, and the objective measures maximal residual uncertainty under retained visible constraints (Kiriukhin, 12 Apr 2026).

The paper proves existence by compactness, uniqueness under a fixed-context-marginal hypothesis $(\mathcal X,\mathcal Y)$7, and a more general strict-concavity characterization: $(\mathcal X,\mathcal Y)$8 is strictly concave on a convex set $(\mathcal X,\mathcal Y)$9 iff no two distinct points in $$\Sigma_{\mathcal X,\mathcal Y} = \ln \frac{p(\tilde{x},\tilde{y}\mid \tilde{c}_x,\tilde{c}_y)} {p(\bar{x},\bar{y}\mid \bar{c}_x,\bar{c}_y)},$$0 are rowwise proportional in every context. Equality in concavity holds iff

$$\Sigma_{\mathcal X,\mathcal Y} = \ln \frac{p(\tilde{x},\tilde{y}\mid \tilde{c}_x,\tilde{c}_y)} {p(\bar{x},\bar{y}\mid \bar{c}_x,\bar{c}_y)},$$1

equivalently $$\Sigma_{\mathcal X,\mathcal Y} = \ln \frac{p(\tilde{x},\tilde{y}\mid \tilde{c}_x,\tilde{c}_y)} {p(\bar{x},\bar{y}\mid \bar{c}_x,\bar{c}_y)},$$2 on a common positive-support face. The paper also derives KKT conditions and an exponential-family form for the optimal kernel,

$$\Sigma_{\mathcal X,\mathcal Y} = \ln \frac{p(\tilde{x},\tilde{y}\mid \tilde{c}_x,\tilde{c}_y)} {p(\bar{x},\bar{y}\mid \bar{c}_x,\bar{c}_y)},$$3

with an additional stationarity-coupling term (Kiriukhin, 12 Apr 2026).

Two global characterization regimes are central. With fixed one-point marginal $$\Sigma_{\mathcal X,\mathcal Y} = \ln \frac{p(\tilde{x},\tilde{y}\mid \tilde{c}_x,\tilde{c}_y)} {p(\bar{x},\bar{y}\mid \bar{c}_x,\bar{c}_y)},$$4, the unique maximizer is the i.i.d. law

$$\Sigma_{\mathcal X,\mathcal Y} = \ln \frac{p(\tilde{x},\tilde{y}\mid \tilde{c}_x,\tilde{c}_y)} {p(\bar{x},\bar{y}\mid \bar{c}_x,\bar{c}_y)},$$5

With fixed stationary $$\Sigma_{\mathcal X,\mathcal Y} = \ln \frac{p(\tilde{x},\tilde{y}\mid \tilde{c}_x,\tilde{c}_y)} {p(\bar{x},\bar{y}\mid \bar{c}_x,\bar{c}_y)},$$6-block law $$\Sigma_{\mathcal X,\mathcal Y} = \ln \frac{p(\tilde{x},\tilde{y}\mid \tilde{c}_x,\tilde{c}_y)} {p(\bar{x},\bar{y}\mid \bar{c}_x,\bar{c}_y)},$$7, the unique maximizer is the $$\Sigma_{\mathcal X,\mathcal Y} = \ln \frac{p(\tilde{x},\tilde{y}\mid \tilde{c}_x,\tilde{c}_y)} {p(\bar{x},\bar{y}\mid \bar{c}_x,\bar{c}_y)},$$8-step Markov extension

$$\Sigma_{\mathcal X,\mathcal Y} = \ln \frac{p(\tilde{x},\tilde{y}\mid \tilde{c}_x,\tilde{c}_y)} {p(\bar{x},\bar{y}\mid \bar{c}_x,\bar{c}_y)},$$9

The gap functional is

$\diss_{\X,\Y} = \Delta s(x,y) - \beta\, \heat_{\X,\Y}(\ft{x},\ft{y}), \qquad \Delta s(x,y) = -\ln p(\rti{x},\rti{y}) + \ln p(\fti{x},\fti{y}).$0

and vanishes exactly at the maximizing completion. This is an explicit conditional-marginal entropy-rate objective in optimization form, with a conditional mutual information gap that measures distance to optimality (Kiriukhin, 12 Apr 2026).

5. Bridge-aware discretization for flows and Schrödinger samplers

The most direct use of the term “conditional-marginal entropy-rate objective” is in bridge-aware discretization for flow and Schrödinger samplers. Let $\diss_{\X,\Y} = \Delta s(x,y) - \beta\, \heat_{\X,\Y}(\ft{x},\ft{y}), \qquad \Delta s(x,y) = -\ln p(\rti{x},\rti{y}) + \ln p(\fti{x},\fti{y}).$1 be the state, $\diss_{\X,\Y} = \Delta s(x,y) - \beta\, \heat_{\X,\Y}(\ft{x},\ft{y}), \qquad \Delta s(x,y) = -\ln p(\rti{x},\rti{y}) + \ln p(\fti{x},\fti{y}).$2 the bridge condition, $\diss_{\X,\Y} = \Delta s(x,y) - \beta\, \heat_{\X,\Y}(\ft{x},\ft{y}), \qquad \Delta s(x,y) = -\ln p(\rti{x},\rti{y}) + \ln p(\fti{x},\fti{y}).$3 the conditional density, $\diss_{\X,\Y} = \Delta s(x,y) - \beta\, \heat_{\X,\Y}(\ft{x},\ft{y}), \qquad \Delta s(x,y) = -\ln p(\rti{x},\rti{y}) + \ln p(\fti{x},\fti{y}).$4 the marginal density after mixing over $\diss_{\X,\Y} = \Delta s(x,y) - \beta\, \heat_{\X,\Y}(\ft{x},\ft{y}), \qquad \Delta s(x,y) = -\ln p(\rti{x},\rti{y}) + \ln p(\fti{x},\fti{y}).$5, $\diss_{\X,\Y} = \Delta s(x,y) - \beta\, \heat_{\X,\Y}(\ft{x},\ft{y}), \qquad \Delta s(x,y) = -\ln p(\rti{x},\rti{y}) + \ln p(\fti{x},\fti{y}).$6 the conditional vector field, and $\diss_{\X,\Y} = \Delta s(x,y) - \beta\, \heat_{\X,\Y}(\ft{x},\ft{y}), \qquad \Delta s(x,y) = -\ln p(\rti{x},\rti{y}) + \ln p(\fti{x},\fti{y}).$7 the marginal vector field. The central identity is

$\diss_{\X,\Y} = \Delta s(x,y) - \beta\, \heat_{\X,\Y}(\ft{x},\ft{y}), \qquad \Delta s(x,y) = -\ln p(\rti{x},\rti{y}) + \ln p(\fti{x},\fti{y}).$8

A score-form version is

$\diss_{\X,\Y} = \Delta s(x,y) - \beta\, \heat_{\X,\Y}(\ft{x},\ft{y}), \qquad \Delta s(x,y) = -\ln p(\rti{x},\rti{y}) + \ln p(\fti{x},\fti{y}).$9

The scheduling signal is the magnitude of this contrast,

$\mathcal X$0

The interpretation is explicit: the first term measures volume change along the endpoint-conditioned bridge, the second measures volume change after endpoint conditions are mixed, and the difference isolates bridge-specific geometry (Trentini et al., 15 May 2026).

The rate is converted into a nonuniform grid by the inverse-CDF rule

$\mathcal X$1

The default schedule regularizes the rate with

$\mathcal X$2

because the raw rate can become too concentrated near singular endpoints. The construction is training-free and inference-time only; it does not alter the generative model itself. The paper also links scheduling to local solver-error heuristics through

$\mathcal X$3

using entropy-rate as a tractable proxy for a $\mathcal X$4-like difficulty profile (Trentini et al., 15 May 2026).

For Gaussian Brownian bridges,

$\mathcal X$5

and the conditional probability-flow field is

$\mathcal X$6

with divergence

$\mathcal X$7

Its magnitude is U-shaped: it blows up near $\mathcal X$8 and $\mathcal X$9, and is zero at $\diss_{\X} = \ln \frac{p(\ft{x})}{p(\rt{x})},$0. This motivates boundary-heavy nonuniform grids. The paper emphasizes that this is a theorem for the Gaussian Brownian bridge, not a universal law for all learned bridges (Trentini et al., 15 May 2026).

Empirically, the objective is supported as a low-budget allocation signal. In trained two-dimensional bridge/flow models, 10-step ODE-Heun MMD improves by 18.1% over linear, and 10-step SDE-Heun improves by 22.7% over linear. On EDM/CIFAR-10 at 5 steps, entropic scheduling achieves FID 186.26 ± 3.97, compared with 200.52 ± 2.91 for linear, 238.03 ± 5.29 for cosine, and 355.06 ± 2.20 for sigmoid. On AlphaFlow proteins, the estimated cond-marg profile places about 31.1% of mass in $\diss_{\X} = \ln \frac{p(\ft{x})}{p(\rt{x})},$1, about 31.1% in $\diss_{\X} = \ln \frac{p(\ft{x})}{p(\rt{x})},$2, and only about 5.0% in $\diss_{\X} = \ln \frac{p(\ft{x})}{p(\rt{x})},$3. The paper also notes that endpoint pLDDT is mixed and not a pure schedule-quality metric (Trentini et al., 15 May 2026).

6. Quantum inequalities and marginal-constrained accumulation

Related frameworks generalize the same structural theme beyond classical stochastic processes. For bosonic additive noise channels, the quantum conditional entropy power inequality considers a tripartite state $\diss_{\X} = \ln \frac{p(\ft{x})}{p(\rt{x})},$4 with $\diss_{\X} = \ln \frac{p(\ft{x})}{p(\rt{x})},$5 and output $\diss_{\X} = \ln \frac{p(\ft{x})}{p(\rt{x})},$6. Its principal statement is

$\diss_{\X} = \ln \frac{p(\ft{x})}{p(\rt{x})},$7

which optimizes to

$\diss_{\X} = \ln \frac{p(\ft{x})}{p(\rt{x})},$8

The proof uses heat-flow interpolation, a conditional Stam inequality, and de Bruijn identities. The same paper applies entropy power inequalities to the quantum Ornstein-Uhlenbeck semigroup and proves the convergence-rate bound

$\diss_{\X} = \ln \frac{p(\ft{x})}{p(\rt{x})},$9

This is not presented as the same objective as the bridge-aware scheduler, but it is a closely related conditional-marginal entropy law in which output conditional entropy is controlled by input and noise entropy powers (Palma et al., 2018).

In quantum cryptography, a different extension appears in the marginal-constrained entropy accumulation theorem. For a channel $H(Z\mid X_t)$00 with fixed marginal input $H(Z\mid X_t)$01, the core quantity is

$H(Z\mid X_t)$02

The paper proves weak additivity,

$H(Z\mid X_t)$03

a chain rule under channel composition,

$H(Z\mid X_t)$04

and strong additivity for tensor products. Its accumulation bound with accept event $H(Z\mid X_t)$05 is

$H(Z\mid X_t)$06

Here the marginal constraint is imposed on each round’s channel input, and the conditional entropy accumulates sequentially. This places conditional-marginal entropic reasoning into the setting of adaptive prepare-and-measure QKD and channel-wise Rényi entropy accumulation (Arqand et al., 4 Feb 2025).

These quantum and cryptographic results show that the conditional-marginal pattern extends beyond classical entropy-rate estimation and sampler discretization. In one branch, it yields lower bounds and convergence statements for conditional entropy under additive noise; in another, it yields chain rules and accumulation theorems under fixed marginal input constraints. A plausible implication is that the central mathematical structure is broader than any one application domain: a conditional quantity is controlled, optimized, or accumulated relative to a marginal constraint, and the residual gap often has an information-theoretic interpretation such as conditional mutual information, divergence, or entropy power (Arqand et al., 4 Feb 2025).