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.
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 Monte Carlo method estimates the expected value by replacing difficult analytical integrals with the average of random samples. It draws independent random samples from the target distribution, computes their running average, and observes the average converging to the true expected value as the number of samples increases.
Conditions: Requires the ability to draw random samples from the target distribution.; Accuracy improves as the number of samples increases.; p(x) must be a valid probability density function.
The Monte Carlo method estimates the expected value by replacing difficult analytical integrals with the average of random samples. It draws independent random samples from the target distribution, computes their running average, and observes the average converging to the true expected value as the number of samples increases.
Conditions: Requires the ability to draw random samples from the target distribution.; Accuracy improves as the number of samples increases.; p(x) must be a valid probability density function.