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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.; $\pi$ is a probability vector and $P$ is the transition matrix.; The target distribution is $\pi$.

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.; $\pi$ is a probability vector and $P$ is the transition matrix.; The target distribution is $\pi$.

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 burn-in period consists of the initial samples generated by an MCMC algorithm before it has converged to its stationary distribution. These samples must be discarded because they are influenced by the arbitrary starting point and do not accurately represent the target distribution.

Conditions: Applies to MCMC sampling.; Requires discarding initial samples before using the rest for inference.; The chain starts from an arbitrary point.

The burn-in period consists of the initial samples generated by an MCMC algorithm before it has converged to its stationary distribution. These samples must be discarded because they are influenced by the arbitrary starting point and do not accurately represent the target distribution.

Conditions: Applies to MCMC sampling.; Requires discarding initial samples before using the rest for inference.; The chain starts from an arbitrary point.

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.

The detailed balance condition requires that for any two states $x$ and $y$, the probability flow from $x$ to $y$ exactly equals the flow from $y$ to $x$. This bidirectional equilibrium guarantees that the target distribution is stationary for the Markov chain.

Conditions: Must hold for all pairs of states $x$ and $y$ in the state space.; $P(x)$ and $P(y)$ are probabilities under the target distribution.; $T(y|x)$ and $T(x|y)$ are transition probabilities.

The detailed balance condition requires that for any two states $x$ and $y$, the probability flow from $x$ to $y$ exactly equals the flow from $y$ to $x$. This bidirectional equilibrium guarantees that the target distribution is stationary for the Markov chain.

Conditions: Must hold for all pairs of states $x$ and $y$ in the state space.; $P(x)$ and $P(y)$ are probabilities under the target distribution.; $T(y|x)$ and $T(x|y)$ are transition probabilities.