Numerical Algorithm: they are used to draw samples, - Coggle Diagram
Numerical Algorithm: they are used to draw samples
Independent sampling: need to know the form of posterior distribution, directly or indirectly
simple MC sampling
frequentist methods applied for estimating mean, 1. strong law of large numbers 2. averaging out E(g(theta) formula
CLT: confidence interval --- normal distribution if M is large enough
as M increases, the accuracy increases
frequentist methods as well, add weights to simple MC sampling
same condition to invoke CLT
the standard error formula is different. the accuracy depends on both M and weights
dependent sampling: based on theta(i-1) and long-run distirbution is the reflection of true distributional space
Gibbs Sampling: need to know full conditional: converge to joint posterior.
dependent sampling works because of the Markov chain properties: the equilibrium distribution reflect the true distribution as M goes to infinity
estimating marginal pdf: for theta1
using theta 1 draws to do kernel density estimation
using theta 2 draws to do R-B estimation
Giddy Gibbs: since we don't know full conditionals, then Gibbs sampling is off the table. but we know the kernel of full conditionals, so we use deterministic method to estimate the normalising constanta c hat. then get the estimation of interested distribution --- CDF hat, use CDF hat to make draws from the approximate full conditionals.
MH: add an extra sub-chain to the Gibbs sampling, need to know the acceptance rate.
standard error for dependent sampling formula: part 6 ---- CLT still apply.
autocorrelation at lag k ---- a measure of how far we are from the accuracy yielded by independnet draws
autocorrelation time k(g) hat: the inflation of the simulation variance due to the autocorrelation in the draws
ESS: the reduction in information from the MCMC sample relative to what would be contained in a asample of M i.i.d. draws
IF = k(g) hat