Microcanonical Optimal E-Variable

• Microcanonical optimal e-variables are evidence measures optimized under strict constraints, ensuring validity through controlled likelihood ratio transformations.
• They are obtained via monotone post-processing of unconstrained likelihood ratios, achieving growth-rate optimality even under bounded or quantized conditions.
• Their applications span sequential testing, maximum entropy models, and privacy-preserving channels, providing robust error control in complex inferential settings.

Searching arXiv for the cited papers to ground the article and citations. A microcanonical optimal e-variable is an optimal e-variable under hard structural constraints, interpreted in analogy with microcanonical ensembles in statistical physics, where quantities are fixed rather than optimized only in expectation. In the contemporary e-variable literature, this notion has two closely related realizations. First, it denotes a growth-rate optimal e-variable subject to constraints such as boundedness, quantization, fixed moments, or local differential privacy; in this sense, the phrase itself is not used explicitly in “Optimal e-variables under constraints,” but the paper identifies exactly this optimize-under-hard-constraints problem and shows that the solution is often obtained by transforming the unconstrained optimal likelihood ratio (Saha et al., 23 Apr 2026). Second, it denotes a growth-rate optimal e-variable for microcanonical maximum entropy models, where sufficient statistics are fixed exactly and the resulting model is uniform on the constraint surface (Giuffrida et al., 1 Sep 2025).

1. E-variables, evidence, and log-growth optimality

An e-variable for a null hypothesis $H_0$ is a nonnegative random variable $E(X)$ satisfying

$\sup_{P_0 \in H_0} \mathbb{E}_{P_0}[E] \le 1.$

Its realized value is an e-value. By Markov’s inequality,

$P_0(E \ge 1/\alpha) \le \alpha$

for all $P_0 \in H_0$, so thresholding at $1/\alpha$ yields a level-$\alpha$ test. In sequential settings, one works with e-processes, that is, nonnegative supermartingales under the null, which preserve validity under continuous monitoring and optional stopping (Saha et al., 23 Apr 2026).

The canonical optimality criterion for e-variables is expected logarithmic growth. In the simple-vs-simple case with densities $p_0,p_1$, the likelihood ratio

$L(X) = \frac{p_1(X)}{p_0(X)}$

is valid under $P_0$ and maximizes $E(X)$0 among all e-variables. In composite settings, one passes to worst-case log-growth: $E(X)$1 When a least favorable distribution pair $E(X)$2 exists, the canonical growth-rate optimal e-variable is the likelihood ratio $E(X)$3 (Saha et al., 23 Apr 2026).

A complementary general result is the numeraire e-variable. For an arbitrary composite null $E(X)$4 and point alternative $E(X)$5, there exists a strictly positive e-variable $E(X)$6 such that

$E(X)$7

Hence

$E(X)$8

so $E(X)$9 is log-optimal. The same construction induces a sub-probability measure $\sup_{P_0 \in H_0} \mathbb{E}_{P_0}[E] \le 1.$0 via $\sup_{P_0 \in H_0} \mathbb{E}_{P_0}[E] \le 1.$1, connecting log-optimal e-variables to reverse information projection (Larsson et al., 2024).

2. The microcanonical viewpoint

The microcanonical interpretation arises when admissible e-variables are restricted by hard structural conditions. “Optimal e-variables under constraints” studies four such classes: local differential privacy, quantization, boundedness, and bounded convex integral constraints. The paper explicitly interprets this as analogous to a microcanonical viewpoint, because the admissible class is restricted by almost-sure range constraints, discrete support, or hard integral restrictions rather than by unconstrained optimization over all e-variables (Saha et al., 23 Apr 2026).

This perspective is also natural on the hypothesis side. “E-variables for hypotheses generated by constraints” defines

$\sup_{P_0 \in H_0} \mathbb{E}_{P_0}[E] \le 1.$2

so null classes are generated by measurable expectation constraints. Equality constraints are encoded by including both $\sup_{P_0 \in H_0} \mathbb{E}_{P_0}[E] \le 1.$3 and $\sup_{P_0 \in H_0} \mathbb{E}_{P_0}[E] \le 1.$4 in $\sup_{P_0 \in H_0} \mathbb{E}_{P_0}[E] \le 1.$5. This makes microcanonical hypotheses—fixed energy, fixed conserved quantities, or hard support restrictions—instances of constraint-generated hypotheses (Larsson et al., 3 Apr 2025).

A microcanonical optimal e-variable is therefore not merely any valid e-variable for such a null. It is a valid e-variable that is optimal relative to a specified criterion—typically worst-case expected log-growth, or more generally expected utility—within a constrained admissible class. This suggests a unifying interpretation: microcanonical optimality concerns evidence variables that are simultaneously safe under the null and extremal within a hard feasibility set (Saha et al., 23 Apr 2026).

3. Optimize-then-constrain

The central structural result is the optimize-then-constrain principle. Suppose $\sup_{P_0 \in H_0} \mathbb{E}_{P_0}[E] \le 1.$6 admit a least favorable distribution pair $\sup_{P_0 \in H_0} \mathbb{E}_{P_0}[E] \le 1.$7, with canonical likelihood ratio

$\sup_{P_0 \in H_0} \mathbb{E}_{P_0}[E] \le 1.$8

Let $\sup_{P_0 \in H_0} \mathbb{E}_{P_0}[E] \le 1.$9 be a constrained class, such as bounded or quantized e-variables. The constrained growth-rate problem is

$P_0(E \ge 1/\alpha) \le \alpha$0

Theorem 6.1 of (Saha et al., 23 Apr 2026) states that if the simple-vs-simple constrained optimizer for $P_0(E \ge 1/\alpha) \le \alpha$1 has the form

$P_0(E \ge 1/\alpha) \le \alpha$2

for some non-decreasing $P_0(E \ge 1/\alpha) \le \alpha$3, then this same $P_0(E \ge 1/\alpha) \le \alpha$4 is constrained GROW-optimal for the full composite problem, and

$P_0(E \ge 1/\alpha) \le \alpha$5

The significance is precise. One first solves the unconstrained canonical problem by identifying the least favorable pair and its likelihood ratio. One then enforces the structural constraint by monotone post-processing of that likelihood ratio. The constrained problem does not require solving for a new least favorable distribution pair. In the terminology suggested by the paper’s interpretation, the microcanonical optimal e-variable is obtained by constraining the canonical one rather than by replacing it (Saha et al., 23 Apr 2026).

This principle also clarifies a frequent misconception. Structural constraints do not, in general, force a new minimax analysis at the level of hypotheses. Under the theorem’s monotonicity condition, the constrained optimum is a post-processing of the unconstrained optimum, not a fundamentally different object.

4. Explicit constrained constructions

The constrained optimizers studied in (Saha et al., 23 Apr 2026) are all monotone transforms of the unconstrained likelihood ratio. In that sense, they are explicit microcanonical optimal e-variables.

Constraint class	Optimal form	Structural effect
Binary quantization	Thresholded step function of $P_0(E \ge 1/\alpha) \le \alpha$6	Discrete support
Bounded range $P_0(E \ge 1/\alpha) \le \alpha$7	Clipped rescaled likelihood ratio	Hard range constraint
Convex integral constraint	Implicit monotone transform of $P_0(E \ge 1/\alpha) \le \alpha$8	Moment-like control
Binary-output LDP	Thresholded randomized mechanism on $P_0(E \ge 1/\alpha) \le \alpha$9	Channel/privacy constraint

For binary quantization, the optimal constrained e-variable takes values in $P_0 \in H_0$0 and has the form

$P_0 \in H_0$1

The levels and threshold satisfy

$P_0 \in H_0$2

and

$P_0 \in H_0$3

This is a genuine hard-level construction: the likelihood ratio determines which of two admissible e-values is assigned (Saha et al., 23 Apr 2026).

For boundedness, with $P_0 \in H_0$4 almost surely and $P_0 \in H_0$5, the unique optimizer is

$P_0 \in H_0$6

where $P_0 \in H_0$7 is chosen so that $P_0 \in H_0$8. This is exactly a clipping operation applied to the canonical likelihood ratio. It is the clearest example of a microcanonical-style e-variable in the sense of an almost-sure hard bound (Saha et al., 23 Apr 2026).

For convex integral constraints, one maximizes $P_0 \in H_0$9 subject to

$1/\alpha$0

where $1/\alpha$1 is strictly convex and superlinear. The optimizer is unique and satisfies

$1/\alpha$2

for some $1/\alpha$3 and $1/\alpha$4. Hence $1/\alpha$5 for a strictly increasing $1/\alpha$6. In the special case $1/\alpha$7,

$1/\alpha$8

This is again a constrained variational transform of the canonical likelihood ratio (Saha et al., 23 Apr 2026).

Local differential privacy is a channel constraint rather than a moment or support constraint, but it fits the same logic. For binary outputs $1/\alpha$9, the optimal $\alpha$0-LDP mechanism thresholds the likelihood ratio: $\alpha$1 The resulting constrained e-variable is the likelihood ratio of the induced Bernoulli marginals. The paper emphasizes that this induced e-value can be written as a randomized post-processing of the unconstrained likelihood ratio (Saha et al., 23 Apr 2026).

5. Constraint-generated hypotheses and maximum entropy microcanonical models

Constraint-generated hypothesis theory gives a broad abstract characterization of microcanonical admissibility. For a null $\alpha$2, the set of all e-variables is

$\alpha$3

where $\alpha$4. For finitely generated hypotheses $\alpha$5, maximal e-variables are of the form

$\alpha$6

with $\alpha$7 satisfying the nonnegativity constraint. Under the paper’s constraint qualification, every such function is maximal. The same framework yields existence and uniqueness of optimal e-variables under a large class of expected utility-based objective functions (Larsson et al., 3 Apr 2025).

This abstract theory is realized concretely in “Testing maximum entropy models with e-values,” where both null and alternative are microcanonical maximum entropy models. For sufficient statistic $\alpha$8, the microcanonical model is

$\alpha$9

with multiplicity

$p_0,p_1$0

For microcanonical null and alternative models equipped with priors $p_0,p_1$1 on sufficient statistics, the exact GRO e-variable is

$p_0,p_1$2

where

$p_0,p_1$3

Under Condition A, namely when there exists $p_0,p_1$4 such that $p_0,p_1$5, this simplifies to

$p_0,p_1$6

The paper also proves directly that

$p_0,p_1$7

and establishes that every microcanonical e-variable is automatically a canonical e-variable for the corresponding canonical model with the same sufficient statistic (Giuffrida et al., 1 Sep 2025).

In this maximum-entropy setting, the term “microcanonical optimal e-variable” is literal rather than interpretive: it is the exact growth-rate optimal e-variable for testing one hard-constraint maximum entropy model against another.

6. Special cases, applications, and limits of the concept

The bounded-mean problem provides a particularly transparent microcanonical example. For testing the mean of a bounded random variable on $p_0,p_1$8, every e-variable for the null $p_0,p_1$9 is pointwise dominated by a coin-betting e-variable

$L(X) = \frac{p_1(X)}{p_0(X)}$0

For several composite alternatives, the GROW- and REGROW-optimal parameters are explicit. The paper interprets these constructions as “microcanonical” because optimization is carried out over all distributions with fixed mean and support; the least favorable alternatives are extreme distributions within that constrained set (Arnold et al., 16 Jan 2026).

A second important example is conditioning in $L(X) = \frac{p_1(X)}{p_0(X)}$1-sample exponential-family tests. Let $L(X) = \frac{p_1(X)}{p_0(X)}$2, where $L(X) = \frac{p_1(X)}{p_0(X)}$3 are sufficient statistics. Under the null hypothesis that all samples share the same parameter, the conditional distribution of $L(X) = \frac{p_1(X)}{p_0(X)}$4 is independent of the common null parameter. The conditional likelihood ratio

$L(X) = \frac{p_1(X)}{p_0(X)}$5

is therefore an e-variable. The paper explicitly interprets conditioning on the sum of sufficient statistics as a microcanonical construction, and in the Gaussian location, Poisson, and Bernoulli cases this conditional e-variable coincides with the globally GRO e-variable (Hao et al., 2023).

The relation between microcanonical and canonical viewpoints is not purely terminological. In minimum-description-length comparisons of canonical and microcanonical models, microcanonical models always have higher likelihood but also higher complexity, so model choice is non-trivial and depends on the empirical values of the constraints. In the thermodynamic limit, the difference in description length per node vanishes for equivalent models but persists when ensembles are non-equivalent (Giuffrida et al., 2023). A plausible implication is that a microcanonical optimal e-variable should not be conflated with a universally superior evidence measure across canonical and microcanonical model classes; optimality is always relative to a specified null, alternative, and admissible class.

Two misconceptions are therefore best avoided. First, a microcanonical optimal e-variable is not just any e-variable defined on a hard-constrained hypothesis; it is an optimizer for a stated utility or log-growth criterion. Second, microcanonical structure does not imply that optimization must be redone from first principles each time a new hard constraint is imposed. The optimize-then-constrain theorem shows that, in many composite problems, the constrained optimum is a monotone post-processing of the unconstrained least-favorable likelihood ratio (Saha et al., 23 Apr 2026).

Under these interpretations, the subject has a coherent core. A microcanonical optimal e-variable is a valid evidence variable adapted to a hard feasibility set—exact sufficient statistics, bounded range, discrete values, moment restrictions, or privacy channels—and chosen so as to be extremal for a formal growth or utility criterion. The concept links least favorable distributions, likelihood-ratio optimality, reverse information projection, maximum entropy modeling, and safe sequential inference into a single constrained evidence framework (Larsson et al., 2024).