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.
A single probability distribution, such as a bell curve, describes uncertainty at one instant but cannot capture temporal dynamics. Markets evolve continuously: returns arrive sequentially, volatility changes, and extreme events cluster.
Conditions: The system being modeled changes over time.; Returns, volatility, or event clustering exhibit temporal structure.
A single probability distribution, such as a bell curve, describes uncertainty at one instant but cannot capture temporal dynamics. Markets evolve continuously: returns arrive sequentially, volatility changes, and extreme events cluster.
Conditions: The system being modeled changes over time.; Returns, volatility, or event clustering exhibit temporal structure.
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.
Volatility clustering arises from simple state dependence in a Markov chain. It does not require heavy-tailed return distributions or complex non-linear dynamics.
Conditions: The underlying process follows a Markov chain with distinct states.; Returns within each state follow the same distribution.
Volatility clustering arises from simple state dependence in a Markov chain. It does not require heavy-tailed return distributions or complex non-linear dynamics.
Conditions: The underlying process follows a Markov chain with distinct states.; Returns within each state follow the same distribution.
Pure random sampling is inefficient in high dimensions because the volume of the space grows exponentially, making it nearly impossible to randomly hit regions of high probability. MCMC solves this by using dependent sampling via a Markov chain, which guides exploration and naturally concentrates samples in regions of high probability.
Conditions: The target distribution is defined over a high-dimensional space.; As dimensionality increases, random sampling becomes inefficient.; MCMC uses a Markov chain designed so that its stationary distribution is the target distribution.
Pure random sampling is inefficient in high dimensions because the volume of the space grows exponentially, making it nearly impossible to randomly hit regions of high probability. MCMC solves this by using dependent sampling via a Markov chain, which guides exploration and naturally concentrates samples in regions of high probability.
Conditions: The target distribution is defined over a high-dimensional space.; As dimensionality increases, random sampling becomes inefficient.; MCMC uses a Markov chain designed so that its stationary distribution is the target distribution.
Transition probabilities encode the likelihood of staying in or switching between states. In the two-state volatility example, high self-loop probabilities (0.8 for Low, 0.7 for High) mean the system tends to remain in its current state.
Conditions: The market is modeled as a two-state Markov chain.; Self-transition probabilities are high relative to switching probabilities.
Transition probabilities encode the likelihood of staying in or switching between states. In the two-state volatility example, high self-loop probabilities (0.8 for Low, 0.7 for High) mean the system tends to remain in its current state.
Conditions: The market is modeled as a two-state Markov chain.; Self-transition probabilities are high relative to switching probabilities.
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.
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.