Statistical Windows in Testing for the Initial Distribution of a Reversible Markov Chain
This work addresses a statistical inference problem for researchers dealing with noisy data from Markov processes, but it appears incremental as it builds on existing minimax rate analysis in hypothesis testing.
The paper tackles the problem of hypothesis testing between two discrete distributions when samples are observed after being altered by a known reversible Markov chain, deriving instance-dependent minimax rates for sample complexity and showing its dependence on the chain's spectral properties. It demonstrates a wide statistical window in sample complexity for different pairs of initial distributions, with results illustrated in concrete examples.
We study the problem of hypothesis testing between two discrete distributions, where we only have access to samples after the action of a known reversible Markov chain, playing the role of noise. We derive instance-dependent minimax rates for the sample complexity of this problem, and show how its dependence in time is related to the spectral properties of the Markov chain. We show that there exists a wide statistical window, in terms of sample complexity for hypothesis testing between different pairs of initial distributions. We illustrate these results in several concrete examples.