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.
The stationary distribution describes the long-run frequency of visiting each state in a Markov chain. It does not mean the system stops moving; rather, it represents a dynamic equilibrium where the probability flows between states balance out.
Conditions: The Markov chain has reached its long-run behavior.; The chain is ergodic (irreducible and aperiodic) for a unique stationary distribution to exist.
The stationary distribution describes the long-run frequency of visiting each state in a Markov chain. It does not mean the system stops moving; rather, it represents a dynamic equilibrium where the probability flows between states balance out.
Conditions: The Markov chain has reached its long-run behavior.; The chain is ergodic (irreducible and aperiodic) for a unique stationary distribution to exist.
The Markov property is the defining assumption of a Markov chain: the future depends on the current state, but not on the full past. It does not mean history is irrelevant; rather, it means that all relevant historical information is summarized by the present state.
Conditions: The process satisfies the Markov property.; The current state contains all information needed to determine future transition behavior.
The Markov property is the defining assumption of a Markov chain: the future depends on the current state, but not on the full past. It does not mean history is irrelevant; rather, it means that all relevant historical information is summarized by the present state.
Conditions: The process satisfies the Markov property.; The current state contains all information needed to determine future transition behavior.
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.