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Abstract In this paper, we propose a new framework to construct confidence sets for a $$d$$-dimensional unknown sparse parameter $${\boldsymbol \theta }$$ under the normal mean model $${\boldsymbol X}\sim N({\boldsymbol \theta },\sigma ^{2}\bf{I})$$. A key feature of the proposed confidence set is its capability to account for the sparsity of $${\boldsymbol \theta }$$, thus named as sparse confidence set. This is in sharp contrast with the classical methods, such as the Bonferroni confidence intervals and other resampling-based procedures, where the sparsity of $${\boldsymbol \theta }$$ is often ignored. Specifically, we require the desired sparse confidence set to satisfy the following two conditions: (i) uniformly over the parameter space, the coverage probability for $${\boldsymbol \theta }$$ is above a pre-specified level; (ii) there exists a random subset $$S$$ of $$\{1,...,d\}$$ such that $$S$$ guarantees the pre-specified true negative rate for detecting non-zero $$\theta _{j}$$’s. To exploit the sparsity of $${\boldsymbol \theta }$$, we allow the confidence interval for $$\theta _{j}$$ to degenerate to a single point 0 for any $$j\notin S$$. Under this new framework, we first consider whether there exist sparse confidence sets that satisfy the above two conditions. To address this question, we establish a non-asymptotic minimax lower bound for the non-coverage probability over a suitable class of sparse confidence sets. The lower bound deciphers the role of sparsity and minimum signal-to-noise ratio (SNR) in the construction of sparse confidence sets. Furthermore, under suitable conditions on the SNR, a two-stage procedure is proposed to construct a sparse confidence set. To evaluate the optimality, the proposed sparse confidence set is shown to attain a minimax lower bound of some properly defined risk function up to a constant factor. Finally, we develop an adaptive procedure to the unknown sparsity. Numerical studies are conducted to verify the theoretical results.
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Optimal Confidence Sets for the Multinomial Parameter
Construction of tight confidence sets and intervals is central to statistical inference and decision making. This paper develops new theory showing minimum average volume confidence sets for categorical data. More precisely, consider an empirical distribution pˆ generated from n iid realizations of a random variable that takes one of k possible values according to an unknown distribution p . This is analogous to a single draw from a multinomial distribution. A confidence set is a subset of the probability simplex that depends on pˆ and contains the unknown p with a specified confidence. This paper shows how one can construct minimum average volume confidence sets. The optimality of the sets translates to improved sample complexity for adaptive machine learning algorithms that rely on confidence sets, regions and intervals.
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- Award ID(s):
- 2023239
- PAR ID:
- 10533098
- Publisher / Repository:
- IEEE
- Date Published:
- ISBN:
- 978-1-5386-8209-8
- Page Range / eLocation ID:
- 2173 to 2178
- Format(s):
- Medium: X
- Location:
- Melbourne, Australia
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/ISIT45174.2021.9517964
