Growth-Rate Optimal E-Variable

• The paper introduces growth-rate optimal e-variables as nonnegative evidential statistics that maximize expected log evidence subject to stringent e-validity constraints.
• Methodologies such as reverse information projection, least favorable distributions, and sequential e-process constructions are employed to optimize log-wealth growth in both fixed and sequential settings.
• Practical implementations span exponential family models, group-invariant tests, and constrained scenarios, highlighting nuances between absolute and relative growth optimality.

A growth-rate optimal e-variable is a nonnegative evidential statistic, or in sequential form an e-process or test supermartingale, that is valid under a null hypothesis and maximizes logarithmic evidence accumulation under an alternative. In fixed-sample formulations, the criterion is typically $\mathbb{E}_{P_1}[\log E]$ for a simple alternative or $$\sup_E \inf_{P_1 \in \mathcal{P}_1}\mathbb{E}_{P_1}[\log E]$$ for a composite alternative; in sequential formulations it becomes an asymptotic log-wealth rate along time or block boundaries. Across recent work, the object appears under closely related names—GRO, GROW, numéraire, strongest e-statistic, or growth-rate optimal e-process—but the central idea is stable: maximize expected log-evidence subject to the e-validity constraint $$\sup_{P_0 \in \mathcal{P}_0}\mathbb{E}_{P_0}[E]\le 1$$ or its stopped-process analogue (Saha et al., 23 Apr 2026, Ram et al., 28 Apr 2026).

1. Core definition and optimization criteria

An e-variable for a null class $\mathcal{P}_0$ is a nonnegative random variable $E$ such that $\sup_{P_0\in\mathcal{P}_0}\mathbb{E}_{P_0}[E]\le 1$. In the single-shot setting, growth-rate optimality is the Kelly-style problem

$$\sup_{E\in \mathcal{E}(\mathcal{P}_0)} \mathbb{E}_{P_1}[\log E]$$

for a simple alternative $P_1$, or the minimax extension

$$\sup_{E\in \mathcal{E}(\mathcal{P}_0)} \inf_{P_1\in \mathcal{P}_1}\mathbb{E}_{P_1}[\log E]$$

for composite alternatives. In this sense, the e-variable is a multiplicative betting factor whose objective is expected log-wealth growth rather than power in the Neyman–Pearson sense (Saha et al., 23 Apr 2026).

Several refinements coexist. In the bounded-mean literature, GRO denotes optimization against a single $Q$, GROW denotes worst-case absolute log-growth over a composite $$\sup_E \inf_{P_1 \in \mathcal{P}_1}\mathbb{E}_{P_1}[\log E]$$0, and REGROW denotes worst-case relative log-growth after subtracting the individually optimal benchmark $$\sup_E \inf_{P_1 \in \mathcal{P}_1}\mathbb{E}_{P_1}[\log E]$$1. The distinction is substantive: in some composite problems GROW yields the trivial e-variable $$\sup_E \inf_{P_1 \in \mathcal{P}_1}\mathbb{E}_{P_1}[\log E]$$2, while REGROW yields a nontrivial optimal betting rule (Arnold et al., 16 Jan 2026).

A distinct but related line studies optimality at the level of e-classes rather than individual e-variables. There, an e-class is majorising if every valid e-variable is pointwise dominated by one inside the class, and the optimal e-class is the set of maximal e-variables. This criterion implies pathwise dominance of capital processes and therefore a strong form of growth-rate optimality without explicitly maximizing $$\sup_E \inf_{P_1 \in \mathcal{P}_1}\mathbb{E}_{P_1}[\log E]$$3 for a fixed $$\sup_E \inf_{P_1 \in \mathcal{P}_1}\mathbb{E}_{P_1}[\log E]$$4 (Grünwald et al., 2024).

2. Reverse information projection, bipolar structure, and least favorable distributions

For simple alternatives against composite nulls, the foundational representation is via reverse information projection. If $$\sup_E \inf_{P_1 \in \mathcal{P}_1}\mathbb{E}_{P_1}[\log E]$$5 is the alternative and $$\sup_E \inf_{P_1 \in \mathcal{P}_1}\mathbb{E}_{P_1}[\log E]$$6 is a convex null class, then the GRO e-statistic is

$$\sup_E \inf_{P_1 \in \mathcal{P}_1}\mathbb{E}_{P_1}[\log E]$$7

where $$\sup_E \inf_{P_1 \in \mathcal{P}_1}\mathbb{E}_{P_1}[\log E]$$8 is the reverse information projection of $$\sup_E \inf_{P_1 \in \mathcal{P}_1}\mathbb{E}_{P_1}[\log E]$$9 onto $$\sup_{P_0 \in \mathcal{P}_0}\mathbb{E}_{P_0}[E]\le 1$$0, i.e. a minimizer of $$\sup_{P_0 \in \mathcal{P}_0}\mathbb{E}_{P_0}[E]\le 1$$1 over $$\sup_{P_0 \in \mathcal{P}_0}\mathbb{E}_{P_0}[E]\le 1$$2. When $$\sup_{P_0 \in \mathcal{P}_0}\mathbb{E}_{P_0}[E]\le 1$$3, this e-statistic uniquely maximizes $$\sup_{P_0 \in \mathcal{P}_0}\mathbb{E}_{P_0}[E]\le 1$$4; when $$\sup_{P_0 \in \mathcal{P}_0}\mathbb{E}_{P_0}[E]\le 1$$5, the classical GRO criterion ceases to discriminate, and the paper “Reverse Information Projections and Optimal E-statistics” extends the RIPr through description gain and shows that the same likelihood ratio is the unique strongest e-statistic in a pairwise log-growth ordering (Lardy et al., 2023).

In composite-vs-composite problems, a parallel representation uses least favorable distributions. If an LFD pair $$\sup_{P_0 \in \mathcal{P}_0}\mathbb{E}_{P_0}[E]\le 1$$6 exists, then the unconstrained worst-case growth-optimal e-variable is the likelihood ratio

$$\sup_{P_0 \in \mathcal{P}_0}\mathbb{E}_{P_0}[E]\le 1$$7

and the minimax value equals $$\sup_{P_0 \in \mathcal{P}_0}\mathbb{E}_{P_0}[E]\le 1$$8. A later extension shows that under additional structural constraints—local differential privacy, quantization, boundedness, or moment restrictions—the constrained optimizer often has the form $$\sup_{P_0 \in \mathcal{P}_0}\mathbb{E}_{P_0}[E]\le 1$$9 for a nondecreasing transform $\mathcal{P}_0$0. The constrained problem can then be solved by an optimize-then-constrain principle rather than by recomputing a new constrained LFD pair (Saha et al., 23 Apr 2026).

The sequential analogue replaces RIPr onto the raw null by reverse projection onto the bipolar effective null. For i.i.d. testing of $\mathcal{P}_0$1 against $\mathcal{P}_0$2, the optimal asymptotic growth rate is

$\mathcal{P}_0$3

not $\mathcal{P}_0$4 in general. This sharpens the role of bipolar closure: e-variables respond to the effective null seen through testing power, not merely the raw model class (Ram et al., 28 Apr 2026).

3. Sequential e-processes and asymptotic wealth growth

An e-process for $\mathcal{P}_0$5 is a nonnegative adapted process $\mathcal{P}_0$6 such that $\mathcal{P}_0$7 for every $\mathcal{P}_0$8 and every finite-valued stopping time $\mathcal{P}_0$9; a test supermartingale is the special case of a nonnegative supermartingale starting at 1. Under an alternative $E$0, the relevant performance criterion becomes asymptotic log-growth,

$E$1

possibly along a reduced filtration $E$2. The central theorem identifies the supremum over all valid wealth processes with $E$3 and shows that for every $E$4 there exists a blockwise test supermartingale attaining $E$5 both in expected log-growth and almost surely along block times (Ram et al., 28 Apr 2026).

The operational construction is finite-horizon repetition. One picks a block length $E$6, finds an $E$7-sample e-variable $E$8 with near-optimal $E$9, and multiplies independent copies over disjoint blocks. This yields a blockwise test supermartingale whose asymptotic growth rate approaches the optimal envelope. The paper emphasizes that test supermartingales on reduced filtrations suffice for all i.i.d. testing problems, and more general e-processes are not required (Ram et al., 28 Apr 2026).

The relation to classical separation is subtle. One always has $\sup_{P_0\in\mathcal{P}_0}\mathbb{E}_{P_0}[E]\le 1$0, and the inequality can be strict: there are explicit examples with $\sup_{P_0\in\mathcal{P}_0}\mathbb{E}_{P_0}[E]\le 1$1 but $\sup_{P_0\in\mathcal{P}_0}\mathbb{E}_{P_0}[E]\le 1$2. Equality holds under weak lower semicontinuity of $\sup_{P_0\in\mathcal{P}_0}\mathbb{E}_{P_0}[E]\le 1$3 at $\sup_{P_0\in\mathcal{P}_0}\mathbb{E}_{P_0}[E]\le 1$4, in particular when $\sup_{P_0\in\mathcal{P}_0}\mathbb{E}_{P_0}[E]\le 1$5 is weakly compact (Ram et al., 28 Apr 2026).

4. Structural characterizations of optimal and admissible e-variables

For nulls generated by measurable constraints, the feasible set of e-variables admits an explicit convex-analytic description. If

$\sup_{P_0\in\mathcal{P}_0}\mathbb{E}_{P_0}[E]\le 1$6

then every e-variable is $\sup_{P_0\in\mathcal{P}_0}\mathbb{E}_{P_0}[E]\le 1$7-quasi-surely of the form $\sup_{P_0\in\mathcal{P}_0}\mathbb{E}_{P_0}[E]\le 1$8 for some $\sup_{P_0\in\mathcal{P}_0}\mathbb{E}_{P_0}[E]\le 1$9 in the weak closure of $$\sup_{E\in \mathcal{E}(\mathcal{P}_0)} \mathbb{E}_{P_1}[\log E]$$0. In finitely generated cases, all e-variables are dominated by

$$\sup_{E\in \mathcal{E}(\mathcal{P}_0)} \mathbb{E}_{P_1}[\log E]$$1

while for one-sided sub-$$\sup_{E\in \mathcal{E}(\mathcal{P}_0)} \mathbb{E}_{P_1}[\log E]$$2 hypotheses they are dominated by mixtures

$$\sup_{E\in \mathcal{E}(\mathcal{P}_0)} \mathbb{E}_{P_1}[\log E]$$3

and for group-symmetry hypotheses by antisymmetrizations

$$\sup_{E\in \mathcal{E}(\mathcal{P}_0)} \mathbb{E}_{P_1}[\log E]$$4

Within each class, growth-rate optimality becomes a finite- or infinite-dimensional log-utility maximization over $$\sup_{E\in \mathcal{E}(\mathcal{P}_0)} \mathbb{E}_{P_1}[\log E]$$5 or $$\sup_{E\in \mathcal{E}(\mathcal{P}_0)} \mathbb{E}_{P_1}[\log E]$$6, and maximal e-variables coincide with the natural admissible objects (Larsson et al., 3 Apr 2025).

For properly constrained hypotheses defined by finitely many regular moment-type constraints, the optimal e-class is the dual e-class

$$\sup_{E\in \mathcal{E}(\mathcal{P}_0)} \mathbb{E}_{P_1}[\log E]$$7

Every member is maximal, every valid e-variable is pointwise dominated by one of them, and any sequential betting strategy using arbitrary e-variables can be matched or improved pathwise by restricting to the dual class. This yields a dominance-based completeness notion that is stronger than mere expected optimality (Grünwald et al., 2024).

5. Canonical model classes and explicit constructions

Exponential families supply the most developed fixed-sample theory. For a regular exponential-family null $$\sup_{E\in \mathcal{E}(\mathcal{P}_0)} \mathbb{E}_{P_1}[\log E]$$8 and a simple alternative $$\sup_{E\in \mathcal{E}(\mathcal{P}_0)} \mathbb{E}_{P_1}[\log E]$$9, one can generate a second exponential family $P_1$0 with the same sufficient statistic. If the covariance matrices satisfy the “simple case” relation $P_1$1, then the RIPr lies inside the null family and the GRO e-variable reduces to a simple-vs-simple likelihood ratio $P_1$2, where $P_1$3. In anti-simple regimes the RIPr prior is generally nondegenerate; for Gaussian nulls and alternatives it is exactly Gaussian, and in general it is approximately Gaussian at scale $P_1$4 around the KL-matching mean (Grünwald et al., 2024, Hao et al., 2024).

The same literature compares four constructions: RIPr, COND, UI, and sequentialized RIPr. For $P_1$5-dimensional null and alternative exponential families, the e-power of UI tends to be smaller by a term of $P_1$6 than that of the COND e-variable, whereas RIPr and COND are exactly equal in the Gaussian anti-simple case and asymptotically equal up to $P_1$7 in general under the paper’s conditions. This is why the conditional construction is identified as the practical winner despite RIPr’s formal optimality (Hao et al., 2024).

Group-invariant testing provides a different route to exact GROW e-statistics. When the null and alternative form group models and the group is amenable, the likelihood ratio of a maximally invariant statistic is GROW and relatively GROW among all e-statistics, invariant or not. Wijsman’s representation writes this statistic as a Bayes factor with a right Haar prior on the group, and the resulting sequence is a nonnegative martingale under the null, yielding anytime-valid tests under optional stopping and continuation (Pérez-Ortiz et al., 2022).

For bounded mean testing, the minimal complete e-class is the coin-betting family

$P_1$8

This allows explicit GROW and REGROW solutions. In point-vs-point and one-sided problems the optimizing $P_1$9 is available analytically or via a one-dimensional balance condition, whereas in the agnostic alternative problem GROW is trivial but REGROW remains nontrivial. This establishes a concrete separation between absolute and relative growth optimality (Arnold et al., 16 Jan 2026).

Maximum entropy models yield another explicit class. For microcanonical MEMs, the growth-rate optimal e-variable has the exact form

$$\sup_{E\in \mathcal{E}(\mathcal{P}_0)} \inf_{P_1\in \mathcal{P}_1}\mathbb{E}_{P_1}[\log E]$$0

where $$\sup_{E\in \mathcal{E}(\mathcal{P}_0)} \inf_{P_1\in \mathcal{P}_1}\mathbb{E}_{P_1}[\log E]$$1 is the induced marginal of the null sufficient statistic under the alternative universal distribution. The same object remains a valid e-variable in the canonical MEM case and serves as a highly accurate microcanonical approximation to the canonical GRO e-variable, including in $$\sup_{E\in \mathcal{E}(\mathcal{P}_0)} \inf_{P_1\in \mathcal{P}_1}\mathbb{E}_{P_1}[\log E]$$2 contingency-table settings with growing $$\sup_{E\in \mathcal{E}(\mathcal{P}_0)} \inf_{P_1\in \mathcal{P}_1}\mathbb{E}_{P_1}[\log E]$$3 (Giuffrida et al., 1 Sep 2025).

6. Constraints, computation, and limitations

Constrained growth-rate optimality is now developed beyond the unconstrained LR paradigm. Under local differential privacy with binary outputs, the optimal mechanism is a staircase rule based on thresholding the unconstrained likelihood ratio; under two-level quantization the optimal e-variable is a thresholded step function of the LR; under boundedness $$\sup_{E\in \mathcal{E}(\mathcal{P}_0)} \inf_{P_1\in \mathcal{P}_1}\mathbb{E}_{P_1}[\log E]$$4 it is a clipped and rescaled LR; under convex integral constraints it is the monotone transform $$\sup_{E\in \mathcal{E}(\mathcal{P}_0)} \inf_{P_1\in \mathcal{P}_1}\mathbb{E}_{P_1}[\log E]$$5 defined by the KKT equation

$$\sup_{E\in \mathcal{E}(\mathcal{P}_0)} \inf_{P_1\in \mathcal{P}_1}\mathbb{E}_{P_1}[\log E]$$6

When an LFD pair exists, the composite constrained solution is obtained by applying the same transform to the LFD likelihood ratio $$\sup_{E\in \mathcal{E}(\mathcal{P}_0)} \inf_{P_1\in \mathcal{P}_1}\mathbb{E}_{P_1}[\log E]$$7 (Saha et al., 23 Apr 2026).

Several limitations are equally central. First, the simple LR form need not exist for composite nulls; it requires structural conditions such as the covariance relation in exponential-family “simple” cases (Grünwald et al., 2024). Second, the optimize-then-constrain principle can fail when least favorable distributions do not exist (Saha et al., 23 Apr 2026). Third, the RIPr may be a strict sub-probability rather than a full probability measure, especially in infinite or constrained discrete spaces (Lardy et al., 2023). Fourth, absolute GROW can be too conservative for broad composite alternatives, producing $$\sup_{E\in \mathcal{E}(\mathcal{P}_0)} \inf_{P_1\in \mathcal{P}_1}\mathbb{E}_{P_1}[\log E]$$8, whereas REGROW can remain informative (Arnold et al., 16 Jan 2026). These distinctions are not peripheral; they determine whether “growth-rate optimal” means exact log-utility maximization, strongest pairwise evidence growth, or a regret-optimal compromise.

Taken together, the literature now supports a layered view. In the simplest cases, a growth-rate optimal e-variable is just a likelihood ratio. In composite settings, it is characterized by RIPr, bipolar projection, or least favorable distributions. In structured model classes, it can often be written explicitly or reduced to a low-dimensional optimization. Under implementation constraints, it is frequently a monotone post-processing of the unconstrained optimizer. This suggests a unifying principle: growth-rate optimality in e-value theory is fundamentally an information-projection problem, with different technical realizations depending on whether the obstacles are composite uncertainty, sequential validity, geometric structure, or external constraints.