Are MCMC samples independent?
In MCMC the samples are not independent, and so things are not so simple. We need to figure out how to put error bars on our estimate, i.e., estimate the variance of our estimator. We should not use the samples generated during this burn-in period in our estimate of the mean we want to compute.
Are Markov chains Bayesian?
Among the trademarks of the Bayesian approach, Markov chain Monte Carlo methods are especially mysterious. MCMC methods are used to approximate the posterior distribution of a parameter of interest by random sampling in a probabilistic space.
How are Markov chain Monte Carlo methods different from sampling methods?
Unlike Monte Carlo sampling methods that are able to draw independent samples from the distribution, Markov Chain Monte Carlo methods draw samples where the next sample is dependent on the existing sample, called a Markov Chain.
Which is the best method for sampling probability?
The solution to sampling probability distributions in high-dimensions is to use Markov Chain Monte Carlo, or MCMC for short. The most popular method for sampling from high-dimensional distributions is Markov chain Monte Carlo or MCMC — Page 837, Machine Learning: A Probabilistic Perspective, 2012.
Where does the term Monte Carlo sampling come from?
This is referred to as Monte Carlo sampling or Monte Carlo integration, named for the city in Monaco that has many casinos. The problem with Monte Carlo sampling is that it does not work well in high-dimensions.
Why is Monte Carlo sampling a curse of dimensionality?
This is firstly because of the curse of dimensionality, where the volume of the sample space increases exponentially with the number of parameters (dimensions). Secondly, and perhaps most critically, this is because Monte Carlo sampling assumes that each random sample drawn from the target distribution is independent and can be independently drawn.