E-Values, Bayes Risk, Dual Role of Markov's Inequality

Abstract: Two approaches to hypothesis testing, e-value testing and Bayes risk minimisation, both invoke Markov's inequality to control error probabilities. They differ in which distribution certifies the unit-moment condition: the null for Type I error, the alternative for Type II error. The likelihood ratio is not intrinsically an e-value; it acquires that status only relative to the experiment under which its expectation is certified. This note makes the resulting role-reversal symmetry explicit, traces its asymptotic sharpening through the information-theoretic arguments of Barron and Clarke (1994), and situates the duality within the typed evidence calculus of Polson, Sokolov, and Zantedeschi (2026).

• The paper demonstrates that Markov's inequality plays a dual role by controlling Type I errors in e-value testing and Type II errors in Bayes risk minimization.
• It employs Bayesian marginalization and information-theoretic asymptotics to derive finite-sample bounds and elucidates the role of KL divergence in error separation.
• The study also extends to sequential inference, establishing almost-sure convergence and linking validity certificates with efficiency in adaptive testing.

Dual Role of Markov's Inequality in E-value Testing and Bayes Risk

Overview

The paper "E-Values, Bayes Risk, Dual Role of Markov's Inequality" (2604.00337) provides a formal analysis of the symmetry—and critical distinction—between e-value testing and Bayes risk minimization in hypothesis testing. The work elucidates how Markov's inequality operates as a common mechanism for controlling error rates on both sides of the classical hypothesis testing problem. By making explicit the dependence of error control on the certifying measure, the authors clarify the structural and operational duality underlying these inferential frameworks. The paper further tracks the consequences of this duality through information-theoretic asymptotics, notably the Barron–Clarke redundancy bounds, and interprets these results within the typed evidence calculus formalism.

Structural Duality of Markov’s Inequality

The starting point is the observation that the Bayes factor, $B_{10}(x) = p(x|H_1)/p(x|H_0)$, satisfies $\mathbb{E}_{H_0}[B_{10}] = 1$ and $\mathbb{E}_{H_1}[B_{01}] = 1$ (where $B_{01} = 1/B_{10}$). Markov's inequality, in combination with these unit-moment properties, yields finite-sample bounds on Type I and Type II errors without further distributional assumptions.

The duality is instantiated as follows:

• Under $P_{H_0}$, Markov's inequality applied to $B_{10}$ (now an e-variable) certifies control over the Type I error.
• Under $P_{H_1}$, applying Markov’s inequality to $B_{01}$ certifies a bound on Type II error.

The main conceptual message is that the likelihood ratio does not inherently possess evidential force; it acquires inferential meaning relative to the probability measure (experiment) under which its expectation is guaranteed. This duality, though formally symmetric, is operationally asymmetric: null-side validity requires only a nonnegative statistic with unit expectation under $H_0$, while controlling Type II error via Bayes risk requires specification of an explicit alternative.

Asymptotics and the Information-Theoretic Regime

The analysis extends to composite alternatives via Bayesian marginalization. For a composite $H_1$ equipped with a prior $\mathbb{E}_{H_0}[B_{10}] = 1$0, the marginal likelihood $\mathbb{E}_{H_0}[B_{10}] = 1$1 certifies $\mathbb{E}_{H_0}[B_{10}] = 1$2 as an e-value under the induced mixture experiment $\mathbb{E}_{H_0}[B_{10}] = 1$3. Markov's inequality then yields finite-sample uniform bounds on the probability of weak evidence for the alternative under the Bayesian mixture.

Barron–Clarke asymptotics sharpen this picture. Under regularity conditions and for fixed $\mathbb{E}_{H_0}[B_{10}] = 1$4,

$\mathbb{E}_{H_0}[B_{10}] = 1$5

Thus, the log reciprocal Bayes factor exhibits a separation rate governed by KL-divergence, up to redundancy penalties. The finite-sample mixture-level guarantee becomes pointwise exponentially sharp in the large-sample limit—corroborating the measure-determined duality observed at the finite-sample level.

Hierarchical Structure and Implications

The paper maps three inferential regimes:

1. Simple alternative: Exact, pointwise certification for $\mathbb{E}_{H_0}[B_{10}] = 1$6.
2. Composite alternative (finite $\mathbb{E}_{H_0}[B_{10}] = 1$7): Exact certification at the mixture level under $\mathbb{E}_{H_0}[B_{10}] = 1$8.
3. Composite alternative (asymptotic): Pointwise exponential rate under $\mathbb{E}_{H_0}[B_{10}] = 1$9, with rate governed by $\mathbb{E}_{H_1}[B_{01}] = 1$0.

This nested structure demonstrates how the same fundamental mechanism—Markov's inequality—yields progressively refined inferential control as additional structure is imposed.

Importantly, this perspective unifies validity-layer guarantees (finite-sample bounds via Markov/Ville) and efficiency-layer rates (asymptotic exponential separation via KL). The finite-sample Markov bounds serve as validity certificates, while asymptotic analysis characterizes efficiency and typical evidence accumulation.

Sequential and Pathwise Extensions

An important extension, grounded in the prequential supermartingale framework of Dawid, is that the log-likelihood ratio behaves as a supermartingale under the Bayesian mixture. This yields almost-sure convergence results: not only are large deviations for the reciprocal Bayes factor unlikely at any finite sample size, but almost every sequence under $\mathbb{E}_{H_1}[B_{01}] = 1$1 eventually crushes $\mathbb{E}_{H_1}[B_{01}] = 1$2 to zero, confirming the universal coding optimality of the Bayesian mixture [Dawid 1984, Dawid 1987]. This strengthens the interpretation of the Markov validity guarantee and motivates future work on anytime-valid inference for Bayes risk.

Theoretical and Practical Implications

This unification via the dual role of Markov's inequality has both foundational and operational consequences:

• It justifies parallel methodologies (e.g., e-value tests versus Bayes decision rules) as different uses of the same mathematical machinery, distinguished solely by the certifying measure.
• It underscores the importance of explicitly specifying the certifying law in practical deployments of testing protocols, especially in complex or adaptive experimental designs.
• It suggests a research agenda for refining non-asymptotic error bounds using information projections and developing sequential analogues for Bayes risk under complex alternatives.

From an information-theoretic perspective, the connection to redundancy and coding regret provides an interpretable lower bound on inefficiency in predictive modeling and sequential experimentation; in effect, the choice of certifying measure calibrates both statistical error and cumulative coding cost.

Conclusion

The paper delivers a rigorous, technically nuanced account of how Markov’s inequality underlies both e-value hypothesis testing and Bayes risk minimization, with the likelihood ratio serving as a central evidence representation whose inferential meaning is fully determined by the certifying distribution. This perspective clarifies that validity and efficiency in statistical testing are two faces of the same underlying structure, with error control, asymptotic separation, and pathwise optimality unified by the typed evidence calculus. The explicit separation between validity and efficiency layers, and the extension to composite and sequential regimes, lays the groundwork for future advances in principled evidence aggregation and robust risk-bounded inference in high-dimensional and adaptive environments.