Diference between monte calo integration amd simulation
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The author you are quoting is using the term "simple Monte Carlo simulation" to
refer to a situation where the random variables generated are independent and
identically distributed. As an example, suppose that X1,X2,…X1,X2,… are
independent throws of a fair six sided die, so each is equally likely to be 1, 2, 3, 4,
5, or 6. Then by throwing the die five times and averaging, one could get an
unbiased estimate of the average value of the die, which is 3.5.
In a Markov chain simulation, the state is started at one number, say for instance 2,
and then a random change is made to the state. So for instance, it could be that
the state increases by 1 with probability 1/2, or decreases by 1 with probability 1/2
(unless that move would move the state outside of the state space.) So starting at
2, the five states could be 2, 3, 4, 3, 2 or 2, 1, 1, 2, 3, or 2, 3, 4, 5, 4.
Now if you took enough states in the chain, the final state XtXt would look much
like a single throw of the die. However, unlike the independent throws of the
die, |Xt−Xt+1|≤1|Xt−Xt+1|≤1 because the state can only change by at most 1 at each
step. So mathematicians say that the states are correlated. That correlation has to
be compensated for when using such draws to estimate quantities of interest,
such as the average of the die value. It also makes a Markov chain Monte Carlo
(MCMC) algorithm much more interesting from a mathematical perspective!
I hope it will help you
please mark my answer as an BRAINLIEST ANSWER
refer to a situation where the random variables generated are independent and
identically distributed. As an example, suppose that X1,X2,…X1,X2,… are
independent throws of a fair six sided die, so each is equally likely to be 1, 2, 3, 4,
5, or 6. Then by throwing the die five times and averaging, one could get an
unbiased estimate of the average value of the die, which is 3.5.
In a Markov chain simulation, the state is started at one number, say for instance 2,
and then a random change is made to the state. So for instance, it could be that
the state increases by 1 with probability 1/2, or decreases by 1 with probability 1/2
(unless that move would move the state outside of the state space.) So starting at
2, the five states could be 2, 3, 4, 3, 2 or 2, 1, 1, 2, 3, or 2, 3, 4, 5, 4.
Now if you took enough states in the chain, the final state XtXt would look much
like a single throw of the die. However, unlike the independent throws of the
die, |Xt−Xt+1|≤1|Xt−Xt+1|≤1 because the state can only change by at most 1 at each
step. So mathematicians say that the states are correlated. That correlation has to
be compensated for when using such draws to estimate quantities of interest,
such as the average of the die value. It also makes a Markov chain Monte Carlo
(MCMC) algorithm much more interesting from a mathematical perspective!
I hope it will help you
please mark my answer as an BRAINLIEST ANSWER
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