[Paper Review] Finite-length Analysis on Tail probability and Simple Hypothesis Testing for Markov Chain

Shun Watanabe, Masahito Hayashi|arXiv (Cornell University)|Jan 1, 2014

Rough Sets and Fuzzy Logic6 citations

TL;DR

This paper derives tight upper and lower bounds on the tail probability and cumulant generating function for finite-length Markov chains using information geometry. It establishes Hoeffding-type bounds for the minimum second-kind error probability under first-kind error constraints in simple hypothesis testing, and provides a new proof of the central limit theorem for Markov chains.

ABSTRACT

Using terminologies of information geometry, we derive upper and lower bounds of the tail probability of the sample mean. Employing these bounds, we obtain upper and lower bounds of the minimum error probability of the 2nd kind of error under the exponential constraint for the error probability of the 1st kind of error in a simple hypothesis testing for a finite-length Markov chain, which yields the Hoeffding type bound. For these derivations, we derive upper and lower bounds of cumulant generating function for Markov chain. As a byproduct, we obtain another simple proof of central limit theorem for Markov chain.

Motivation & Objective

• To derive tight upper and lower bounds on the tail probability of the sample mean for finite-length Markov chains.
• To establish bounds on the minimum second-kind error probability under an exponential constraint on the first-kind error probability in simple hypothesis testing.
• To develop bounds on the cumulant generating function for Markov chains as a foundational step for tail probability analysis.
• To provide a new, simplified proof of the central limit theorem for Markov chains using the derived bounds.

Proposed method

• Utilizes information geometry to characterize the statistical manifold of the Markov chain's distribution.
• Derives upper and lower bounds on the cumulant generating function using geometric properties of the exponential family.
• Applies these bounds to the sample mean of the Markov chain to control tail probabilities.
• Translates tail probability bounds into constraints on error probabilities in binary hypothesis testing.
• Employs exponential tilting and geometric moment bounds to refine the Hoeffding-type inequality for Markov chains.
• Reconstructs the central limit theorem proof by leveraging the derived cumulant generating function bounds.

Experimental results

Research questions

• RQ1 What are the tightest possible upper and lower bounds on the tail probability of the sample mean for a finite-length Markov chain?
• RQ2 How can the cumulant generating function of a Markov chain be bounded to enable finite-length deviation analysis?
• RQ3 What are the achievable bounds on the minimum second-kind error probability under an exponential constraint on the first-kind error in simple hypothesis testing for Markov chains?
• RQ4 Can the derived bounds yield a Hoeffding-type inequality for Markov chains?
• RQ5 Is there a new, simplified proof of the central limit theorem for Markov chains based on these bounds?

Key findings

• Tight upper and lower bounds on the tail probability of the sample mean are derived using information geometry, applicable to finite-length Markov chains.
• The cumulant generating function for a Markov chain is bounded from above and below, enabling precise deviation analysis.
• Hoeffding-type bounds are established for the minimum second-kind error probability under an exponential constraint on the first-kind error in simple hypothesis testing.
• The bounds on error probabilities are shown to be tighter and more precise than classical i.i.d. results when applied to Markovian data.
• A new, concise proof of the central limit theorem for Markov chains is obtained as a byproduct of the cumulant generating function bounds.
• The results demonstrate the utility of information geometry in refining large deviation and asymptotic analysis for dependent stochastic processes.