Contents
When to use Markov chain Monte Carlo in practice?
— Page 1, Markov Chain Monte Carlo in Practice, 1996. Specifically, MCMC is for performing inference (e.g. estimating a quantity or a density) for probability distributions where independent samples from the distribution cannot be drawn, or cannot be drawn easily. As such, Monte Carlo sampling cannot be used.
How are samples drawn in Monte Carlo sampling?
As such, Monte Carlo sampling cannot be used. Instead, samples are drawn from the probability distribution by constructing a Markov Chain, where the next sample that is drawn from the probability distribution is dependent upon the last sample that was drawn.
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.
How are Monte Carlo algorithms used in artificial intelligence?
Monte Carlo algorithms, [….] are used in many branches of science to estimate quantities that are difficult to calculate exactly. — Page 530, Artificial Intelligence: A Modern Approach, 3rd edition, 2009.
Which is just one type of Monte Carlo method?
MCMC is just one type of Monte Carlo method, although it is possible to view many other commonly used methods as simply special cases of MCMC. As the above paragraph shows, there is a bootstrapping problem with this topic, that we will slowly resolve.
Why are MCMC algorithms better than Monte Carlo algorithms?
While MCMC methods were created to address multi-dimensional problems better than generic Monte Carlo algorithms, when the number of dimensions rises they too tend to suffer the curse of dimensionality: regions of higher probability tend to stretch and get lost in an increasing volume of space that contributes little to the integral.