--- title: Growth-Rate Optimal E-Variable url: https://www.emergentmind.com/topics/growth-rate-optimal-e-variable type: topic --- # Growth-Rate Optimal E-Variable 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 [2604.21680] [2604.25280]. ## 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 [2604.21680]. 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 \(\mathcal{Q}\), and REGROW denotes worst-case relative log-growth after subtracting the individually optimal benchmark \(GRO(Q)\). The distinction is substantive: in some composite problems GROW yields the trivial e-variable \(E \equiv 1\), while REGROW yields a nontrivial optimal betting rule [2601.11347]. 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 \(\mathbb{E}_Q[\log E]\) for a fixed \(Q\) [2412.17554]. ## 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 \(P\) is the alternative and \(\mathcal{C}\) is a convex null class, then the GRO e-statistic is \[ \hat{E} = \frac{dP}{d\hat{Q}}, \] where \(\hat{Q}\) is the reverse information projection of \(P\) onto \(\mathcal{C}\), i.e. a minimizer of \(D(P\|Q)\) over \(Q\in\mathcal{C}\). When \(D(P\|\mathcal{C})<\infty\), this e-statistic uniquely maximizes \(\int \log E\, dP\); when \(D(P\|\mathcal{C})=\infty\), 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 [2306.16646]. In composite-vs-composite problems, a parallel representation uses least favorable distributions. If an LFD pair \((P_0^*,P_1^*)\) exists, then the unconstrained worst-case growth-optimal e-variable is the likelihood ratio \[ L^*(X)=\frac{dP_1^*}{dP_0^*}(X), \] and the minimax value equals \(\mathbb{E}_{P_1^*}[\log L^*]\). A later extension shows that under additional structural constraints—local differential privacy, quantization, boundedness, or moment restrictions—the constrained optimizer often has the form \(E^*=\psi(L^*)\) for a nondecreasing transform \(\psi\). The constrained problem can then be solved by an optimize-then-constrain principle rather than by recomputing a new constrained LFD pair [2604.21680]. The sequential analogue replaces RIPr onto the raw null by reverse projection onto the bipolar effective null. For i.i.d. testing of \(\mathscr{P}\) against \(Q\), the optimal asymptotic growth rate is \[ d_\star(Q,\mathscr{P}) = \lim_{n\to\infty}\frac{1}{n}a_n(Q,\mathscr{P}), \qquad a_n(Q,\mathscr{P}) = \inf_{R\in (\mathscr{P}^n)^{\circ\circ}} \mathrm{KL}(Q^n\|R), \] not \(\mathrm{KL}_{\inf}(Q,\mathscr{P})\) 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 [2604.25280]. ## 3. Sequential e-processes and asymptotic wealth growth An e-process for \(\mathscr{P}\) is a nonnegative adapted process \((E_n)\) such that \(\mathbb{E}_{P^\infty}[E_\tau]\le 1\) for every \(P\in\mathscr{P}\) and every finite-valued stopping time \(\tau\); a test supermartingale is the special case of a nonnegative supermartingale starting at 1. Under an alternative \(Q\), the relevant performance criterion becomes asymptotic log-growth, \[ \mathcal{R}_Q(W)=\limsup_{k\to\infty}\frac{1}{t_k}\mathbb{E}_{Q^\infty}[\log W_{t_k}], \] possibly along a reduced filtration \(0=t_00\) there exists a blockwise test supermartingale attaining \(d_\star(Q,\mathscr{P})-\varepsilon\) both in expected log-growth and almost surely along block times [2604.25280]. The operational construction is finite-horizon repetition. One picks a block length \(m\), finds an \(m\)-sample e-variable \(E^\star\in (\mathscr{P}^m)^\circ\) with near-optimal \(\mathbb{E}_{Q^m}[\log E^\star]\), 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 [2604.25280]. The relation to classical separation is subtle. One always has \(d_\star(Q,\mathscr{P})\le \mathrm{KL}_{\inf}(Q,\mathscr{P})\), and the inequality can be strict: there are explicit examples with \(\mathrm{KL}_{\inf}(Q,\mathscr{P})>0\) but \(d_\star(Q,\mathscr{P})=0\). Equality holds under weak lower semicontinuity of \(R\mapsto \inf_{P\in\mathscr{P}}\mathrm{KL}(R\|P)\) at \(Q\), in particular when \(\mathscr{P}\) is weakly compact [2604.25280]. ## 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 \[ \mathcal{P}(\Phi)=\left\{\mu:\int f\, d\mu \le 0 \text{ for all } f\in\Phi\right\}, \] then every e-variable is \(\mathcal{P}(\Phi)\)-quasi-surely of the form \(1+f\) for some \(f\) in the weak closure of \(\operatorname{cone}(\Phi)-\mathcal{L}_p^\Phi\). In finitely generated cases, all e-variables are dominated by \[ E_\pi(x)=1+\sum_{i=1}^d \pi_i g_i(x), \] while for one-sided sub-\(\psi\) hypotheses they are dominated by mixtures \[ E_\pi(x)=\int_\Lambda e^{\lambda x-\psi(\lambda)}\,\pi(d\lambda), \] and for group-symmetry hypotheses by antisymmetrizations \[ E_f(x)=1+f(x)-f_\pi(x). \] Within each class, growth-rate optimality becomes a finite- or infinite-dimensional log-utility maximization over \(\pi\) or \(f\), and maximal e-variables coincide with the natural admissible objects [2504.02974]. For properly constrained hypotheses defined by finitely many regular moment-type constraints, the optimal e-class is the dual e-class \[ \mathcal{E}_{\mathcal{H}}^\vee=\{\,1-\lambda\cdot \Phi:\lambda\in\Lambda_\Phi\,\}, \qquad \Lambda_\Phi=\{\lambda:\sup_x \lambda\cdot\Phi(x)\le 1\}. \] 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 [2412.17554]. ## 5. Canonical model classes and explicit constructions Exponential families supply the most developed fixed-sample theory. For a regular exponential-family null \(\mathcal{P}\) and a simple alternative \(Q\), one can generate a second exponential family \(\mathcal{Q}\) with the same sufficient statistic. If the covariance matrices satisfy the “simple case” relation \(\Sigma_q^{\mathrm{gen}}(\mu)-\Sigma_p(\mu)\preceq 0\), then the RIPr lies inside the null family and the GRO e-variable reduces to a simple-vs-simple likelihood ratio \(q/p_{\mu^*}\), where \(\mu^*=\mathbb{E}_Q[X]\). 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 \(1/n\) around the KL-matching mean [2404.19465] [2409.11134]. The same literature compares four constructions: RIPr, COND, UI, and sequentialized RIPr. For \(d\)-dimensional null and alternative exponential families, the e-power of UI tends to be smaller by a term of \((d/2)\log n + O(1)\) 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 \(o(1)\) in general under the paper’s conditions. This is why the conditional construction is identified as the practical winner despite RIPr’s formal optimality [2409.11134]. 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 [2208.07610]. For bounded mean testing, the minimal complete e-class is the coin-betting family \[ E_\alpha(x)=1+\alpha(x-\mu_0). \] This allows explicit GROW and REGROW solutions. In point-vs-point and one-sided problems the optimizing \(\alpha\) 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 [2601.11347]. Maximum entropy models yield another explicit class. For microcanonical MEMs, the growth-rate optimal e-variable has the exact form \[ S^{\mathrm{GRO}}_{\mathrm{mic}}(\mathbf{x}) = \frac{\Omega_0(\mathbf{c}_0(\mathbf{x}))}{\Omega_1(\mathbf{c}_1(\mathbf{x}))} \cdot \frac{W_1(\mathbf{c}_1(\mathbf{x}))}{W_0^*(\mathbf{c}_0(\mathbf{x}))}, \] where \(W_0^*\) 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 \(2\times k\) contingency-table settings with growing \(k\) [2509.01064]. ## 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 \(c_1\le E\le c_2\) it is a clipped and rescaled LR; under convex integral constraints it is the monotone transform \(\psi(L)\) defined by the KKT equation \[ \frac{L(X)}{E^*(X)}=\lambda+\gamma \phi'(E^*(X)). \] When an LFD pair exists, the composite constrained solution is obtained by applying the same transform to the LFD likelihood ratio \(L^*\) [2604.21680]. 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 [2404.19465]. Second, the optimize-then-constrain principle can fail when least favorable distributions do not exist [2604.21680]. Third, the RIPr may be a strict sub-probability rather than a full probability measure, especially in infinite or constrained discrete spaces [2306.16646]. Fourth, absolute GROW can be too conservative for broad composite alternatives, producing \(E\equiv 1\), whereas REGROW can remain informative [2601.11347]. 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. Source: https://www.emergentmind.com/topics/growth-rate-optimal-e-variable