The stationary distribution is a probability distribution that remains unchanged as the Markov chain evolves. It is important for MCMC because when the chain reaches equilibrium, the long-run fraction of time spent in each region exactly matches the target probability, allowing the chain to act as a perfect sampler.
Conditions: The Markov chain must reach equilibrium.; π is a probability vector and P is the transition matrix.; The target distribution is π.
The stationary distribution is a probability distribution that remains unchanged as the Markov chain evolves. It is important for MCMC because when the chain reaches equilibrium, the long-run fraction of time spent in each region exactly matches the target probability, allowing the chain to act as a perfect sampler.
Conditions: The Markov chain must reach equilibrium.; π is a probability vector and P is the transition matrix.; The target distribution is π.
A Markov chain is a mathematical system that undergoes transitions from one state to another according to certain probabilistic rules. Its defining characteristic is the memoryless property: the probability of transitioning to any particular state depends solely on the current state and time elapsed, not on the sequence of events that preceded it.
Conditions: The future state depends only on the present state.; Transition probabilities are defined between states.; The system is a Markov chain.
A Markov chain is a mathematical system that undergoes transitions from one state to another according to certain probabilistic rules. Its defining characteristic is the memoryless property: the probability of transitioning to any particular state depends solely on the current state and time elapsed, not on the sequence of events that preceded it.
Conditions: The future state depends only on the present state.; Transition probabilities are defined between states.; The system is a Markov chain.
A trace plot visualizes the sequence of sample values generated by an MCMC chain over iterations. To determine stationarity, look for the point where the values stop converging and begin to fluctuate around a stable equilibrium, indicating the end of the burn-in period.
Conditions: Used for diagnosing MCMC convergence.; Requires plotting sample values against iteration numbers.; The plot shows both the burn-in and stationary phases.
A trace plot visualizes the sequence of sample values generated by an MCMC chain over iterations. To determine stationarity, look for the point where the values stop converging and begin to fluctuate around a stable equilibrium, indicating the end of the burn-in period.
Conditions: Used for diagnosing MCMC convergence.; Requires plotting sample values against iteration numbers.; The plot shows both the burn-in and stationary phases.