Confidence Sets in Time-Series Filtering

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.