tuitions of mcmc AR(1) example Simulation results(stationarity, CLT, autocorrelation) 0 04 -10 0200040006000800010.000 0200040006000800010000 100200300400500 Ti Ime
Intuitions of MCMC • AR(1) example – Simulation results (stationarity, CLT, autocorrelation)
tuitions of mcmc · Variance estimation Nonoverlapping Batch Means m nonoverlapping batches with each length of b bk 从=5∑8x-N(") 2 1.0 b=元∑单b一pn)2 abk and ab k+1 are not significantly correlated 5 0.0 -0.5 4681012
Intuitions of MCMC • Variance Estimation – Nonoverlapping Batch Means • 𝑚 nonoverlapping batches with each length of 𝑏 ∼ 𝒩 𝜇, 𝜎 2 𝑏 , 𝜎ෝ 2 𝑏 = • 𝜇 Ƹ 𝑏,𝑘 and 𝜇 Ƹ 𝑏,𝑘+1 are not significantly correlated
tuitions of mcmc · Variance estimation Initial Sequence Methods · Using o var(X川+2∑ covig(x,g(x+) directly usually wont converge k=1 TK= Y2k Y2k+1 is shown to be strictly positive strictly decreasing and strictly convex Geyer, 1992) Method: find the largest index m st Tk>0, k=0,.,m define Im+1=0, define k b tk to be the greatest convex minorant of k b tk, then 02m=-+2∑k k=0
Intuitions of MCMC • Variance Estimation – Initial Sequence Methods • Using directly usually won’t converge • is shown to be strictly positive, strictly decreasing and strictly convex (Geyer, 1992). • Method: find the largest index 𝑚 s.t. define Γ𝑚+1 = 0, define to be the greatest convex minorant of , then
tuitions of momo Variance estimation Initial Sequence Methods 100 ,· 100 150 Index half lag
Intuitions of MCMC • Variance Estimation – Initial Sequence Methods
tuitions of mcmc The practice of mcmc Black Box MCmc nothing is known except the outputs Pseudo-Convergence The chain appears to converge when it has not. The parts of the state space are poorly connected. multimodality One Long run vs Many Short runs (to avoid pseudo-convergence) Burn-in (comes from electronics throwing away some iterations at the beginning of an mcmc run
Intuitions of MCMC • The Practice of MCMC – Black Box MCMC (nothing is known except the outputs) – Pseudo-Convergence The chain appears to converge when it has not. The parts of the state space are poorly connected. “multimodality” – One Long Run vs. Many Short Runs (to avoid pseudo- convergence) – Burn-in • (comes from electronics) throwing away some iterations at the beginning of an MCMC run