Confidence Sets in Time-Series Filtering

TL;DR

This paper introduces a method for constructing confidence sets in finite-alphabet stationary ergodic time series filtering, ensuring probabilistic coverage and optimal exponential growth rate related to the signal's conditional entropy.

Contribution

It proposes a novel approach to confidence set construction in time-series filtering with proven optimal growth rate tied to conditional entropy.

Findings

Confidence sets include the true signal with probability γ.

Size of confidence sets grows exponentially at the rate of the conditional entropy.

The growth rate of the confidence sets is proven to be optimal.

Abstract

The problem of filtering of finite-alphabet stationary ergodic time series is considered. A method for constructing a confidence set for the (unknown) signal is proposed, such that the resulting set has the following properties: First, it includes the unknown signal with probability $\gamma$, where $\gamma$ is a parameter supplied to the filter. Second, the size of the confidence sets grows exponentially with the rate that is asymptotically equal to the conditional entropy of the signal given the data. Moreover, it is shown that this rate is optimal.