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On the optimality of coin-betting for mean estimation

This paper establishes the optimality of the coin-betting formulation for mean estimation and testing by characterizing all valid e-variables and e-processes and identifying the minimal complete class of admissible strategies within this framework.

Original authors: Eugenio Clerico

Published 2026-05-08

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Eugenio Clerico

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The "Mean" Mystery

Imagine you are trying to guess the average weight of apples in a giant, mysterious orchard. You can't weigh them all at once; you have to pick them one by one. As you pick more apples, you want to update your guess about the average weight.

In statistics, this is called estimating the mean. But there's a catch: you need to be sure your guess is reliable, even if you decide to stop looking at apples at any random moment (this is called "sequential testing").

The Old Way: The "Coin-Betting" Game

Recently, researchers found a clever way to do this using a game. Imagine you are a gambler at a casino, but instead of betting on red or black, you are betting on the average weight of the apples.

The Setup: You pick a specific guess for the average weight (let's say 100 grams).
The Bet: You bet on whether the next apple you pick will be heavier or lighter than 100 grams.
The Rule: If your guess (100g) is actually the true average, the game is "fair." You shouldn't be able to win a lot of money. If you do win a massive amount of money, it proves your guess (100g) was wrong.
The Result: By running this game for many different guesses (90g, 95g, 100g, etc.), you can cross out the ones that let you win too much. The guesses you can't cross out form a "confidence sequence"—a shrinking list of likely averages that is guaranteed to contain the true answer.

This method is called coin-betting because it's like betting on a coin flip, but the "coin" is continuous (the apple's weight).

The Problem: Are We Using the Best Tools?

The paper asks a very specific question: Is this coin-betting game the absolute best way to do this?

Imagine you have a toolbox full of different betting strategies. Some are simple (like the coin-betting game), and some are complicated (using complex math formulas like "Hoeffding's inequality").

The Hoeffding Strategy: This is like using a heavy, clunky hammer. It works, but it's not very precise.
The Coin-Betting Strategy: This is like a laser-guided scalpel. It's precise and efficient.

The author, Eugenio Clerico, wanted to prove that the "scalpel" (coin-betting) isn't just good, but that it is optimal. In other words, you cannot find a better tool in the toolbox that will give you a sharper, more accurate result without breaking the rules of the game.

The Key Discovery: "Majorizing"

To prove this, the author invented a way to compare tools. He called it "majorizing."

Think of it like this:

Imagine two players, Alice and Bob.
Alice uses the Coin-Betting tool.
Bob uses any other tool (like the Hoeffding hammer).
The Rule: No matter what apple comes out next, Alice's tool will always earn her at least as much "wealth" (evidence against the wrong guess) as Bob's tool. Sometimes, Alice will earn strictly more.

If a tool can beat or match every other possible tool in the entire toolbox, it is called a "majorizing" tool. If it is the smallest set of tools that can do this, it is the "optimal" tool.

The Paper's Main Claim:
The author proves that the Coin-Betting formulation is the optimal tool.

It is the "simplest" set of rules that cannot be beaten.
Any other method you try to use for this specific problem is either worse than coin-betting or is just a clumsy version of it.
If you use coin-betting, you aren't losing any statistical power; you are using the most efficient method possible.

Two Different Scenarios

The paper looks at two different types of orchards:

The Independent Orchard (The Simple Case):
Every apple is picked independently. The weight of the current apple doesn't depend on the previous one.
- Result: Coin-betting is the undisputed champion here. It is the perfect, optimal strategy.
The Dependent Orchard (The Complex Case):
The apples might be related. Maybe if the first apple is heavy, the second one is likely to be heavy too (a "conditional mean").
- Result: Even in this complex, messy scenario, coin-betting remains the optimal strategy. It still beats all other methods.

The One Exception:
The paper notes one tiny catch. If you assume the apples are not just dependent, but strictly identical and independent (a very strict rule where the distribution never changes at all), the coin-betting method is still great, but it's no longer the only perfect method. There are other weird, symmetrical strategies that work just as well in that specific, rigid case. But for the general, real-world cases where we just want to know the average, coin-betting is the king.

Why Does This Matter?

You might ask, "Why do we need to prove the hammer is the best hammer?"

The author explains that knowing the "best" tool simplifies everything.

Simpler Math: Instead of trying to design a new, complex betting strategy from scratch, statisticians can just use the coin-betting rules. They know they can't do better.
Confidence: It gives a mathematical guarantee that they aren't missing out on a "super-strategy" that would give them tighter, more accurate results.
Efficiency: It tells us that the "Hoeffding" style tools (the heavy hammers) are unnecessary. We can throw them away and just use the coin-betting scalpel.

Summary

Imagine you are trying to find a needle in a haystack.

The Goal: Find the true average weight of apples.
The Method: A betting game where you try to prove a guess is wrong.
The Discovery: The author proved that the specific "coin-betting" way of playing this game is the perfect, unbeatable strategy.
The Analogy: It's like discovering that the Swiss Army Knife is the only tool you need to survive in the wild; any other tool is either useless or just a worse version of the Knife.

The paper doesn't tell you how to bet (which specific numbers to pick), but it proves that the structure of the game itself (the coin-betting rules) is the best possible framework for solving this problem.

Technical Summary: On the optimality of coin-betting for mean estimation

Problem Statement
The paper addresses the problem of sequentially testing the mean of a bounded real random variable and constructing confidence sequences. While recent algorithmic approaches have utilized "coin-betting" games to derive tight confidence sequences for bounded variables (Orabona and Jun, 2023; Waudby-Smith and Ramdas, 2023), the theoretical justification for why this specific formulation is optimal has largely relied on heuristic arguments. The paper seeks to rigorously formalize the claim that the coin-betting approach yields the strongest possible guarantees among all e-variable-based frameworks for this specific testing problem.

Methodology and Framework
The author introduces a novel perspective on optimality within the context of e-variables and e-processes. Rather than focusing on the optimality of a single e-variable (e.g., log-optimality against a specific alternative), the paper defines optimality at the level of sets of e-variables (e-classes).

Key definitions include:

Majorising e-class: An e-class $\mathcal{E}$ majorises another class $\mathcal{E}'$ if for every $E' \in \mathcal{E}'$ , there exists an $E \in \mathcal{E}$ such that $E$ pointwise dominates $E'$ (i.e., $E(x) \geq E'(x)$ for all $x$ ).
Optimal e-class: A majorising e-class that is contained within every other majorising e-class. If it exists, it corresponds to the set of all maximal e-variables (those not strictly dominated by any other e-variable).
Admissibility Connection: The paper notes that this framework is an adaptation of the classical statistical problem of identifying a minimal complete class of tests (Lehmann and Romano, 2022), where maximal e-variables correspond to admissible tests.

The analysis proceeds in three stages:

Single-round e-variables: Testing hypotheses where observations are independent and identically distributed (i.i.d.) with a fixed mean.
Multi-round e-variables: Testing hypotheses over a fixed horizon $T$ where the conditional mean is fixed, but dependence between observations is allowed.
E-processes: Extending the results to sequential testing with arbitrary stopping times under the fixed conditional mean hypothesis.

Key Results

Optimality for Single-Round E-variables (Theorem 1):
For the hypothesis $H_\mu$ that a distribution on $[0, 1]$ has mean $\mu$ , the paper proves that the coin-betting e-class $\mathcal{E}^{cb}_\mu = \{x \mapsto 1 + \beta(x-\mu) : \beta \in [(\mu-1)^{-1}, \mu^{-1}]\}$ is the unique optimal e-class.
- Proof Sketch: The author constructs a function $F_\mu$ that pointwise dominates all valid e-variables. It is shown that any e-variable not in the coin-betting class is strictly dominated by some element within the coin-betting class. Conversely, every element in the coin-betting class is maximal.
- Implication: Restricting the player to the coin-betting class does not sacrifice statistical power; any strategy using a general e-variable can be matched or outperformed by a strategy using a coin-betting e-variable.
Optimality for Multi-Round E-variables (Theorem 2):
When testing the hypothesis $H^T_\mu$ that a sequence of length $T$ has a fixed conditional mean $\mu$ (allowing for arbitrary dependence structures), the class of T-round coin-betting e-variables (products of single-round coin-betting functions) is the optimal e-class.
- Note: This result holds for fixed conditional means but fails if the hypothesis is restricted to i.i.d. observations. The paper demonstrates that for i.i.d. hypotheses, the set of maximal e-variables is strictly larger than the set generated by coin-betting, meaning coin-betting is not optimal in the i.i.d. setting.
Optimality for E-Processes (Theorem 3):
Extending to the setting of e-processes (processes valid for any stopping time) under the fixed conditional mean hypothesis, the coin-betting e-process class (sequences of wealth processes derived from coin-betting) is proven to be the optimal e-process class.

Significance and Claims
The paper claims to provide the first rigorous characterization of the optimality of the coin-betting formulation. Its primary contributions are:

Explicit Characterization: The results provide a direct and explicit description of the set of all valid e-variables and e-processes for testing the (conditional) mean of a bounded random variable. Specifically, any valid e-variable/process is one that is majorised by a coin-betting e-variable/process.
Theoretical Justification: It moves beyond heuristic justifications (such as the "exactness" property where expectations equal 1) to prove that the coin-betting class constitutes a minimal complete class of tests.
Strategic Simplification: By identifying the optimal e-class, the framework simplifies the design of testing strategies. For instance, it justifies the use of Bayesian aggregation techniques (like universal portfolios) over the low-dimensional parametric space of coin-betting strategies, ensuring that no statistical power is lost by restricting the search space to this class.
Distinction from Prior Work: The author distinguishes this work from recent characterizations of e-variables for linear constraints (Clerico, 2024; Larsson et al., 2025). While those works imply the single-round result, the paper argues its approach is more direct and tailored to the specific setting, providing intuition via geometric arguments (visualized in Figure 1). Furthermore, this paper extends the characterization to e-processes and multi-round variables, which the cited works do not fully address in this specific context.

Limitations and Scope
The paper explicitly notes that the optimality of coin-betting does not hold under the stricter i.i.d. hypothesis, where the set of maximal e-variables is larger. Additionally, the paper focuses on defining the optimal class of e-variables rather than designing specific betting strategies, deferring the construction of concrete strategies to existing literature (Orabona and Jun, 2023; Waudby-Smith and Ramdas, 2023). The results are presented for variables in $[0, 1]$ but are noted to extend to any bounded closed real set.