Stationarity detection in the initial transient problem
ACM Transactions on Modeling and Computer Simulation (TOMACS)
Algorithms for random generation and counting: a Markov chain approach
Algorithms for random generation and counting: a Markov chain approach
Polynomial-time approximation algorithms for the Ising model
SIAM Journal on Computing
Convergence assessment techniques for Markov chain Monte Carlo
Statistics and Computing
Randomization and Derandomization in Space-Bounded Computation
CCC '96 Proceedings of the 11th Annual IEEE Conference on Computational Complexity
A Complete Promise Problem for Statistical Zero-Knowledge
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Simulated Annealing in Convex Bodies and an 0*(n4) Volume Algorithm
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries
Journal of the ACM (JACM)
Monte Carlo Statistical Methods (Springer Texts in Statistics)
Monte Carlo Statistical Methods (Springer Texts in Statistics)
Fast Algorithms for Logconcave Functions: Sampling, Rounding, Integration and Optimization
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Relationships between nondeterministic and deterministic tape complexities
Journal of Computer and System Sciences
CRYPTO'05 Proceedings of the 25th annual international conference on Advances in Cryptology
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An important problem in the implementation of Markov Chain Monte Carlo algorithms is to determine the convergence time, or the number of iterations before the chain is close to stationarity. For many Markov chains used in practice this time is not known. There does not seem to be a general technique for upper bounding the convergence time that gives sufficiently sharp (useful in practice) bounds in all cases of interest. Thus, practitioners like to carry out some form of statistical analysis in order to assess convergence. This has led to the development of a number of methods known as convergence diagnostics which attempt to diagnose whether the Markov chain is far from stationarity. We study the problem of testing convergence in the following settings and prove that the problem is hard computationally: - Given aMarkov chain that mixes rapidly, it is hard for Statistical Zero Knowledge (SZK-hard) to distinguish whether starting from a given state, the chain is close to stationarity by time t or far from stationarity at time ct for a constant c. We show the problem is in AM ∩ coAM. - Given a Markov chain that mixes rapidly it is coNP-hard to distinguish from an arbitrary starting state whether it is close to stationarity by time t or far from stationarity at time ct for a constant c. The problem is in coAM. - It is PSPACE-complete to distinguish whether the Markov chain is close to stationarity by time t or still far from stationarity at time ct for c ≥ 1.