Spectral Theory for Reversible Chains detailed balance equation: π(x)P(c,y)=π(y)P(y,c) 不则 S is symmetry x=y S=ΠPΠ-1 x丰y qPt=q(I-1ST)=qlΠ-1StΠ
Spectral Theory for Reversible Chains detailed balance equation: ⇡(x)P(x, y) = ⇡(y)P(y, x) r⇡x ⇡y P(x, y) = r⇡y ⇡x P(y, x) S(x, y) = r⇡x ⇡y let P(x, y) ) S is symmetry S = ⇧P⇧ ⇧ 1 (x, y) = (p⇡x x = y 0 x 6= y qPt = q(⇧1S⇧) t = q⇧1St ⇧
Spectral Theory for Reversible Chains detailed balance equation: π(x)P(c,y)=π(y)P(y,c) x=y S=IIPII-1 x丰y qPt=q(Π-1STI)t=qlΠI-1StΠ △x(t)=lp克-πlTv=l(ex-π)PtIl1 =l(e-元)Π-1sTml1≤点ax 1-πx 不x whereλmax=max{λ2l,ldnl}
Spectral Theory for Reversible Chains detailed balance equation: ⇡(x)P(x, y) = ⇡(y)P(y, x) S = ⇧P⇧ ⇧ 1 (x, y) = (p⇡x x = y 0 x 6= y qPt = q(⇧1S⇧) t = q⇧1St ⇧ t maxr1 ⇡x ⇡x x(t) = kpt x ⇡kT V = k(ex ⇡)Pt k1 = k(ex ⇡)⇧1St ⇧k1 where max = max{|2|, |n|}