Assessing significance in a Markov chain without mixing.

Area of Science:

• Statistical modeling
• Markov chain analysis
• Computational statistics

Background:

• Detecting deviations from stationary distributions is crucial in analyzing Markov chains.
• Existing heuristic methods lack rigorous statistical guarantees.
• Assessing outlier states requires robust statistical tests.

Purpose of the Study:

• To introduce a novel statistical test for identifying states not originating from a stationary distribution in reversible Markov chains.
• To provide a rigorous method for detecting outlier states based on their values.
• To establish a statistically sound approach for significance testing under the null hypothesis.

Main Methods:

• Developing a statistical test based on random walks from a presented state.
• Proving the significance of observing an outlier state during such a walk.
• Analyzing the test's properties under the null hypothesis of stationary distribution.

Main Results:

• The proposed test rigorously establishes significance at a specified level (p-value).
• Observing the presented state as an outlier during a random walk from itself is statistically significant.
• The test's significance level is shown to be optimal in general cases.

Conclusions:

• The new statistical test offers a rigorous and efficient method for detecting non-stationary states in reversible Markov chains.
• The test has potential applications in fields requiring rigorous outlier detection, such as identifying gerrymandering.
• This work provides a best-possible statistical significance test for outlier detection in Markov chains.