English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We present a new statistical test to detect that a presented state of areversible Markov chain was not chosen from a stationary distribution. Inparticular, given a value function for the states of the Markov chain, we wouldlike to demonstrate rigorously that the presented state is an outlier withrespect to the values, by establishing a $p$-value for observations we makeabout the state under the null hypothesis that it was chosen uniformly atrandom.  A simple heuristic used in practice is to sample ranks of states from longrandom trajectories on the Markov chain, and compare these to the rank of thepresented state; if the presented state is a $0.1\%$-outlier compared to thesampled ranks (i.e., its rank is in the bottom $0.1\%$ of sampled ranks) thenthis should correspond to a $p$-value of $0.001$. This test is not rigorous,however, without good bounds on the mixing time of the Markov chain, as onemust argue that the observed states on the trajectory approximate thestationary distribution.  Our test is the following: given the presented state in the Markov chain,take a random walk from the presented state for any number of steps. We provethat observing that the presented state is an $\varepsilon$-outlier on the walkis significant at $p=\sqrt {2\varepsilon}$, under the null hypothesis that thestate was chosen from a stationary distribution. Our result assumes nothingabout the structure of the Markov chain beyond reversibility, and we constructexamples to show that significance at $p\approx\sqrt \varepsilon$ isessentially best possible in general. We illustrate the use of our test with apotential application to the rigorous detection of gerrymandering inCongressional districtings.

Assessing significance in a Markov chain without mixing