Finite-Sample Expansions for the Optimal Error Probability in Asymmetric Binary Hypothesis Testing

Khái niệm cốt lõi

The authors derive new sharp bounds and accurate nonasymptotic expansions with explicit constants for the best achievable error probability in asymmetric binary hypothesis testing based on independent and identically distributed observations.

Tóm tắt

The content discusses the problem of binary hypothesis testing between two probability measures P and Q. The authors consider the asymmetric version of the problem, where different requirements are placed on the two error probabilities.

Key highlights:

1. New sharp bounds are derived for the best achievable error probability of such tests based on independent and identically distributed observations.
2. Accurate nonasymptotic expansions with explicit constants are obtained for the error probability, using tools from large deviations and Gaussian approximation.
3. Examples are shown indicating that, in the asymmetric regime, the approximations suggested by the new bounds are significantly more accurate than the approximations provided by either normal approximation or error exponents.
4. The authors revisit the binary hypothesis testing problem between P and Q, and consider the case of hypothesis tests between two product measures P^n and Q^n, corresponding to a sequence of i.i.d. observations.
5. The authors derive finite-sample bounds that provide a stronger version of the asymptotic expansions obtained in previous works using Stein's lemma, error exponents, and saddlepoint approximations.
6. The new bounds demonstrate the utility of the approximation as it provides the most accurate estimates of the error probability compared to other approaches, especially in the regime of small error probabilities.

Thống kê

The following sentences contain key metrics or important figures used to support the author's key logics:
The best achievable performance among all deterministic or randomised tests can be described as the smallest possible value, e^_1(ε), of the first error probability e_1 over all tests whose second error probability, e_2, satisfies e_2 ≤ ε.
The smallest achievable first error probability e^_1,n(ε) decays exponentially with the sample size n, log e^_1,n(ε) = -nD(P||Q) + o(n), as n → ∞, where D(P||Q) denotes the relative entropy between two probability measures P and Q.
When the maximum allowed value ε > 0 of the second error probability is required to decay to zero exponentially fast, the first probability of error can also decay to zero at an exponential rate as n → ∞, log E^_1,n(δ) = -nD(δ) + o(n), as n → ∞, where D(δ) is the optimal error exponent.

Trích dẫn

"The best achievable performance among all deterministic or randomised tests can be described as the smallest possible value, e^_1(ε), of the first error probability e_1 over all tests whose second error probability, e_2, satisfies e_2 ≤ ε."
"The smallest achievable first error probability e^_1,n(ε) decays exponentially with the sample size n, log e^_1,n(ε) = -nD(P||Q) + o(n), as n → ∞, where D(P||Q) denotes the relative entropy between two probability measures P and Q."
"When the maximum allowed value ε > 0 of the second error probability is required to decay to zero exponentially fast, the first probability of error can also decay to zero at an exponential rate as n → ∞, log E^_1,n(δ) = -nD(δ) + o(n), as n → ∞, where D(δ) is the optimal error exponent."

Thông tin chi tiết chính được chắt lọc từ

Finite-sample expansions for the optimal error probability in asymmetric binary hypothesis testing

by Valentinian ... lúc arxiv.org 04-16-2024

https://arxiv.org/pdf/2404.09605.pdf

Yêu cầu sâu hơn

How can the new finite-sample bounds be extended to other hypothesis testing problems beyond the asymmetric binary case considered in the article

The new finite-sample bounds derived in the article for the optimal error probability in asymmetric binary hypothesis testing can be extended to other hypothesis testing problems by considering different setups and scenarios. The key lies in adapting the methodology and techniques used in the paper to suit the specific characteristics of the new hypothesis testing problem. This may involve adjusting the bounds, constants, and approximations based on the requirements and constraints of the new problem. By understanding the underlying principles and mathematical frameworks presented in the article, researchers can apply similar approaches to analyze and derive bounds for error probabilities in various hypothesis testing scenarios, not limited to asymmetric binary cases.

What are the potential applications of the accurate nonasymptotic expansions derived in the paper, and how can they be leveraged in practical settings

The accurate nonasymptotic expansions obtained in the paper have several potential applications across different fields. In practical settings, these expansions can be leveraged to improve the performance and efficiency of hypothesis testing algorithms. By providing more precise estimates of error probabilities, the expansions can enhance decision-making processes in areas such as signal processing, communication systems, and statistical analysis. Additionally, the expansions can aid in optimizing sample sizes, improving the reliability of hypothesis testing outcomes, and reducing computational complexity. Industries such as telecommunications, healthcare, finance, and engineering can benefit from the practical applications of these accurate nonasymptotic expansions in enhancing decision-making and problem-solving processes.

What are the connections between the error probability bounds in binary hypothesis testing and the sample complexity results in other areas of information theory and machine learning

The error probability bounds in binary hypothesis testing and sample complexity results in information theory and machine learning are interconnected through their fundamental principles and mathematical frameworks. In binary hypothesis testing, the bounds on error probabilities provide insights into the trade-off between the probabilities of making Type I and Type II errors. These bounds are essential in determining the performance of hypothesis testing algorithms and the accuracy of decision-making processes. On the other hand, sample complexity results in information theory and machine learning focus on the minimum number of samples required to achieve a certain level of accuracy in learning tasks. By understanding the relationships between error probabilities and sample complexity, researchers can optimize the design of learning algorithms, improve model performance, and enhance the efficiency of data-driven decision-making processes. The insights gained from error probability bounds in hypothesis testing can inform the development of sample-efficient learning algorithms and contribute to advancements in machine learning and information theory.